References
Software see at the beginning of Chaps. 19 and 24.
General References
[GR I ] Abramowitz, M. and I. A. Stegun (eds.). Handbook of Mathematical Functions. 10th printing, with corrections. Washington. DC: National Bureau of Standards. 1972 (also New York: Dover, 1965).
fGR2] Cajori, F., History of Mathematics. 5th ed. Reprinted. Providence. RI: American Mathematical Society. 2002.
[GR3] Courant,   R.   and   D.   Hilbert,   Methods of
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[GR7] Kreyszig, E., Introductory Functional Analysis with Applications. New York: Wiley, 1989.
[GR81 Krcyszig, E., Differential Geometry. Mineola, NY: Dover. 1991.
[GR9] Kreyszig, E. Introduction to Differential Geometry and Riemannian Geometry. Toronto: University of Toronto Press, 1975. IGR10] Szegö, G.. Orthogonal Polynomials. 4th ed. Reprinted. New York: American Mathematical Society, 2003.
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Part A. Ordinary Differential Equations (ODEs) (Chaps. 1-6)
See also Part E: Numeric Analysis
[Al I Arnold, V. I., Ordinary Differential Equations. 3rd
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[A5] Churchill, R. V., Operational Mathematics. 3rd ed.
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[A8] Dong, T.-R. et al., Qualitative Theory of Differential
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NJ: Princeton University Press, 1941. [A 151 Zwillinger, D., Handbook of Differential Equations.
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Part B. Linear Algebra, Vector Calculus (Chaps. 7-10)
For books on numeric linear algebra, see also Part E: Numeric Analysis.
[B1J Bellman, R., Introduction to Matrix Analysis. 2nd
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[B5] Greub, W. H., Linear Algebra. 4th ed. New York: Springer, 1996.
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[B71 John. A. W., Matrices and. Tensors in Physics. 3rd ed. New York: Wiley, 1995.
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[B8] Lang, S., Linear Algebra. 3rd ed. New York:
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Part C. Fourier Analysis and PDEs (Chaps. 10-11)
For books on numerics for PDEs see also Part E: Numeric Analysis.
[C11 Antimirov. M. Ya., Applied Integral Transforms.
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Applications. 3rd ed. New York: McGraw-Hill, 2000. [C31 Carslaw, H. S. and .1. C. Jaeger, Conduction of Heat
in Solids. 2nd ed. Reprinted. Oxford: Clarendon, 1986. LC4J Churchill, R. V. and J. W. Brown, Fourier Series
and Boundary Value Problems. 6th ed. New York:
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Differential Equations. Mineola, NY: Dover, 2001. [C6] Hanna, J. R. and J. H. Rowland, Fourier Series,
Transforms, and Boundary Value Problems. 2nd ed.
New York: Wiley, 1990. [C7] Jerri, A. J., The Gibbs Phenomenon in Fourier
Analysis, Splines, and Wavelet Approximations. Boston:
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[C8] John, F., Partial Differential Equations. New York:
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1976.
[C10] Widder, D. V., The Heat Equation. New York:
Academic Press, 1975. [Cll] Zauderer, E., Partial Differential Equations of
Applied Mathematics. 2nd ed. New York: Wiley. 1989. [CI2] Zygmund. A. and R. Fcffcrman, Trigonometric
Series. 3rd ed. New York: Cambridge University Press,
2003.
Part D. Complex Analysis (Chaps. 13-18)
[Dl] Ahlfors, L. V., Complex Analysis. 3rd ed. New-York: McGraw-Hill, 1979.
[D2] Bieberbach, L., Conformed Mapping. Providence, RI: American Mathematical Society, 2000.
[D31 Henrici, P., Applied and Computational Complex Analysis. 3 vols. New York: Wiley, 1993.
[D4] Hille, E.. Analytic Function Theory. 2 vols. 2nd ed. Providence, RI: American Mathematical Society, 1997.
|D5| Knopp, K., Elements of the Theory of Functions. New York: Dover, 1952.
[D6] Knopp, K., Theory of Functions. 2 parts. New York: Dover, 1945, 1947.
[D7] Krantz, S. G., Complex Analysis: The Geometric
Viewpoint.   Washington,   DC:   The Mathematical
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Springer, 1993. [D9] Narasimhan, R., Compact Riemann Surfaces. New
York: Springer, 1996. [D10] Nehari, Z„ Conformal Mapping. Mineola, NY:
Dover, 1975.
I Dil J Springer. G., Introduction to Riemann Surfaces. Providence, RI: American Mathematical Society, 2002.
Part E. Numeric Analysis (Chaps. 19-21)
[Ell Ames, W. F., Numerical Methods for Partial Differential Equations. 3rd ed. New York: Academic Press, 1992.
[F.2] Anderson, E„ et al„ LAPACK User's Guide. 3rd ed.
Philadelphia: SIAM, 1999. |E3J Bank, R. E., PLTMG. A Software Package for
Solving Elliptic Partial Differential Equations: Users'
Guide 7.0. Philadelphia: SIAM, 1994. [E4] Constanda, C, Solution Techniques for Elementary
Partial Differential Equations. Boca Raton, FL: CRC
Press, 2002.
[E5] Dahlquist, G. and A. Björck, Numerical Methods.
Mineola, NY: Dover, 2003. [E6] DeBoor. C..A Practical Guide to Splines. Reprinted.
New York: Springer, 1991. |E7] Dongarra, J. J. et al., LIN PACK Users Guide.
Philadelphia: SIAM, 1978. (See also at the beginning of
Chap. 19.)
[E81 Garbow, B. S. et al., Matrix Eigensystem Routines:
EISPACK Guide Extension. Reprinted. New York:
Springer, 1990. |F.9| Golub,  G.  H.  and C.  F.  Van Loan, Matrix
Computations. 3rd ed. Baltimore, MD: Johns Hopkins
University Press, 1996. [RIO] Higham, N. .1., Accuracy and Stability of Numerical
Algorithms. 2nd ed. Philadelphia: SIAM, 2002. [Ell] IMSL (International Mathematical and Statistical
Libraries), FORTRAN Numerical Library. Houston, TX:
Visual Numerics, 2002. (See also at the beginning of
Chap. 19.)
[El2] IMSL,  IMSL for Java.  Houston,  TX: Visual
Numerics, 2002. [E13] IMSL, C Library. Houston, TX: Visual Numerics,
2002.
[E14] Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations. Philadelphia: SIAM, 1995.
[El5] Knabncr, P. and L. Angerman, Numerical Methods for Partial Differential Equations. New York: Springer,
2003.
[EI6] Knuth, D. E., The Art of Computer Programming. 3 vols. 3rd ed. Reading, MA: Addison-Wesley, 2005.
App. 1
[El 71 Kreyszig. E., Introductory Functional Analysis with
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multigrid theory. Lecture Notes in Pure and Applied
Mathematics 157. New York: Dekker, 1994, pp. 225-242. [El91 Kreyszig, E.. Basic ideas in modern numerical
analysis and their origins. Proceedings of the Annual
Conference of the Canadian Society for the History and
Philosophy of Mathematics. 1997, pp. 34-45. [E20] Kreyszig, E., and J. Todd, QR in two dimensions.
Elemente der Mathematik 31 (1976), pp. 109-114. |E21| Möllensen, M. E., Geometric Modeling. 2nd cd.
New York: Wiley, 1997. [E221 Morion, K. W., and D. F. Mayers, Numerical Solution
t>f Partial Differential Equations: An Introduction. New
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[E24] Overton, M. L., Numerical Computing with IEEE
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IE27] Varga. R. S., Matrix Iterative Analysis. 2nd ed. New
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Springer, 2004. [E29] Wilkinson,   J.   H.,   The  Algebraic Eigenvalue
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[Fl] Bondy, J. A., Graph Theory with Applications.
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Springer. 2000. [F4] Diwekar, U.M., Introduction to Applied Optimization.
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[F81 Harary, F., Graph Theory. Reprinted. Reading, MA: Addison-Wesley, 2000.
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APPENDIX 2
Answers to
Odd-Numbered Problems
Problem Set 1.1, page 8
1. (cos ttx)Itt + c 3. ex2'2 + c       5. First order
7. Second order 9. Third order
11. v = i tan (2x + nir), n = 0, ±1, ±2, • • • 13. v = 15. (A) No. (B) No. Only y = 0.
17. y" = g, y' = gt, y = gt2/2
19. y" = k, y' = kt + 6, y = |/tr2 + 6t, y(60) = 1800/: + 360 = 3000, k = 1.47,
y'(60) = 1.47 • 60 + 6 = 94 [m/sec] = 210 [mph] 21. ekH =\,H= (In       = (1011 In 2)/1.4 = 1570 [years]
Problem Set 1.2, page 11
11. v = -(2/tt) cos \ttx + c 15. v = x(l - In x) + c
17. Verify the general solution y2 + i1 = c. Circle of radius 3V2 19. mtr- = mg - /w2, u' = 9.8 - u2, u(0) = 10. v' = 0 gives the limit V9!8 = 3.1 [meter/sec].
Problem Set 1.3, page 18
3. cos 2y dy = 2 dx, y = ^ arcsin {Ax + c)     5. y2 + 36.x2 = c, ellipses 7. dy/y = cot 7rx dx, y = c(sin -n-x)1'" 9. y = tan (c - e~"xlTf)
ll.r=roe-*2 13. / = /0e-Rl/L
15. y = exNlx + 5 17. y = 4 In jc
19. y = Vln (x2 - 2x + e)
21. y' = (y - fo)/(.v - a), y -   = c(x - a)
23. y0(?'c = 2y0, ek = 2 (1 week), <?2,f = 22 (2 weeks), e4fc = 24
25. y = y0ew = y0<ra00012131 = y0«-ao«nai»«0P = 0.62y0; 62%; cf. Example 2.
27. y' = — Ary, y = y0e~kt, e~5k = 0.5, A; = -(In 0.5)/5 = 0.139, t= -(In 0.05)/0.139 = 22 [min]
29. T(0) = 10, T = 23 - 13efct, 1(2) = 23 - 13c2'1' = 18, k = -0.478, T = 22.8 gives t = [In (-0.2/-13)]/(-0.478) = 8.73 [min].
31. h = ^f2/2, / = V2ft/g, v = gt = gVWg = V2gh 33. y' = 0 - (2/800)y, y = 200<?-°'0025t, / = 300 [min], y(30Q) = 94.5 [lb| 35. (A) is related to the error function and (C) concerns the Frcsnel integral C(x); see App. 3.1.   (D) y = 2xy + 1, y(0) = 0
App. 2   Answers to Odd-Numbered Problems
A5
Problem Set 1.4, page 25
1. Exact. x4 + y4 = c
3. Exaci. u = cos ttx sinh y + k(y), uy = cos mc cosh v + k , k = 0.
Ans. cos ttx sinh y = c 5. Exact. 9x2 + 4y2 = c
7. Exact, M„ = Nr = -2<T20, u = re~20 + k(9), ue = -2re~20 + k', k' = 0.
Ans. re~2e = c, r = ce2" 9. Exact, u = y/x + sin 2x + k(y), uy = \lx + k' = l/x — 2 sin 2y. Ans. ylx + sin 2x + cos 2y = c 11. Not exact. F = l/x2 by Theorem 1. ->'/x2 clx + \lx dy = d(ylx) = 0. A«.v. v = ex 13. -3y2/x4 rfx + 2ylxa dy = d(y2/x3) = 0. y = cv3/2 (semicubical parabolas) IS. Exact, a = e2** cos y + k(y), uy = —e2x sin v + k', k' = 0. Ans. e2x cos y = c, c = 1
17. Not exact. Try R. F — e~x, e~x(cos cox + <o sin ojx) dx + dy = 0, u = y + /(.v),
m,. = l = e-x(cos wx + w sin wx), u = y + I = y — e~x cos cox = C, c = 0 19. a = ex + /<(>•), Ma = ik' = -1 + ey, k = -y + ey. Ans. ex - y + ey = c 21. U = C, \Ax2 + Cxy + \Dy2 = c
Problem Set 1.5, page 32
3. y = ce~z-5x + 0.8 5. y = 2.6e~1-2ax + 4
7. y = x + c (if it = 0). y = ce~fcT' + e2kxBk if ik # 0
9. Separate, y - 2.5 = c cosh4 1.5x 11. y = 2xec<KS 2x
13. y = sin 2x + c/sin2 2x, c =1 15. y = e1/x(x2 + c), c = 4.1
T7Ty =_(F+ I cosh 10x)/.v3. Note (x3v)' = 5 sinh 10x.
6.5
19. y = 1/m, u = ce~"'--
5.7
21. u = y"2 = ex\\ + ce2x). c = 3, h(0) = 4
23. Separate variables, y2 = 1 - cecos *, c = -lie
25. y = Ry + k, y = ceRt - kIR, c = y0 + kIR. y0 = 1000, R = 0.06,
t = 65 - 25 = 40, k = 1000. y = $178 076.12. Start at 45 gives
y0[(l + l/0.06>006*20 - 1/0.06] = 41.988732y0 = 178 076.12, y0 = k = $4241.05. 27. y' = 175(0.0001 - y/450), y(0) = 450 ■ 0.0004 = 0.18,
y = 0.135<r0-38891 + 0.045 = 0.18/2,
e-o.3889t = (()09 _ o.045)/0.135 = 1/3,
t = (In 3)/0.3889 = 2.82. Ans. About 3 years 29. y' = A - ky, y(0) = 0, y = A(l - e~kt)lk
31. y' = By2 - Ay = By(y - AIB), A > 0, B > 0. Constant solutions y = 0, y = AIB.
y' > 0 if y > AIB (unlimited growth), y < 0 if 0 < y < AIB (extinction).
y = AI{ceAl + ß), y(0) > AIB if c < 0, y(0) < A/ß if c > 0. 33. y' = y - y2 - 0.2?, y = 1/(1.25 - 0.75e-afit), limit 0.8, limit 1 35. y' = y - 0.25y2 - O.ly = 0.25y(3.6 - y). Equilibrium harvest 3.6,
y = 18/(5 + ce-0'9') 37. (y, + y2Y + p(yi + y2) = (yL' + pyj + (y2' + py2) = 0 + 0 = 0 39. (>'! + y2)' + p(yx + y2) = fa' + py,) + (y2' + Av2) = r + 0 = r 41. Solution of cyx' + pcyy — c(y/ + py\) = er
App. 2   Answers to Odd-Numbered Problems
43. CAS Experiment (a) v = x sin (1/x) + ex. c = 0 if y(2/v) = 2/tt. y is undefined at x = 0, the point at which the "waves" of sin (1/x) accumulate; the factor x makes them smaller and smaller. Experiment with various ^-intervals, (b) y = xm[sin (1/x) + e]. y(2/ir) = (2/'tt)""'. n need not be an integer. Try n = |. Try n —    1 and see how the "waves" near 0 become larger and larger.
45. y = uy*, y + py = u y* + uy*' + puy* = u'y* + u(y*' + py*) = ay* + u • 0 = r, u = rly* = reSp *», u = J efp dx r dx + e. Thus, y = wyh gives (4). We shall see that this method extends to higher-order ODEs (Sees. 2.10 and 3.3).
Problem Set 1.6, page 36
1. y = 4, y  = -1/4, y = -x!4 + c*
3. ylx = c, y'lx = y/x2, y' = y/x, y = -xly, v2 + x2 = c*, circles
5. 2xy + x2y' = 0. v' =    2y/x. y = x/(2y), y2 - x2/2 = c*, hyperbolas
7. ye-*12 = c, / = xy, y' = -l/(xy), yy' = -1/x, y2/2 = -In |x| + C**,
x = c*e~y2r2, bell-shaped curves (with x and y interchanged) 9. y = — 4x1 y, y' = y/4x, 4 In |y| = In \x\ + e**, x = c*y4, parabolas 11. xe-ylA = c, y' = 4/x, y = -x/4, y = -x2/8 + c* 13. Use dyldx = \l(dxldy). (y - 2x)ex = c, (y' - 2 + y - 2x)ex = 0, y' = 2 — y + 2x, dxldy = — 2 + y — 2x is linear, dxldy + 2x = y - 2, x = c*e_2y + y/2 - 5/4 15. m = c, m.,.Jx + Mydy = 0, y = —ujuy. Trajectories y = iiy/ux. Now v = c*, Ujudx + uac/y = 0, y' = —vx/vy. This agrees with the trajectory ODE in u if wx = vy (equal denominators) and uy = — 6'.r (equal numerators). But these are just __the Cauchy-Riemann equations.
17. 2x + 2yy' = 0, y = —xly. Trajectories y' = y/x, In |y| = In |x| + c**, y = c*x. 19. y' = -4x/9y. Trajectories y' = 9y/4x. y = c*xm (c* > 0). Sketch or graph these curves.
Problem Set 1.7, page 41
1. In |x — Xq| < a; just take b in    = /j/A^ large, namely, /? = aA".
3. No. At a common point (xi, y'i) they would both satisfy the "initial condition"
yiXf) = yL, violating uniqueness. 5. y' = fix, y) = r(x) — p(x)y; hence df/dy = —p(x) is continuous and is thus
bounded in the closed interval \x — x0| s a. 1. R has sides 2a and 2b and center (1,1) since y(l) = 1. In R,
f = 2y2 g 2(b + l)2 = K, a = WJf = bl(2{b + l)2). rfa/rffo = 0 gives b= 1, and
aopt = bIK = 1/8. Solution by dyly2 = 2 t/x, etc., y = 1/(3 - 2x). 9. |1 + y2| g ^ = 1 + b2, a = &/A", rfa/J/? = 0, fe = 1, a = 1/2.
Chapter 1 Review Questions and Problems, page 42
11. dy/(y2 + |) = 4 r/x, 2 arctan 2y = 4x + c*, y = | tan (2x + c) 13. Logistic ODE. y = 1/m, y' = -u'/u2 = 4/k - 1/w2, u = c*e"te + | 15. dyl(y2 + 1) = x2 dx, arctan y = x3/3 + c, y = tan (x3/3 + c) 17. Bernoulli, y + xy = xly, u = y2, m  = 2yy' = 2x — 2xu linear,
u = e~xA (JexA 2x dx + c) = ] + ee~x\ y = vu. Or write
yy  = — x(y2 — 1) and separate.
App. 2   Answers to Odd-Numbered Problems A7
19. Linear, y = * siaxdx + c) = cecos* + 1. Or by separation.
21. Not exact. Use Theorem 1, Sec. 1.4; R = 2/x, F = x2; the resulting exact ODE is
3x2 sin 2y dx + 2x3 cos 2y dy = d(x3 sin 2y), .v3 sin 2y = c. Or by separation,
cot 2y dy = — 3/(2x) dx, etc., sin 2y = ca-3. 23. Exact, u = J M dx = sin at - x2 + k, «„ = x cos at + k' = N, k = y2,
sin at - x2 + v2 = c. 25. Not exact, i?* - 1 in Theorem 2, Sec. 1.4, F* =     Exact is
<?w sin (v — x) dx + ev[cos (y — x) — sin (y — x) \ dy = 0.
it = fMdx = ev cos (y - jc) + k, uv = <?a(cos (y - x) - sin (y - a)) +     = N,
ev cos (y — a) = c. 27. Separation, y2 + a2 = 25 29. Separation, y = tan (a + c), c = —\tt 31. Exact, u = a2v2 + cos x + 2y = c, r = h(0, 1) = 3 33. y = a/v. Trajectories y' = -y7x, y = c*/x by separation. Hyperbolas. 35. y = y0 <•>'<*, <?4fe = 0.9, it = A In 0.9, *kt = 0.5.
/ = (In 0.5)/*; = (In 0.5)/[(ln 0.9)/4] = 26.3 [days] 37. ekt = 0.01, t = (In 0.01)/* = 175 [days] 39. y = — 4x/y. Trajectories y = cxxlfA or a = c2y4
41. Logistic ODE y' = Ay ~ By2, y = \lu, u  + An = +B, u = c<?_A* + BIA 43. ,4 = amount of incident light. A thin layer of thickness Ax absorbs A/4 = — fc4Ax (—k = constant of proportionality). Thus A4/Av = — kA. Let Ax—* 0. Then A' = — kA, A = A0e~kx = amount of light in a thick layer at depth a from the surface of incidence.
Problem Set 2.1, page 52
1. y = 2.5e4x + 0.5e"4-T      3. y = e~x cos x 5. y = 4x2 + 7/x2
7. Yes 9. Yes if a * 0 11. No
13. No 15. F{x, z, z) = 0 17. y = c,efcx + c2
19. y cfeWy = 4s, y = (clX + c2)_1/3
21. (dz/dv)z = —z3 sin v, — 1/c = —dx/dy = cos v + cx, x = —sin v +      + c2
23. y V = 2, y = §(r + 1)3/2 - J, y(3) = f, y'(3) = 4
25. y" = At', z' = kz, z =        = y\ cx = 1, y = (e** - 1)/*
Problem Set 2.2, page 59
1. y = Clelx + c2^-x 3. y = (c, + c2x)eZ5x
5. y = de0-ftr + cze~llx 7. y = ea5z(A cos 1.5x + ß sin l.5x)
9. y = de3-5* + c2e~L5x 11. y = A cos 3ttx + B sin 3ttx
13. y = Cle1Zx + c2e-12a; 15. y" - 3y' + 2y = 0
17. y" - 2V3 y' + 3y = 0 19. y" - 16y = 0
21. y = 4e3x - 2<Ta: 23. y = e~2x(2 cos x - sin a)
25. y = 2 + e~nx 27. y = (2 - 4x)<Ta25a;
29. y = <?-°lx(3.2 cos 0.2x + 1.6 sin 0.2x)     31. y = 4e5x - 4<TS* 33. >'! = e~x, y2 = O.GOle* + e~x
35. Write £ = e~ax'2, c = cos wa, s = sin wx. Note that E' = -\aE, c  = -w.v, s' = wc. Substitute, drop E, collect c-terms, then ^-terms, and use of = b — \a2, to get c(b - \a2 + \a2 - w2) + s{-a<o + \aa + \aw) = 0 + 0 = 0.
A8
App. 2   Answers to Odd-Numbered Problems
Problem Set 2.3, page 61 1. 0, 0, -2 cos x
5. - I2a:! + 9x2 + 8x -2,-28 sin 4x 7. v = (d + c^e-2* 11. y = c^-3-1* + c2e-:,:
3. -0.8a-3 + 6a-2 + 0.4, 0, e0Ax — 4 cos 4x, 0
9. >• = t'"3-^^ cos 2a- + B sin 2x) 13. y = A cos 4.2a>x + B sin 4.2g«
Problem Set 2.4, page 68
1- y = )'o cos wo? + (.Vq/wq) sin w0r. At integer / (if cü0 = it), because of periodicity. 3. mLd" = —mg sin 0 = —ingO (tangential component of W = mg), ß" + co02ö = 0,
co0/(2ir) = VglLlilTT). 5. No, because the frequency depends only on klm. 7. (i) Greater by a factor V'3. (ii) Lower
9. co* = [«02 - c2/(4m2)]U2 = oi0[] - c2/(4mk)]V2 « w0(l - c2/8mfe) = 2.9583 11. 2Wto* since Eq. (10) and y' = 0 give tan {<a*t — 8) = —a/(o*; tan is periodic with period 7r/<w*.
13. Case (II) of (5) with c = \/4m~k = V4 ■ 500 ■ 4500~ = 3000 [kg/sec], where 500 kg
is the mass per wheel. 15. v = [y„ + (u0 + ay0)t]e-at, y = [1 + (v0 + L)/]«"*; (ii) u0 = -2, -3/2, -4/3,
-5/4, -6/5
17. v = 0 gives t\ = —c2e~2ßt, which has one or no positive zero, depending on the initial conditions.
Problem Set 2.5, page 72
1. d*3 + c2x~2 3. (d - c2 In |a-|).v4
5. x[A cos (In |*|) + B sin (In |x|)] 7. Clx1A + c2x16
9. d*0'1 + c2x0-9 11. 3a-2 - 2a-3
13. x~°-5[2 cos (10 In |x|) - sin (10 In \x\)]     15. 2a-3 + 10
Problem Set 2.6, page 77
1. y" - 0.25y = 0, W = -1 3. / - 2ky' + k2y = 0, W = e2te
5. jc2v" + 0.5xy' + 0.0625V = 0, W = a"05 7. x2y" + xv' + 4y = 0, W = 21 x 9. x2v" - 0.75v = 0, W = -2 11. y" - 6.25? =0,W= 2.5
13. y" + 2y' 4- 1.64y = 0, W = 0.8c_2x       15. y" + 5y' + 6.34y = 0, W = 0.3e~
17. y" + 7.67ry' + \4.44n2y = 0, W = £-7-6™
Problem Set 2.7, page 83
1. de"x' + c2e"2x + 2.5<?2x 3. Cle4x + c2e"4a- + 2.4.w4'r - 4«*
5. d<?2x + c2e~'3x - x3 - 3a- - 0.5 7. e-3%4 cos 8x + B sin 8x) + ex(cos 4x + \ sin 4x) 9. ,,«• 0Ax + r.,,-"1" + 20xe° 1  - 20xe ,u 11. d cos 1-2.« + c2 sin 1.2.v + lOx sin 1.2x
13. e~2x(A cos x + B sin x) + 5x2 - 8x + 4.4 - 1.6 cos 2x + 0.2 sin 2x 15. 4x sin 2x
A9
17. e-°lx(\.5 cos 0.5x - sin 0.5x) + 2e05x 19. 2e~3x + 3<?4x - I2x3 + 3x2 - 6.5x
Problem Set 2.8, page 90
1. -0.4 cos 3r + 7.2 sin 3? 3. -12.8 cos 4.5? + 3.6 sin 4.5?
5. 0.16 cos 2? + 0.12 sin 2l 7. j| cos 3t - jg sin 3?
9. Cie-£/2 + c2e-'3tl2 - f cos ? - § sin ? 11. (ci + c2?)(?"3t/2 - I cos 3? - sin 3? 13. e~lr"(A cos ? + B sin ?) + 4 + 0.8 cos 2? - 6.4 sin 2? 15. 0.32e_t cos 5? + 0.68 cos 3? + 0.24 sin 3? 17. 5e~41 - 4e~2t - 0.3 cos 2? + 0.1 sin 2? 19. e'1 5t(0.2 cos t - 1.1 sin /) + 0.8 cos ? + 0.4 sin t
Problem Set 2.9, page 97
1. Li' + RI = E, I = (EIR) + ce-RtlL = 2.4 + ce~sm 3. Rl' + IIC = 0, I = cy'-*/(rc' 5. 7 = 5(cos t ~ cos 10?)/99
7. 70 is maximum when S = 0; thus C = l/(w2L). 9. R > Äerit = 2VI/C is Case I, etc. 11. 0
13. Cle-ZOi + c2e~lm + 16.5 sin 10? + 5.5 cos 10?
15. E' = -e'4t(7.605 cos \x + 1.95 sin \t), I - <?"01t(A cos |f + B sin |r)
- e_4t cos I?
17. £(0) = 600, /'(0) = 600, / = e~3'(-100 cos 4? + 75 sin 4?) + 100 cos t 19. (b) R = 2 O, L = 1 H, C = 1/12 F, £ = 4.4 sin 10? V
Problem Set 2.10, page 101
1. A cos x + B sin x — x cos x 4- (sin x) In |sin x| 3. CjX + c2xz — x cos x
5. (cos x)(Ci + sin x — In |sec x + tan xj) + (sin x)(c2 - cos x) = (c1 — In |sec x + tan x|) cos x + c2 sin x
7. (c'x + fx) sin x + (c2 + In [cos x|) cos x
9. (c1 + c2x)ex + x2 + 4x + 6 - e*(In |x| + 1) 11. cL cos 2x + c2 sin 2x + |x cosh 2x 13. cxx + c2x2 — x sin x
15. A cos x + B sin x + ypl + yp2, >'pl as yp in Example 1, yp2 = 24 sin 5x
17. u" + 1.1 — 0 by substitution of y = ?<x_1/2. Vt = x~1/2 cos x, y2 = x_1/2
1
2
form.
-1      -1 Icy -t _ -1 Jcy
sin x, yv — — gx    cos x + gx   '  sin x from (2) with the ODE in standard
Chapter 2 Review Questions and Problems, page 102
9. cveAx + c2e"2': - 1.1 cos 6x - 0.3 sin 6x 11. erix(A cos 3x + ß sin 3x) - | cos 3x + | sin 3x
App. 2   Answers to Odd-Numbered Problems
13. v, = x3, y2 = x-4, r = x-5, W = -7x-2, yv = -^x"3 - |x-3 = -gx-3 15. yi = ex, y2 = xex, W = e2x, yp = exl(2x)
17. >'i = ex cos x, y2 = ex sin x, W = <?21', yp = -xex cos x + e*(sin x) In |sinx| 19. y = 4e2x + 2e_7'T 21. y = 9x~4 + 6x6
23. y = e-2* - 2e~3x + 18x2 - 30x + 19    25. y = \x3 + 4x2 - 5x"2 27. y = -16 cos 2/ + 12 sin 2/ + 16(cos 0.5/ - sin 1.5?).
Resonance for ü}I{2tt) = 2/(2if) = Mir 29. w = 3.1 is close to <w0 = V'k/m = 3, y = 25(cos 3/ — cos 3.1?). 31. R = 9 iL L = 0.5 H, C = 0.025 F, E = 17 sin 6t V, hence 0.5/" + 9/' + 40/
= 102 cos 6t,   I = -8.16e"8t + 7.5e~101 + 0.66 cos 6t + 1.62 sin 6t 33. E' = 220 ■ 314 cos 314?, / = e-50t(A cos 150/ + B sin 150?) + 0.847001 sin 314/
- 1.985219 cos 314/
Problem Set 3.1, page 111
7. Linearly independent 11. x|x| = x2 if x > 0, linearly dependent 13. Linearly independent 17. Lineaily independent
9. Linearly dependent
15. Linearly independent 19. Linearly dependent
Problem Set 3.2, page 115
1. /" - 6y" + 11/ - 6y =0 3. >•"' - y = 0
5. yw + 4y" = 0 7. C\ + c2 cos x + c3 sin x
9. cxex + ic2 + csx)e-x 11. cxex + c2ca+v¥)* + c3el
13. e0-25* + 43e~°-lx +12.1 cos O.lx - 0.6 sin O.lx
15. 2.4 + f-lfia:(cos 1.5x - 2 sin 1,5*)
17. y = cosh 5.v — cos 4x
19. y = r,x 2 + c2x + c3x2. VV = 12/x2
Problem Set 3.3, page 122
1. (d + c2x)e2x + c3^2:r - 0.04e"3x + x2 + x + 1
3. c, cos |x + c2 sin |x + x(c3 cos \x + cA sin \x) — \e~x sin |x
5. cix05 + c2x + c3x15 + O.lx55
7. cx cos x + c2 sin x + c3 cos 3x + c4 sin 3x + 0.2 cosh 2x
9. y - (4 - x2)e3'' - 0.5 cos 3x + 0.5 sin 3x 11. x-2 - x2 + 5x4 + x(ln x + 1) 13. 3 + 9er2x cos 9x - (1.6 - \.5x)ex
Chapter 3 Review Questions and Problems, page 122
7. ct + c2xV2 + c3x~1/2 9. c,e-^x + c2e05* + c3<r1J
11. c-ix2(| In x - |) + c2x2 + c3x + c4 + |x7 13. cxe~x + ex/2(c2 cos (|V3x) + c3 sin (|V3x)) + %ex'2 15. (d + c2x)ex + c3e~x + 0.25xV        17. -0.5X"1 + 1.5x"5 19. cos It + e3x - 0.02 cosh x
App. 2   Answers to Odd-Numbered Problems
All
Problem Set 4.1, page 135
1. Yes   5. y[ = 0.02(-yi + y2), y'2 = O.Q2(y, - 2y2 + y3), y'% = 0.02(y2 - y3) 7. c, = l,c2 = -5 9. 3 and 0
11- >'[ = J2, y'z = 4}'!, >'! = cLe~2t + c2e2t = y, y2 = yi
13. yi = y2, y2 = y2, eigenvalues 0, 1, y, = c\ + c2ef, y2 = yi = y'
15. yi = y2, y2 = 0.109375.v! + 0.75y2 (divide by 64), yi = Cle_0125* + c2e0815'
Problem Set 4.3, page 146
1. >, = Cle-6t + c2enl\ y2 = -2c,e-6' + 2c2em 3. yT = c\e2t + c2, y2 = c^e2' - c2
5. y, = c\em + c2e-4'* = (c1 + c2) cos 4/ + ř(ci - c2) sin At = /l cos 4? + B sin 4/,   y2 = k^il - ic2e~4it
— (fcj — ;'c2) cos 4r + /(/ci + ;V:2) sin At = B cos 4? — v4 sin At, A = c3 + c2, ß - í(Cl - c2)
7. v-! = 2C] + c2f"6t, v2 = -Cl + c3e"6t, y3 = -c, + 2(c2 + c3)e"6f 9. Vl = CleL8t + 2c2e^-91 + 2c2e~x%t, y2 = 2Clelst + c2e~0M - 2c3e-1M, y3 = 2c1eliit - 2c2e-°M + c^"18*
11. yx = 10 + 6e2', y2 = -5 + 3e2t
13. y1 = 2Ae.-t - 2e2M, y2 = 1.8c-ť + 2e25t
15. .Vl = 2e14'r"' + 10, y2 = 5e14J5i - 4
17. y2 = yi + yi, y'2 = yi + y[ = -yi - y2 = ->'i - (yi + yi), y" + 2y[ + 2yL = 0, y\ = e~\A cos / + B sin f), y2 = y'i + >'i = e~\B cos t - A sin t). Note that r2 = Vi2 + V22 = e"2t(^2 + B2).
19. I, = 4Cle-2m + c2e~50t, I2 = -clťT200í - 4c2<r50t
Problem Set 4.4, page 150
1. Saddle point, unstable, yx = c^-4* + c2e4t, y2 = -2r1e"4t + 2c2eu 3. Unstable node, y1 = Cje' + c2£>3t, y2 = —c2e* 4- c2ď3í 5. Stable and attractive node, yx = t^e-34 + c2e~r>t, y2 = c^-34 - c2e~54 7. Center, stable, y-, = A cos 4r + B sin 4i, y2 = -2B cos 4r + 2A sin 4/ 9. Saddle point, unstable, y1 = c1e3i + c2e~\ y2 = cx<?3t — c2e~l 11. >'! = y = c1efet 4- c2e"'rt, y2 = y', hyperbolas Z:2)'!2 - y2  = const 13. y = e_2404 cos ř + B sin f), stable and attractive spirals 17. For instance, (a) -2, (b) -1, (c) -\, (d) 1, (e) 4.
Problem Set 4.5, page 158
1. (0, 0), yi = y2, y2 = 3yls saddle point; (0, -1), yt = % y2 = -1 + y2, y[ = -y2, % = 3yl3 center
3. (0, 0), yi = 4y2, y'2 = 2yt, saddle point; (2, 0), yx = 2 + yl5 y2 = y2, yi = 4y2, y2 = —2^, center
5. (0, 0), yi = -yj + y2, y2 = -y2 - y2, stable and attractive spiral point; (-2, 2), >'i = -2 + yi, y2 = 2 + y2, yi = -yx - 3y2, y2 = -y, - y2, saddle point
App. 2   Answers to Odd-Numbered Problems
7- >'i = )'2- % = _.Vi() _ 4>'i)' (0, 0), y[ = y2, };2 = ~yi, center;
(l, 0), >'! = | + ylt j2 = y2,    = y2, % = (-| - y1)(-4j1), y:', = y1; saddle 9. (gTi ± 2n7T, 0) saddle points; (—\tt ± Inir, 0) centers.
Use —cos (±f7T + Jt) = sin (±yi) ~ ±.yi-11. yi = y2, y2 = -v,(2 + y1)(2 -      (0, 0), y'2 = -4yu center; (-2, 0), y2 = 8&,
saddle point; (2, 0), y2 = 8yx saddle point 13. y"ly' + 2y'/y = 0, In y + 2 In y = c, y'y2 = y2y\2 = const 15. y = A cos f + £ sin t, radius Va2 + B2
Problem Set 4.6, page 162
3. y\ = A cos At + B sin At + §§, y2 = 5 cos At — A sin 4* - fr 5. yx = Cle4t + c2e~3t + A, y2 = ty4* - 2.5c2£>-3t - 10 7. yx = 2c1<^9t + c2e'^ - 90? + 28, y2 = c^-9' + c2e~4t - 126r + 14 9. v, = c-y + 4c2e2t    3t    4    2e~\ y2 = -cxel - 5c2e2t + 5/ + 7.5 + e~l 11. y, == 3 cos 2r - sin 2r + r + 1, y2 = cos It + 3 sin 2/ + 2/ - | 13. y1 = 4e_t - 4e* + e2t, y2 = -4(Tf + t 15. y'! = 7 - 2e2t + e3t - Ae~3\ y2 = -e2t + 3e~3t 17. /{ + 2.5(4 - 4) = 845 sin t, 2.5(72 - j[) + 25/2 = 0, lx = (95 + 162.5/)e_5t - 95 cos t + 312.5 sin t, I2 = (-30 - 162.5/)«"r" + 30 cos t + 12.5 sin t 19. /{ + 2(/i - I2) = 200, 2(/2 - 4) +8/2 + 2j/s*= 0. h = 2c ^ + 2c2e^t + 100,
I2 = (1.1 + Va4T)c1eAlt + (1.1 - V0Al)c2e^\ A: = -0.9 + V041, A2 = -0.9 - V'0.41
Chapter 4 Review Questions and Problems, page 163
11. y: = c^ + c2e~*\   y2 = 2c!e8t - 2c2e"8t. Saddle point
13. j>] = Cje* + c2<?~6,;,   y2 = c,e' - 6c2<?~8(. Saddle point
15. yx = Cie131 + c2e~3',   y2 =   r,e7r,/ + 0.75c2(r3t. Saddle point
17. yx = Cj^e5' + c2e'\   y2 = c^e3' — c2e'. Unstable node
19. yt = e~'(A cos 2f + B sin 2/),   y2 = e~f(B cos 2l - A sin 2r). Stable and
attractive spiral point 21. y,     <y' - r2<?_t + <?2t + e_2t,   y2 = -c2e-t - 1.5e"2t 23. y, = C]gf + c2e_2t - 6e_t - 5,   y2 = -cxef - 2cze~2t + \0i~' + 6 25. yL = cte3t + c2e~l + t2 - 2t + 2,   y2 = cxe3t - c2<?_t - t2 + It - 2 27. A saddle point at (0, 0) 29. /j = 4e-40t - e~wt,   I2 = -e"40t + 4e"10t 31. («17, 0) center for even n and saddle point for odd n 33. Saddle points at (0, 0) and (|, |), centers at (0, |) and (|, 0)
Problem Set 5.1, page 170
1. e0(l + .y + \x2 + •••) = «0^'^
3. a00 ~ 2x2 + fx4 - + •■-) + fliCx - %x3 + ^x5 - + •••) = a0 cos 2jc + \ax sin 2x
App. 2   Answers to Odd-Numbered Problems
A13
5. «0(1 + \x)
7. «0 + a0x + (|ö0 + i)x2 + • • • = a0ex + ex - x - l = cex - x-l,c = aQ+\
9. a0 + a,x + 2aix2 + ' " ' = «o ~~ a\ + «i^* 11. 5 = § - Ax + 8x2 - fx* + fx4 - ^ffx5,   5(0.2) = 0.69900 13. s = J + J* - ix3 + ^x5,   5(1) = 0.73125
15. s —  1  + x — x2 — §x3 + |a'4 + 24^5'     '?(|) = ffl
Problem Set 5.2, page 176
1. \c\ 3. 2 (as function of t = (x - 3)2). Ans. V5
5. 0 7. 2
9. 1 11. 77
17. flod - 4v4 - 4v5----) + ai(x + ix2 + ix3 + ±x* - ±x5----)
19. a0 + av(x - Ix3 + §x5 - ±x' + ±x» - rf^x11 +----)
21. flo(] - ix2 - £x4 + ^|xfi +••■) + ch(x - ix3 - ±x5 + j^x7 + ■■■) 23. a0( 1 + x2 + x3 + x4 + x5 + x6 + ■ ■ ■) + axx
Problem Set 5.3, page 180
3. P6(x) = ^(231xe - 315x4 + 105x2 - 5),
Pi(x) = ie(A29x7 - 693x5 + 315x3 - 35x) 7. Set x = az. y = cxPn(xla) + c2Qn(x/a)
15. Pa] = Vi - x2, Pa1 = 3xVl - x2, P22 = 3(1 - x2), P42 = (1 - x2)(105x2 - 15)/2
Problem Set 5.4, page 187
, ,  ,  x2      x4 sinh x 1       x      x3 cosh x
1- >'i = 1 + — + — +■•• = - ,   y2 =- + — + — + ■■■= —
3!      5! x x      2!      4! x
~ .      v2        1 144      36      x2 25x4
3-yi = 1 " 72 + ^84*      ••••V-9V'lnA      .v- - ^ + T     T024 +
1 x y3
5. r(r - 1) + 4r + 2 = 0, r2 = -1, r2 = -2; v, =--- +---+ • ■ •
■     x     6 120
1       1      x2 x4
y2 = -Tj--H-----[--•■■
'2     x2      2      24 720
7. Euler-Cauchy equation with r = x + 3, y1 = (x + 3)5, y2 = y: In (x + 3) 9.b0= 1, c0 = 0, r2 = 0, yi = «T* y2 = e~* In x 11. y, = l/(x + 1), y2 = 1/x
13. b0 = h2,c0 = 0, n = i r2 = 0, y, = x1/2(l + 2x + 2x2 + |xB + ■••),
y2 = 1 + 2x + 2x2 + • • •
15. yx = (x - 4)7, y2 = (x - 4)~5 (Euler-Cauchy with t = x - 4)
I
6
17. y, = x + x3 - ix4 + 4x5 - ^x6 + • •    y2 = l + 3x2 - ±x3 + fx4
^-x6 - + •
A14
App. 2   Answers to Odd-Numbered Problems
19. v = c^Xi i 5; x) + c2VxF(l, 1, |; x)
21. v = A(l - 4x + fx2) + ÄVxF(-§, §,1; x)
23. y = ClF(2, -2, -1; f - 2) + c2(» - 2)3/2F(l, -J, f; r - 2)
Problem Set 5.5, page 197
1. Use (7b) in Sec. 5.2.
3. 0.77958 (exact 0.76520), 0.19674 (0.22389), -0.27651 (-0.26005),
-0.39788 (-0.39715), -0.17038 (-0.17760), 0.15680 (0.15065), 0.30086 (0.30008), 0.16833 (0.17165)
5. y = dJu(te) + c2i_„(Ax), v * 0, ±1, ■ • •
7. y = ClJ„(Vx) + c2J_v{Vx).. v # 0, ±1, • ■ ■
9. y — c1x/1(2x), Ju J_1 linearly dependent 11. y = x~v[CiJj[x) + c2J_v(x)\, v + 0, ±1, • • • 13. y = CiJJx3) + c2/_„(x3), v * 0, ±1, • • • 15. y = c1Vx/1(2Vx), JL,      linearly dependent 17. y = x1/4/x(|x1/4), y1( /_! linearly dependent 19. y = x2/5(c1./8/5(4x1/4) + 62./_8/5(4x1/4))
21. Use (24b) with v = 0, (24a) with v = 1, (24d) with v = 2, respectively.
23. Jn(x{) = JnXx2) = 0 implies x1_n/„(x1) = x2~nJn(x2) = 0 and [x_n/n(x)]' = 0
somewhere between xx and x2 by Rolle's theorem. Now use (24b) to get
Jn+,(x) = 0 there. Conversely, Jn+](x3) = in+1(x4) = 0, thus
x3K+1/n+|(xH) = x4*l+1/K+1(x4) = 0 implies Jn{x) = 0 in between by Rolle's
theorem and (24a) with v = n + 1. 25. Integrate the formulas in (24).
27. Use (24a) with v = 1, partial integration, (24b) with v — 0, partial integration. 33. CAS Experiment, (b) x0 = 1, x, = 2.5, x2 = 20. approximately. It increases with n. (c) (14) is exact,   (d) It oscillates,   (e) Formula (24b) with v = 0
Problem Set 5.6, page 202
l.y = ClJ5(x) + c2Y5(x) 5. y = ClJ2(x2) + c2F2(x2) 9. y = x3(Cl/3(x3) + c2F3(x3)) 13. Set x = « in (1), Sec. 5.5, to get Sec. 5.5.
3.y = CjJ0(Vx) + c270(Vx) 7. y = x-5(Cl75(x) + c2Y5(x)) 11. Set H(1) = /t//<2), use (10). present ODE (12) in terms of s. Use (20),
Problem Set 5.7, page 209
3. Set x = ct + k. 5. x = cos 9, dx = —sin 9 d9, etc.
7. Am = (m7r/5)2, m = 1, 2, • ■ • ;      ym = sin (mirx/5)
9. Am = [(2m + l)ir/2L]2, m = 0, 1, • • • ;      yTO(x) = sin [(2m + 1)ttx/2LJ 11. A,m = m2, m = 0,1, • * ■ ;      y0 = 1, ym = cos mi, sin mx, m — 1, 2, ■ • ■ 13. k = k,m from tan k = —k.      Am = &m2, m = 1, 2, ■ ■ • ;      ym = sin fcmx 15. A„, = m2, m = 1, 2, • ■ ■ ;      ym = x sin (m In |x|) 17. p = e8x, q — 0, r = e8x, Am = m2; ym = e~4x sin mx, m = 1, 2, ■ ■ • 19. A,re = {mir)2, y.m = x cos m7Tx, x sin m-7rx, m = 0, 1, • • •
App. 2   Answers to Odd-Numbered Problems
A15
Problem Set 5.8, page 216
1. 1.6P4(x) - 0.6P0(x) 3. §P3(x) - §P2(x) + g/^Cx) - 4P0(x)
7. -0.4775/51(x) - 0.6908P3(x) + 1.844P5(x) - 0.8234P7(x) + 0.1544P9(x) + ■ • • ,
n/0 = 9. Rounding seems to have considerable influence in Probs. 6-15. 9. 0.3799P2W + 1.673P4(x) - 1.397P6(x) + 0.3968P8(x) + • • • , m0 = 8
11. 1.175P0(x) + 1.104P1(x) + 0.3575P2(x) + 0.0700P3(x)----,     = 3 or 4
13. 0.7855Po(x) - 0.3550P2(x) + 0.09()0P4(x) - • • - , m0 = 4
15. 0.1212PoW - 0.7955P2(x) + 0.9600P4(x) - 0.3360P6(x) + • ■ • , m0 = 8
17. (c) am = (2//12(a0.m))(./1(a0,m)/a()m) = 2/(a0,m./1(a0,n))
Chapter 5 Review Questions and Problems, page 217
11. e3x, e~3x, or cosh 3x, sinh 3x 13. ex 1 + x
15. e~:"'2, xe~x2 17. e~x, e~x In x
19. 1/(1 - x2), x/(l - x2) or 1/(1 - x), 1/(1 4- x)
21. y = c1Jvi(6x) 4- <:,/_x-(6.vl 23. y = c^.ix2) + c^Y^x2)
25. y = \rx[cvIv^kx2) + c2./_1/4(|fo-2)]
27. ATO = (2mTr)2, y0 = 1, ym = cos 2mirx, sin 2rmrx, m = 1, 2, • • ■
29. y = C\J\(kx) + c^Y^kx), c2 = 0, v(l) = c1/1(fe) = 0, k — km = ax m (the positive
zeros of /J, >•„,, = ./^a, mx) 31. 1.813P0(x) 4- 2.923f1(x) 4- 1.759P2(x) 4- 0.663P3(x) 4- 0.185P4(x) 4- • ■ • 33. 0.693P0(x) - 0.285P2(x) 4 0.144P4(x) - 0.091P6(x) 4- • • • 35. 0.25P0(x) + 0.5Pv(x) + 0.3l25P2(x) - 0.0938P4(x) 4- 0.0508P6(x) + • • ■
Problem Set 6.1, page 226
2       2 .v s-2
1 —     — 3 - 5 -
" s3      s2 ' s2 4- 4tt2 ' (s - 2)2 - 1
s cos 6 — co sin 0 e3a 1
7- -2-2- 9- - 11. -5-
sz + oj2 s 4- 2/? 4- 4
19.--— 21.         = 2(1) - <Af) = - - - (1 - ^2s) =
s 5 5
23. Set c< = /J. Then = f e_st/(cf) dK = f <r(s/c)p/(p) dplc = F{slc)lc.
29. 4 cos 77-/ - 3 sin irt 31. 1 - 1 f2 4- ^r4 33. sin —
2      2 L
35. 2 - 2<?-4t 37. (eV3t - <Tv'Bt)/(V3 4- V5)
1           r-        -t 3.8 5«
39. —=- sin VSt - £-°f 41. -^ 43.
V5 (* - 2.4)2 (i- + a)2 4- w2
a(s + k) + b (s + A)2 + 1
51. 3e~2t sin 5t 53. e~57rt sinh tt?
45. —-~- 47. 3.5/V 49. V/2rVtv'5
App. 2   Answers to Odd-Numbered Problems
Problem Set 6.2, page 232
I 2a>2 2a2
1 - 3 - 5 -
(s - kf ' s(s2 + 4u>2) ' s(s2 - 4a2)
its
1 -
(s    + -gTT )
9. Use shifting. Use cos2 a = \ + § cos 2a; use cos2 a + sin2 a = 1.
Am\ (2.?2 + l)/[2s(s2 + 1)] 11. 0 + |)F = -1 + 17 • 2/U2 + 4), y = 7e-m + 2 sin 2t - 8 cos 2? 13. (s2 - |)7 = 4s, y = 4 cosh I?
15. (s2 + 2s + 2)7 = s - 3 + 2 • 1, 7 = (s + 1 - 2)/[(s + I )2 + 1],
y = e-t(cos f — 2 sin t) 17. (s2 + 7s + 12)7 = 3.5.v - 10 + 24.5 + 21/0 ~ 3), y = \eAt + §<>-4t + ^e~3' 19.    + 1.5)2K = s + 31.5 + 3 + 54/.v4 + 64/.?,
Y = 1/(5 + 1.5) + 1/(5 + 1.5)2 + 24/,v4 - 32/.v3 + 32/.v2,
y = (1 + t)e-1M + 4t3 - \6t2 + 32t 21. t = t + 2, Y = 4l(s - 6), y = 4e6t, y = 4e6(T~2)
23. t = t + 1, (s - 1)0 + 4)Y - 4s + 17 + 670 - 2), y = 3et~1 + e2a_1) 25. (b) In the proof, integrate from 0 to a and then from a to x and see what happens, (c) Find ££(/) and it(j') by integration and substitute them into (1*).
27. 2 - 2e~m 29.      (ekt - 1) - -        31. cosh V5/1 - 1
k k
33. | sinh 2? - |r
2s
5\    «.     / 1 10
Problem Set 6.3, page 240
3. (1 - e2-2s)l(s - 1)
(2       2       1 \ / 2       4 4,
7-    (-e_s - ^     M;- •i) ' \s2 s,
11.   T20*, (f_3s + e_6s) 13. —— (e~2s+2^ - eris 47T)
s    +  tt s - tt
15. 0 if t < 4, t - 4 if t > 4 17. sin / if 2tt < f < 8tt, 0 elsewhere
19. 0 if t < 2, (t - 2)4/24 if t > 2 21. u(t - 3) cosh (2r - 6)
23. e~l sin t 25. e~2t cos 3r + 9 cos 2t + 8 sin 2?
27. sin 3f + sin t if 0 < t < tt and § sin 3t if r > tt
29. r - sin t if 0 < / < 1, cos (f - 1) + sin (r - 1) - sin t if f > 1
31. ef - sin / + u(t - 2ir)(sin r - g sin 2t)
33. t = 1 + J, y" + 4y = 8(1 + t )2(1 - w(? - 4)), cos 2t + 2t2 - 1 if r < 5,
cos 2/* + 49 cos (2t - 10) + 10 sin (2t - 10) if t > 5 35. fty' + r//C = 0, Q = S£(q), q(0) = CV0, i = q'(t), R(sQ - CV0) + QIC = 0,
q = CV0e-tma
100        100    , , / 1 1 \
37. 10/ + -/ =        e'2\ I = e_2s---, i = 0 if t < 2 and
.9 s2 \ s      s + 10 /
1 - e-10«-2)if/> 2
App. 2   Answers to Odd-Numbered Problems
A17
39. i = e-20t + 20t - 1 + u(t - 2)[-20/ + 1 + 39e-20ct-2)]
41. 0.1/' + 25/ = 490e-5*[l - u(t - 1)]. i = 20(e_5t - <?~250t) + 20u(t - \)[-e~F
_|_ e-2S0t + 245|
43. i = (10 sin 10r + 100 sin t)(u{i - tt) - u{t - 3tt)) 45. i' + 2i + 2 \ i(t) (1t = 1 - w(r - 2), / = (1 - <»-2s)A>2 + 2,v + 2), / = e"* sin t - M(f - 2) e"t+2 sin (f - 2)
47. i = 27 cos / + 6 sin t - e~\21 cos 3f + II sin 3;)
+ u(t - 2tt) [-27 cos t - 6 sin t + e~ct_2ir)(27 cos 3/ + 11 sin 3?)]
Problem Set 6.4, page 247
l. v = 10 cos / if 0 < t < 2tt and 10 cos t + sin f if f > 2tt
3. y = 5.5/ + 4.5<T* + 5(e*_1/2 - e"tH l/2)«(f - \) - 50(et_l - e-'+1)u(' - l)
5. v = O.lfe* + e~2*(-cos I + 7 sin f)]
+ 0.l«(r - lO)[-e* + e_2'+30(cos (t - 10) - 7 sin (i - 10))] 7. y - l + |e_t sin 3r + u(t - 4)|-l + e_t+4(cos (3r - 12) + | sin (3/ - 12))]
- fu(t - 5)e_t+5 sin (3f - 15) 9. y = 5f — 2 — 50«(r — 7r)f~' + 7T sin 2?. Straight line, sharply deformed between tt
and about 8
11. y = (0.4r + L.52)e< + 0.48^4t + 1.6u(t - 2)[-<?' + e_4t+1°]
Problem Set 6.5, page 253 l.t 3. e* - / - 1
5. — sin cot 7. — (fkt - e tet) = — sinh At
w 2« A
9. l(eM - e~5t) 11. l(i - \ cos 2f) = \ sin2 /
13. f - sin t 15. |(cosh 3/ - 1)
19. Y = 3/((s2 + 4)(.s2 + 9)), y = 0.3 sin 2r - 0.2 sin 3f
21. (x2 4- 9)Y = 4 + 8(1 + e^/fs2 + 1), y = sin / + sin 3r if f < tt, f sin 3/ if / > tt 23. 0 if 0 < t < 1, § [ sin (2(t - 1))     = -| cos (2/ - 2) + | if t > 1
25. y = 2e"2* - e"4' + (<?-2tl2 - ?-4j+4)«(f - 1) + (e"2tt4 - e-4t+s)u(t - 2) 27. y - 1 * y = 1, y = el 29. y - y * sin f = cos f, Y = 1/s. y = 1
31. Y(l + \ls2) = Ms, y = cos t 33. Y(\ + 2l(s -!)) = (,$- I)"2 y = sinh t
Problem Set 6.6, page 257
1.
5.
9.
(s - 1 f
2s + 4 (.v2 + 4s + 5)2 2üj(3s2 - w2) (s2 + co2f
3.
2 cos
II.
(s2 + co2)2 24s2 + 128 (,v2 - 16)3
Is cos A + (s2 - 1) sin k (s2 + 1?
Al 8
App. 2   Answers to Odd-Numbered Problems
13. 6r<-' 17. t2ekt
15. fe"
sin ?
19. In ,v - In (s - 1); (g* - \)/t
Problem Set 6.7, page 262
1. y, - -e~* sin 7,   y2 = e~' cos / 3. y, = 2e~u - 4e2t,    y2 = e~4t - 8e2t
5. y, = le'* + 4e-2t + \t - i   y2 = -3e~* - 4e"2t - |? + | 7. y, = e~\2 cos 2? + 6 sin 2?) + ?2,   y2 = 10e_t sin 2? - f2 9. >>! = 4 cos 5r + 6 sin 5/ - 2 cos r - 25 sin t,  y2 = 2 cos 5t - 10 sin 5t + 20 sin f 11. Vj = -cos f + sin t + 1 + u(t - !)[-] + cos (/ - 1) - sin (f - 1)] y2 = cos t + sin t -1 + w(? - 1)[1 - cos (r - 1) - sin (t - 1)|
e* + iu(t - 1)(-
13. Vi = 2«(f - 2)(<?4t 15. ya = -e 21
+ \u{t - i)(-e-2l+?' + e') 17. _Vl = 3 sin 2t + 8e"3'
et+%   y2 = e2t + u(t-e-2t+3 + e%
2)(e4
3e2
v2
3 sin 2t
t.
5e
y2 = -e-2t
-3t
i + 2et' 6) 4e*
19. \j = <?' - e_t,   y2 = f',   y3 = e 25. 4?i + 8(4 - j3) + 2z{ = 390 cos t,   8i2 + 8(i2 -"8t + 42 cos f + 15 sin t,   U = -26e~n
h) + 4(2 = 0, iL
-26t
\6c
8e"
+ 18 cos t + 12 sin t
Chapter 6 Review Questions and Problems, page 267
11.
1
(.s - 3)2
13.
17.
s(s* 2.V2
4)
(s - l)(s2 + 4)
23. 10 cos tV'2
29. te~2t sin t
35. 20 sin t + u(t - 1)|1 37. 10 cos 2t -\ sin 2t -I
41. e    + u(t - ir)[L2 cos t - 3.6 sin t 43. u(t - l)(t - l)e2t"2 45. y, = e* , 2« 2 47. y, = |e_t sin 2/. y2
ie * - i cos t
19.
25. 3e"2t sin 4t
31. (J2 - l)w(z - 1)
- cos(r - l)j 4u(t - 5) sin (2t - 10)
2e~t+7T
4u(t - 2)(2 - t)e2'-4
\ sin t,   V;> = — e* l
49. yj = e2t,   y2 = e2t + e<
e l(cos 2/    ^ sin 2/) = e2t + e' 2s')/[^ + 10)1,   i = 0.1(1 - e~w
15. e-™ I —
21.
a — b
(s — ä)(s — b)
27. u(t - 2)(5 + 4(1 - 2))
77
33. -g (<Wf
39. e
Sin trJiT)
'(7 cos 3f + 2 sin 3/)
0.8e2'-2^
I cos t
51. / = (1 - e~'AS)/[s(s + 10)1, i = 0.1(1 - e~mt) + (Uu(t 53.1 = e"2t(76 cos 4f - 42 sin 4r) - 76 cos 20f + 16 sin 20t 10(4 - ^) = 100 t2,   304' + 10(4' - '{) + 100(2 =
55. i[
ii
(% + 4t)c~at + \0t2
5-
i2 = (I + 2t)e-M + 2t
sin t
2)[-l + c-
Problem Set 7.1, page 277
	" 6	3"		" 36	-6"		"-36	6~
1.	-2	11	, same.	-36	18		36	-18
	.-7	6_		_-54	0_		_ 54	()_
App. 2   Answers to Odd-Numbered Problems
A19
3. Undef
-6 6 9
48 -2' 38 -44 67   -15
6 -6 -9
undef.
36 -12 72
0 24 60
48' 24 -48
same,
	" 66"		"   0 "		" 6.5"
7.	0		3.2	, same,	-0.8
	_-33_		.-4.2.		-21.
9. -5x2 = -3
-5x, + 2x2 = 4 -3xj + 4x2 = 0
-0.3 -4.9 -3.6
5.0 -3.4" 1.8 3.8 3.5 0.4.
	" 2"	
, undef.,	38	
	_-22.	
Problem Set 7.2, page 286 24 49
-43
54 10 -46
74 19 -29
-74 -5 51
236 -92 -12
-92 38 6
-12 6 6
5. [20   -3   -71,1-62   34 2],
54 10 74 19 74 -5 134 -50 94 -29 -134 63
'565'
525 790
-46' 29 51.
18'
19.
44 64
-72
same
64 110 114
	"15	0	40"		"-310	170	10"	
7.	3	0	8	, 31,	-62	34	2	, same
	_ 6	0	16 J		_-124	68	4_	
	" 337		8 -	160"		" 257	68	-188"
9.	252		49	-68	, same,	232	97	-96
	.-308		52	233.		_-248	-16	265.
-72" 114 126.
App. 2   Answers to Odd-Numbered Problems
11.
324     32 -320 244     38 -322 -244   -10 366 4324      1520 -4816 3636      1242 -4518
216 280 -280
-104 -132 140
-104'
-68 76
7060 7548 -8140
960 -5120" 1246 -5434 -1090 6150.
-3700
■1046 5002
13. 83, 166, 593, 0 19. (d) AB = (AB)T
BTAT = BA; etc. (e) Ans. If AB = -BA.
21. Triangulär arc Uj. + U2, UjU2, U/, Lx + L2, LXL2, Lx2. 23. [0.8   1.2]T, [0.76   1,24|T, 10.752 1.2481T 27. p — 1110   45   80JT, v = [92000 86300]T
Problem Set 7.3, page 295
3. x = 0.2, y = 1.6 7. x = 4, y = 0, z = -2 11. y = 2x + 3z + 1, x, z arb. 15. w= 3,x = 0,y = -2,z =
1. X = 2.5, y = -4.2 S.x = 0, y = -2, 7 = 9 9. x = 3y + 2, y arb., z = —y + 6
13. iv = 1, y = 2z — x, x, z arb. ia. n = j, x = u, y = — z,
17. /! = (Rx + R2)E0/(RXR2),   J2 = EaIRx,   I3 = E0/R2 [Amps] 19. Ix - 72 - 4 = 0, (3 + 2 + 5)/! + 10/2 = 95 + 35, 10/2 - 5/3 = 35, Ix = 8, 4 = 5, /3 = 3 Amps
21.     + x4 = 500, xx + x2 = 800, x2 + x3 = 1100, x3 + x4 = 800, x, = 500 - x4, X;> = 300 + x4, x3 = 800 — x4, x4 arbitrary
Problem Set 7.4, page 301
-2];[1   0 -3]T 4   0   7], [0 1]T
0 5], [0 3 0   4], [0 2
1. 1, [1 3. 3, [1
[0 0 5. 2, [3 7. 2, [8 9. 3, [1 11. 4, [1 13. No 19. Yes 29. No 35. 1, [5
-2   1   3], [0   0   5   1051; [~2   4   5JT, [0   1 5]T,
4]; [3 0 5]T, [0 3 0]; [8   0   4   0]T, [0
0 3 0 0
0], [0 0], [0
8   -37], [0 0 0   0], [0   0 1 15. No 21. (c) 1 31. 1, \-i
4]T
2   0 4]T
74   296]; same transposed 0], [0   0   0   11; same transposed 17. Yes
27. 2, [1   -1   0], [0   0 1] 1] 33. No
Problem Set 7.7, page 314
5. 107
9. -66.88
13. u3 + v3
19.x = 23. 3
w  — 3uvw 1.2, y = 0.8, z = 3.
7. cos (a + ß) 11. 0 15. 4 21. 1
App. 2   Answers to Odd-Numbered Problems
Problem Set 7.8, page 322
I.
1.80	-2.32
_-0.25	0.60
1 0	0"
-2 1	0
3 -4	1
cos 2d sin 2d
-sin 26' cos 20
9. A~
'2 0 5 1 .0 1
11. No inverse
15. (A2)-
(A )2
4 15
-1 -5 I -5 4 9.
19. AA"1 = I. (AA-1)-1 = (A-1)-^-1 = I. Multiply by A from the right.
21. del A 23. det A
1. C,
C-12 — C21 — C:
1. c,9
33 —
-2, Ca
the other Q> are zero
C
3, C
-4, C
33
Problem Set 7.9, page 329
1. Yes, 2, [3   5   0]T, [2   0   -5|T 3. No
5. Yes, 2, [0   0   0   1   0]T, [0   0   0   0   1 ]T 0 1"
7. Yes, 1. 9. No
.-1 0.
11. Yes, 2, xe~x, e~x
13. [1 0|T, [0 1]T: [1 1]T, [-1 If; [1 Of, [0 15. jtj = -0.6yx + 0.4y2
x2 = -0.8.V! + 0.2y2 19. xj = 5y1 + 3y2 — 3ya
x2 = 3>i + 2y2 - 2y3
*3 = 2>'1 ~   >'2 + 2j3 21. V56 25. 2
If
17. X] = 2y1 + y2
x2 = 5\\ + 3y2
23. 16 V5
29. 4«! - 3y2 = 0, v = ± [|
Chapter 7 Review Questions and Problems, page 330
11. x = 4. y = 7 I5.* = iy=-J,z = § 19. x = 2z, y = 4, - arbitrary
13. x = y + 6, z = y, y arbitrary
17. x = 7, y = -3 21. 0
8.0 -3.6
23. 638, 0, 0
25.
-3.6 1.2
2.6 2.4
1.2 2.4 9.0.
	"12	0	6"		^-20
27. 14. 14,	28	0	14	29. [-20   9 -3],	9
	_ 4	0	2_,		-3
A22
App. 2   Answers to Odd-Numbered Problems
31. 2, 2 35. 2, 2
33. 2, 2
37 J-
■"• 51
4 _-5
	"-5	10	5"		" 72	-72	132
j^. 45	11	5	-2	*a* 12	31	-32	59
	. 23	-10	4_		.-19	20	-35
43. Ix = 33 A, I2 = 11 A, Ig = 22 A 45. Ix = 12 A, /2 = 18 A, I3 = 6 A
Problem Set 8.1, page 338 1. -2, [1   0]T; 0.4, [0 1]T
3. 4, 2xx + (-4 - 4)x2 = 0, say, xx = 4, x2 = 1; -4, [0 1]T 5. -4, [2   9]T; 3, [1   1]T   7. 0.8 + 0.6/, [1   -i]T; 0.8 - 0.6;, [1 ;]T 9. 5, [1   21T;0, [-2   1]T 11.4, [1   0   0]T; 0, [0   1   0]T; -1, [0   0 if 13. —(A3 - 18A2 + 99A - 162)/(A - 3) = -(A2 - 15A + 54); 3, |2   -2   1 ]T;
6, [1   2   2]T; 9, [2   1 -2]T 15. 1, [-3   2   10]T;4, [0   1   2]T; 2, [0   0 1]T 17. -(A3 - 7A2 - 5A + 75)/(A + 3) = -(A2 - 10A
5, [3   0   1]T, [-2   1 0]T 19. -(A - 9)3; 9, [2   -2   1]T; defect 2
21. A(A3 - 8A2 - 16A + 128)/(A - 4) = A(A2 - 4A - 32); 4, [-1   3   1 1]T;
-4, [1   1   -1   -1]T;0, [1   1   1   1]T; 8, [1   -3   1 -3]T 23. 2, [8   8   -16   1]T; 1, [0   7   0   4]T; 3, [0   0   9   2]T, -6, [0   0   0 1]T 25. (A + 1)2(A2 + 2A - 15); -1, [1   0   0   0]T, [0   1   0 Of;
-5, [-3   -3   1   1]T, 3, [3   -3   1 -1]T 29. Use that real entries imply real coefficients of the characteristic polynomial.
25); -3, [1   2 -1]T;
Problem Set 8.2, page 343
	~-l	0		"l		"0"
1.			; -1,		; l,	
	0	1_		_0_		.1
; any point (x, 0) on the x-axis is mapped onto (—x, 0), so that [1   0]T is an eigenvector corresponding to A = —1.
A point on the x-axis maps
	"1 0		"l		"0"
3. (x, y) maps onto (x, 0).		; l,		; o,	
	.0 0_		.0.		.1.
onto itself, a point on the y-axis maps onto the origin.
5. (x, >•) maps onto (5x, 5_y). 2X2 diagonal matrix with entries 5.
7. -2, [1   -1]T, -45°; 8, [1   11T, 45°
9.2,|3   -1JT, -18.4°; 7, [1   3]T, 71.6° 11. 1, [-1/V6   l], 112,2°; 8, [l    1/VrjJ, 22.2° 13. 1, [1    1]T, 45°; -5, [1    -1]T, -45° 15. c[15   24   50]T, c>0
17. x = (I - A)_1y = [0.73   0.59   1.04]T (rounded)
19. [1   1   1]T 21.1.8 23.2.1
App. 2   Answers to Odd-Numbered Problems
A23
Problem Set 8.3, page 348
3. No 5. A-' = (-AT)_1 = -(A_1)T
7. No since del A = det (AT) = det (-A) = (-1)3 det A = -det A = 0. 9. Orthogonal, 0.96 ± 0.28* 11. Neither, 2, 2, defect 1
13. Symmetric, 9, 18, 18 15. Orthogonal, 1, i, -i
17. Symmetric, a + 2b, a — b, a — b
Problem Set 8.4, page 355
			"1	2"		7 0"
1. [1	2]\ [2 -	if; x =				
			2	-1_		.0 2_
			^4	0"		
3. [1	-1]T, [1	1]T, D =				
			.0	6.		
5. [2	-UT, [2	If: diag (	-2,	4)		
7. [1	0   0]T, [1	-2 If	. to	1   Of, diag (1, 2, 3)		
9. [0   3   2f, [5   3   Of, [1   0   2f. diag (45, 9, -27)
	"16	-7		"1		"l		-2		0"
13.			; -5,		; 2,		; x =			
	_42	-19.		_3_		_2_		0_		_-l_
	" -30	-72"		9		" 12		-3		n-3
15.			■ 2,		; 0,		; x =			
	40 3	32_		_-4j		_-5_				- 6_
	"-95	18	-144"		r 4"		r 6"		" 3"
17.	24	-2	36	; 4,	— 2	; -2,	-1	; 1,	0
	66	-12	100_		.-3.		_-4j		_-2_
	"-2"		r o"		" 0"
x =	-4	>	— 2	»	0
	-6		-4		-2
19. C
21. C
23. C
25. C =
27. C
1 12"
.12 -6.
3 -4
.-4 -3
4 V3 V3 2
1 -6"
-6 1.
12 16
16 12
, lO.v'!2 - 15y5
5, x
y, hyperbola
5>'f - 5y22 = 0, x =
>>'i2 + 5y22 = 10, x
7yj2 - 5y22 = 35, x
28y!2 - 4y22 = 112, x =
0.8 0.6 0.6 -0.8 2/V5 L/VŠ 1/V5 2/V5
1/2      V 3/2 -VŠ/2 1/2 l/V'2 1/V2" -1/V2 1/V2_ 1/V2 1/V2 1/V2 -1/V2
y, straight lines y, ellipse y, hyperbola y, hyperbola
A24
App. 2 Answers to Odd-Numbered Problems Problem Set 8.5, page 361
V2, \ -i i + V2T
3. (ABC)T = CTBTAT = C1(B)A
5. Hermitian, 3 + V2, [-i   1 - V2T; 3
7. Hermitian, unitary, 1. [l <
9. Skew-Hermilian, 5i, [1   0   0]T, [0   I   1]T; -5i, [0   1 -1]T 11. Skew-Hermitian, unitary, i, [1   0   l]T, [0   1   0]T; -/. [1   0 -1]T 13. Skew-Hermitian, —66/ 15. Hermitian, 10
/V'2lT;-l,[l   i + 1V2Y
Chapter 8 Review Questions and Problems, page 362 3 2"
-4 -3
15.
	3 2		5	0"
A		=		
	_-4 -3.		_()	-7_
		1/3	2/3"		"1	2		(		0"
II.				A			=			
		2/3	-1/3.		2	l_		_0		9_
	2 9	l 9	o"		4	1		2		~6
13.	1 y	0	9	A	1	-2		4	=	0
	0	2 9	1 9_		-2	4	-	1_		0
-1.0
4.8
0" 5.0
19. 1.1 Vi2
,5,-1
1. ellipse
17.
0
-3 0
-2
0 0 0 0 0 0
-4.50' -2.75 4
, 4, 0,
Problem Set 9.1, page 370
1. 2, -4, 0; V2Ö; [l/V'5, -2/V5, 0] 5. -8, -6, 0; 10: [-0.8, -0.6. 0]
9. (i §, |); V^7/8 13. [4, -2, 0], [-2, 1, 0], [-1. \ 17. [28, -14,-14] 23. (5.5, 5.5, 0). (i i f) 27. [-8, -2. 4]; V84 31. [-9, 0, 01, [0, -2, 01, [0, 0, -25     25 1     [ 20 V" 2 ' Vi J V 2
01
3. -1, 0, 5; V26; 7. (7, 5. 0); VlO
35
1J. Yes. 33. |p 20
\/2
11. (0, 1,|); V37/2 15. |10, -5, -15] 19. [-2, 1, 8], [6, -3 25. [0, 0, 9]; 9 29. v = [0, 0, -9] q + u 5
1/V26, 0, 5/V26]
24]
45
V2
V2
6. Nothing 37. |w|/(2 sin a)
Problem Set 9.2, page 376
''241
1.4 3.
5. [12, -8. 41, |-18, -9, -36J 9.-4,4 11.-24 17. |a + hf + |a - b|2 = a-a + 2a«b + b'b 19. 0 21. 15
7. 17
15. Use (1) and |cos -y| = 1-(a«a - 2a«b + b«b) = 2|a|2 + 2|b|2 23. Orthogonality. Yes
App. 2   Answers to Odd-Numbered Problems
25. 2, 2, 0, -2 27. 79.11° 29. 82.45°
31. 54.74° 33. 54.79°, 79.11°, 46.10° 35. 63.43°, 116.57°
37. 3 39. 1.4
41. If |a| = |b| or if a and b are orthogonal
Problem Set 9.3, page 383
1. [0, 0, -10J, [0, 0, 10]      3.|-4,-8,26| 5. [0, 0, -60]
7. -20, -20 9. 240 11. [19, -21, 24], V1378
13. [10, -5, -1] 15. 2 17. 30, -30
19. -20, -20 25. [-2. 2, 0| x [4, 4, 0] = -16k. 16
27. [1, -1, 2] x [1, 2, 3] = [-7, -1. 3], V59 29. [0, 10, 0] X [4, 3, 0] = [0. 0, -40], speed 40 31. |[7, 0, 0] x [1, 1, 0]| = 7 33. |V3
35. [18, 14, 26]; 9x + ly + 13z = c, 9 • 4 + 7 ■ 8 + 13 • 0 = 92 = c 37.16 39. c = 2.5
Problem Set 9.4, page 389
1. Hyperbolas 3. Hyperbolas 5. Circles
7. Ellipses; 288, 100, 409; elliptic ring between the ellipses
x2 y2 x2
+ —-—r = 1 and -s- + v2 = 1.
(2/3r      (1/2)2 (4/3 r
9. Ellipsoids 11. Cones 13. Planes
23. [8x. 0, yz], [0, 0, xz], [0, 18z. xy]; [0, z. y], [z, 0, xj, [y, x, 0]
Problem Set 9.5, page 398
1. [4 + 3 cos /,   6 + 3 sin t] 3. [2 - t,   0,   4 + t\
5. [3,   -2 + 3 cos t,   3 sin f] 7. [a + 3/,   b -It,   c + 5f]
9. Wl cos t,   sin f,   sin f] 11. Helix on (x - 2)2 + (y - 6)2 = r2
13. Circle (x - if + (y + if = 1, z = 5   15. x4 + y4 = 1
17. Hyperbola xy = 1
23. r' = [-5 sin t,   5 cos t,   0], u = [-sin t,   cos t,   0], q - [4 - 3w,   3 + 4w, 0] 25. r' = [sinh r,   cosh t\, u = (cosh 2t)~V2 |sinh     cosh /], q = [| + Aw,   % + 5w] 27. Vr'-r' = cosh /, Í = sinh 1 = 1.175 29. Start from r(r) = [t, f(t)]. _
33. v = r' = [l.   It,   0], |v] = Vl + 4r2, a = [0,   2, 0| 35. v(0) = 2u,R\, a(0) = ~w2R\
37. 1 year = 365 ■ 86400 sec, R = 30 ■ 365 • 86400/2tt = 151 • 106 [km]. |a| = a2R
= \\f/R = 5.98 • 10"6 [km/sec2] 39. R - 3960 + 80 mi = 2.133 ■ 107 ft, g = |a| = ofR = \\\2/R, |v| = VgR =
Vó.61 • 108 = 25700 [ft/sec] = 17500 [mph| 43. r(/) = [t,   yd),   0], r' = [1,   y\   0], r'«r' = 1 + y'2, r" = [0,   y",   0], etc. 47. 3/(1 + 9/2 + 9?4)
App. 2   Answers to Odd-Numbered Problems
Problem Set 9.6, page 403
1. w' = 2V2(sinh 4;)/(cosh 4t)m
3. w' = (cosh r)sinh t_1((cosh2 t) In (cosh t) + sinn2 f)
5. w' = 3(2r4 + r8)2(8f3 + 8r7) 7. e4" sin2 2v, |e4" sin 4u
9. -2(«2 + u2)"3«, -2(«2 + u2r3v
Problem Set 9.7, page 409
1. [2x,   2yl 3. [1/y,   -x/y2] 5. [y + 2,   je - 2]
7. [6,   4,   4| 9. [-1.25,   0] 11. [0, -e]
13. [-4,   2J 15. [-18,   24] 17. [48, -36]
19. [6,   4] 21. [-6, -12]
23. [-0.0015, 0,   -0.0020] 27. \a,_b, c]
29. [8,   6,   0] 31. [108, J08,   108] 33. V2/3
35. 7/3 37. 2e2/Vl3
39. fx2 + |v2 - 2z2 41. -4^'3
Problem Set 9.8, page 413
1. 3(x + v)2 3. 2(x + xz + z) 5. (v + x + 1) cos xy
7. 9x2y V
9. [Vl, v2, v3\ = r' = [x\ y', z'l = [y, 0, 0], z = 0, z = c3, y' = 0, y = c2, x = y = c2, x = c2r + cx. Hence as t increases from 0 to 1, this "shear flow" transforms the cube into a parallelepiped of volume 1.
11. div (w Xr)-0 because i\. v2, v3 do not depend on x, y, z, respectively.
13. (b) (JvOx + i.i'r,), + (fv3\ = f[(Vl)x + (v2)v + (t'3y + fxV! + fyv2 + fzv3, etc. (c) Use (b) with v = V^.
15. 4(x + y)/(y - x)3 17. 0 19. exyz(y2z2 + x2z2 + x2y2)
Problem Set 9.9, page 416
1. [0,   0,   4x - 1] 3. [0,   0,   2ex sin y] 5. [0,   0,   -4y/(x2 + y2)]
9. curl v = [ — 2z,   0,   0], incompressible, v = r' = fx',   y',   z'] — [0,   z2, 0],
' 2 _      2 2.i
x = c1( z = c3, y  = z  - c3 , y = c3 t + c2 11. curl v -   [0,   0,   —2], incompressible, x = y, y' = —x, z = 0, z — ca,
y dy + x dx = 0, x2 + y2 = c 13. Irrotational, div v = 1, compressible, r = [c^,   c2e~f, c3ef] 17. 0, 0, [xy - zx,   yz - xy,   zx - yz.\ 19. 0, 0, 0, ~2yz2 - 2z.r2 - 2xy2
Chapter 9 Review Questions and Problems, page 416
11.1-1,   9,   24] 13.0,1-43,   54,   3], [43,   -54, -3]
15. [0,   0,   -740], [0,   0,   -740] 17. [-24,   3,   -398], [114,   95, -76]
19. -495, -495 21. 90°, 95.4°
23. If u X v = 0. Always 25. [vx, v2, -3]
27. 3.4 29. If y > \-n, \n
A27
33. 45/6 35. No
37. 0, 2y2 + (z + xf 39. [-1,  1,  -1], [-2z,  ~2x, -2y]
41. 0, 2x2 + 4y2 + 2z2 + 4xz 43. 488/V3323 45. 0
Problem Set 10.1, page 425
1. F(r(f)) = [125f6,   I3,   0], 16448/7 = 2350       3. 0 + 160 5. F(r(/)) = [cosh t sinh2 f,   cosh2 t sinh f], 93.09 7. F(r(/)) = [?,   cos t,   sin r], 6tt 9. F(r(/)) = [cosh \t,   sinh \t,   em], 0.6857 11. F(r(f)) = [<?',   el\   e% e2 + 2e4 - 3
15. 17/3 17. [36tt,   |(8t7)3, 36tt]
Problem Set 10.2, page 432
1. sin xy, 1 3. -\e~^+^\ 0 5. exz + y, -2
7. x2y + cosh z, 392 11. sinh ac 13. No
15. cea - aeb 17. \a2bc2 19. No
Problem Set 10.3, page 438
3. I   \x — x3 — (x2 — x5)] dx = ^ 5. \ cosh 6 — cosh 3 + \
A ,.77/2
7. f |(e3x - e~x) dx = \eX2 + \e~A - § 9. f (esin y cos y - cos y) dy = e - 2 11.      (2x2 4- |) dx = § 13. Vl - x2 dy dx = \
-1 0 0
15. x = lb, y = \h 17. lx = bh3l\2, ly = b3h/4
19. /a = (a + /?)/i3/24, ly = /?(a4 - />4)/(48(a - b))
Problem Set 10.4, page 444
1. 2x3y - 2xy3, 81 - 36 = 45 3. 3x2 + 3y2, 1875 77/2 = 2945
5. e*-» - ex+y, -±e3 + \e2 + e_1 - §     7. 2x - 2y, -56/15
9. 0 (why?) 11. Integrand 4. Ans. 40tt
13. y from 0 to \x, x from 0 to 2. Ans. cosh 2 — \ sinh 2
15. y from 1 to 5 - x2. Ans. 56 19. 4e4 - 4
Problem Set 10.5, page 448 1. Straight lines, k
3. x2la2 + y2/b2 = 1, ellipses, straight lines, [—b cos v,   a sin u, 0]
5. z — (c/a)V x2 + y2, circles, straight lines, [—acu cos y,   —acw sin u, «2m]
7. x2/9 + _y2/16 = z, ellipses, parabolas, [ — 8m2 cos v,   —6u2 sin v, I2u\
9. x2/4 + y2/9 + z2/16 = 1, ellipses, [12 cos2 v cos u,   8 cos2 v sin u,   6 cos v sin i>]
13. [10m,   IOv,   1.6 - 4« + 2d], [40,   -20, 100]
15. [—2 + cos v cos u,   cos u sin u,   2 + sin u], [cos2 v cos «,   cos2 u sin u,   cos u sin v]
App. 2   Answers to Odd-Numbered Problems
17. [u,   v,   3o%[0,   -6v, 1]
19. [it cos V,   3h sin v,   3u], [ —9u cos v,   — 3u sin v, 3h]
21. Because r„ and r„ are tangent to the coordinate curves v = co/wť and w = const,
respectively. 23. [m,   V,   ií2 + v2], Ň = [~2%   -Iv, 1]
Problem Set 10.6, page 456
1.-64                           3.-18 5.-12877
7. 2t7                            9. |a3 11. 17/7/4 15. 140V6/3                  17. 12877V2/3 = 189.6
19. r^772(373/2 - 53/2) = 22.00 25. lirh 27. 77/z4/V/2                      29. 77/2 + 2irh3ß
Problem Set 10.7, page 463 1. 8a3/?3c3/27 3. 6
7. 23477
13. 77/!5/10
21. 0
9. 2a5/3 17. 108 77 23. 8
5. 42§77 11. /ia477/2 19. 21677 25. 38477
Problem Set 10.8, page 468
1. Integrals 4 ■ 1' 1 (x = 1), 4 • 1 • 1 (y = 1),   8 - 1 - 1 (z = 1), 0 {x = y = z = 0) 3. 2 (volume integral of 6y2), 2 (surface integral over x = 1). Others 0 5. Volume integral of 6y2 - 6x2 is 0. 2 (.v — 1), -2 (y = 1), others 0. 7. F = [jc,   y,   z], div F = 3, In (2), Sec. 10.7, F*n = |F||n| cos 4>
= Vx2 + y2 + z2 cos 4> = r cos 4>. 9. F = [x,   0,   0], div F = 1, use (2*), Sec. 10.7, etc.
Problem Set 10.9, page 473
1. [0,   8z,   16] «[0,   -1,   1], ±12
3. [~ez,   -ex,   ev]'[-\,   -1,   1], ±{e2 - 1)
5. S: [u,   v,   v2], (curl F)'N = -4ue2u2, ±(4 - 4e2)
7. (curl F)«n = 3/2, ±3ö2/2 9. The sides contribute a, 3a2/2, -a, 0.
11. curl F = [0,   0,   6], 2477 13. (curl F)«n = 2x - 2y, 1/3
15. -77/4 17. (curl F)*N = 77(cos tjx + sin Try), 2
19. Ft = L-sin B,   cos 0] • [-sin 6,   cos 0] = 1, 2tt, 0
Chapter 10 Review Questions and Problems, page 473
11. Exact, -542/3 17. By Stokes, ±1877 23. 0, Aal377 29. By Gauss, IOO77 35. Direct, 5(e2 - 1)
13. Not exact, <?4 - 7 19. By Stokes, ±12t7 25. 8/7, 118/49 31. By Gauss, AOabc
15. By Green, 1152 77 21. 4/5, 8/15 27. Direct, 5 33. Direct, 77/z
App. 2   Answers to Odd-Numbered Problems
A29
Problem Set 11.1, page 485
3. 277/«, 2tt/m, k, k, kln, kln
12/1 1 \
13. — + —   cos x — — cos 3a- + — cos 5a —h • • • I 2      tt \ 3 5 /
tt 4    / 1 1
15. — 4- —   cos x + — cos 3a + — cos 5a + 2      tt \ 9 25
TT      2  / l 1
17. — — —   cos x + — cos 3x + — cos 5x + 4      tt \ 9 25
1 1
+ sin a — — sin 2a + — sin 3x — + ■ • •
2 3
4 / 1 1
19. — —   cos a + — cos 3x + — cos 5a tt \ 9 25
/ 1 1
+ 21 sin x + — sin 3a + — sin 5a + - ■ \ 3 5
„   1    9      / 1 1
21. — tt  - 41 cos a - — cos 2a + — cos 3a--h • • •
1    2     4 1 4 1
23. — tt — — cos x — — cos 2x + -cos 3x + — cos 4x — • • •
6 tt 2 27tt 8
29. /' = 2a, /" = 2J1 = 0,j[ = -4ttJ" = 0,an= — |- — ) (-4tt) cos «tt, etc.
«tt \ n/
Problem Set 11.2, page 490
,   4 /     Tjx      1       3tta      1 5tta
1. — I sin — + — sin - + — sin -
tt \      2 3 2        5 2
14/ I 1
3. —--—~   cos tta — — cos 2tta + — cos 3tta — +
3      ttz \ 4 9
TT        TT   y 1
7^ cos 27ta + —j cos 4ttx + -j-^ cos 6ttx + • ■ •
14/ 1 1
7. Rectifier, —--j- (cos irx + — cos 3ttx -I--cos 5ttx + •
2      7t   \ 9 25
111 \
COS ttx--- cos 2ttx + — cos 3ttx - — cos 4ttx -I--
4 9 16 /
ttx      1 1        3ttx       1        5ttx 1
COS — + - COS ttx + - cos -y- + — cos — + — cos 3ttx
9.	2	4
	--	H--ö
	3	tt2
11.	3	4
	—	---
	4	tt2
	+	")
	3	1
13.	—	+ - c
	8	2
8
15. Translate by \. 17. Set x = 0.
A30
App. 2   Answers to Odd-Numbered Problems
Problem Set 11.3, page 496
1. Even, odd, neither, even, neither, odd 3. Odd 5. Neither 7. Odd 9. Odd
77 4   / 1 1
11. — + —   cos x + — cos 3x + — cos 5x + 2      77 \ 9 25
4  / 1 1
13. — I sin x — — sin 3x + — sin 5x 77 \ 9 25
4  /     77jc      1       377X      1 5ttx
IS. 1 — — I sin -       + — sin - + — sin - + •
77 \      2       3        2       5 2
4   /        77X 1 377X 1 577X
17. (a) 1, (b) -   sin — + - sin — + - sin — + 77 \      2       3 2        5 2
77X 1 377X 1 577X
19. (a) 1 + —5- I cos — + — cos-        + — cos- +
772 \      2       9         2       25 2
4 /      77X       1 1        3ttx 1
(b) —   sin- + — sin ttx + — sin- + — sin 277X
77 \      2       2 3        2 4
3 2    / 77x 1 377X 1 577X 1 777x
21. (a) — — — I cos — — — cos- + — cos- — — cos - H--•
2t7\2       3 2       5 2       7 2
6   /        77X 1 1 3t7X 1 577X 1
(b) —   sin-       — — sin 77x + — sin-        + — sin-        — — sin 377X +
77\23 3        2       5        2 9
L 4L   ( ttx 1 377x 1 577X \
23. (a) — — —t; I cos — + — cos- + — cos- + • ■ • I
W2t72\L9 L        25 L )
2L   (        ttx 1 277x 1 377x
(b) —   sin — — — sin-        + — sin- —
77 \     L      2        L       3 L
tt       4   t 1 1
25. (a) — + — I cos x + — cos 3x + — cos 5x + K    2      tt \ 9 25
/ 1 1
(b) 21 sin x + — sin 2x + — sin 3x + ■ \ 2 3
Problem Set 11.4, page 499 3. Use (5).
9.i 2
(-Dn
n= — oo
13. 77/2   — er
n— -co m#0
2i
y gC2n4 l)ix
tt 2n + 1
n— — oo
11.
2 2
n= — sc n¥=0
(-if
App. 2   Answers to Odd-Numbered Problems
A31
Problem Set 11.5, page 501
3. (Q,05«)2 in Dn changes to (0.02ra)2, which gives C5 = 0.5100, leaving the other
coefficients almost unaffected. 5. y = cx cos cot + c2 sin cot + A(co) cos t, A(co) = l/(co2 - 1) < 0 if to1 < 1 (phase
shift!) and > 0 if co2 > 1
N
7. v = cY cos cot + c2 sin <w? + 2 —?—~—5" cos «f
, or — «
77        4  /     1 1/9
9. y = Ci cos cot + c2 sin     + —^ + —   —5-cos I + —5-cos 3?
\ 2w2      77 \ «2 - 1 co2 - 9
1 1
11. v = Ci cos cot + fo sin cot + —5- — -5- cos 21
2 2w2 1 ■ 3(co2 - 4)
1
cos At — ■ • •
3-5(w2 - 16) 13. The situation is the same as in Fig. 53 in Sec. 2.8.
3c
64 + 9c2 64 + 9c2
15. y =----5- cos 3f - -=■ sin 3?
17. y = 2 (- ^ cos + 0     "2)K sin «*), D„ = (1 - „2)2 + „2c2
^ ui 240(10 - «2)
19. /(f) = y (A- cos nf + ß„ sin nf),   A_ = (-If 11--- ,
»-i » ö« 4.1 2400
Bn = (-l)K+1 ——, Dv = (10 - «2)2 + lOOra2 nu„
Problem Set 11.6, page 505
/        i (-iyv+1 \
h F = 21 sin x - — sin 2x + • ■ • +--- sin Nx 1, £* = 8.1, 5.0, 3.6, 2.8, 2.3
77 4    / 1 1 \
3« F = — - —   cos x + — cos 3x + — cos 5x + •••,£* = 0.0748, 0.0748, 2      77 \ 9 25 /
0.0119, 0.0119, 0.0037
2      4/1 1 1
5. F = —---—- cos 2x +--cos Ax H--cos 6x + ■
77      77 \ 1 • 3 3-5 5-7
E* = 0.5951, 0.0292, 0.0292, 0.0066, 0.0066
4/1 1 \
7. F = — I sin x + — sin 3x + — sin 5jc + • • • I, E* = 1.1902, 1.1902, 0.6243,
0.6243, 0.4206   (0.1272 when N = 20) „   8 / 1 1 \
9. —   sin x + — sin 3x + —- sin 5x + •••.£* = 0.0295, 0.0295, 0.0015, 77 \ 27 125 /
0.0015, 0.00023
A32 App. 2   Answers to Odd-Numbered Problems
Problem Set 11.7, page 512
1
(see Example 3), etc. "°
r i
1. f(x) = ue x (x > 0) gives A = I   e K cos wf dv = -j , B
3. ,f(x) = We_x gives A = 1/(1 + w2).
2 '
2\
5. Use / = {it 12) cos v and (11) in App. 3.1 to get A = (cos (7w/2))/(l - w2)
r03 sin aw cos xw 2 r -aw                  9. —
co r-\ co i • 1
2   r   sin aw cos xw 2 r   cos w + w sin w — 1
7. — -dw 9. — -=-cos xw dw
tt  ■>()                 w tt j0 w
ttw
—2 sin xw dw
tt jq    1 — w
2   r   cos 7TW +1 2   r sin
11. — -o— cos     <^vv 15- — 1—
77 Jn         1 — w tt  j0 1 -
2   r   77w — sin 7tw 2   r wa
17. — -s-        sin xw dw 19. — —
77  j0                w tt j0
Problem Set 11.8, page 517
\   77   \ w
7. V77/2 cos iv if 0 < w < 77/2 and 0 if w > tt/2 9. Yes, no
/ 2   / sin 2w — 2 sin w \ ,-
1.  / — I- 5. Vtt/2 e~x   (x > 0)
11. V(2/t7) w/(w2 + tt2)
13. W's(xe-x2'2) = 9^(-(e_*,/2)') = iv^c(<r*%) = we""-'2'2
_    ,      _ ^ [2       a2 [2
/ 2 w V 77 a2 +
\ 77  a"5 + \ 77
19. In (5) for f(ax) set ax = v.
Problem Set 11.9, page 528
3. lk(e-'dyw - 1)/(V2t7w) 5. \ '(2,tt)k (sin w)/w
7. [(1 + iw)e-'iw - 1]/(V2t7w2) 9. V(2/tt)i(cos w - l)/w
11. lc-«;2/2
13. (e** - e-*")/(iwV277) = V2/77(sin Mv)/w
Chapter 11 Review Questions and Problems, page 532
4k I 1 1
11. —   sin 77x + — sin 3t7x + — sin 57tx + • ■
77 \ 3 5
/ .   x      1 1   .   3x      1 1 5.v
13. 4 sin — — — sin x + — sin — — — sin 4x + — sin —
\     2      2 324 52
8     / 77x 1 377X 1 577X
15. —, I sin — — — sin - + — sin - — + ■
772 \      2       9 2        25 2
2      4/1 1 1
17. - — - - cos 1677x + - cos 3277x + - cos 4877X
77      77 \ 1 -3 3-5 5-7
App. 2   Answers to Odd-Numbered Problems
A33
tt2 111
19. — — cos 2x + — cos 4x — — cos 6x + — cos 8.x — + 12 4 9 16
21. 77/4 by Prob. 11
23. 7t2/8 by Prob. 15
25. - [f(x) + /(-*)], - [fix) - /(-*)]
27. 77
je      1 3jc cos — -I- — cos — 2      9 2
I 5x
— cos — 25 2
29. 8.105, 4.963, 3.567, 2.781, 2.279, 1.929, 1.673, 1.477
7t2 /   cos /
31. y = Ci cos (at + C2 sin cot + ——» — 4
3ü>2
1      cos 4t
16    of - 16
of - 1      4 o)z
cos 2/       1    cos 3/ - +
9 oß-9
33.
1   r00 (cos vv + w sin vv — I) cos vvx + (sin vi' — w cos vv) sin wx
1
77 Jn 2
sin w cos vf
„2
2 r w 35. — -
77  J0 W
4   r°° sin 2w — 2vv cos 2vi
37. i/
sin wx dw
cos k>jc dw
tt ■>„
39. . /
\   77      VT + 4
Problem Set 12.1, page 537
1. u = Cxix) cos 4y + cz(x) sin 4y 3. u = c\ix) + c2{x)y
5. u = <ia l<—' + exyl(x +1) 7. u = cix) exp (iy2 cosh x)
9. m = cxix)y + c2(.v)y-2 11. m = c{x)ey + /i(y)
15. c = 1/4 17. Any c
19. 77/4 21. Any c and w 27. w = 110 - (110/ln 100) In (x2 + y2)   29. u = cxx + c2(y)
Problem Set 12.3, page 546
1. A' cos 277? sin 277x
8* /         .           1             . 1 \
3. —3 I cos 77? sin 77x + —- cos 3 77? sin 3 ttx + ——- cos 577? sin 577.1 + ■ • • I
4    t                       1 1
5. -—x   cos 77/ sin ttx — — cos 3tt? sin 2>ttx + — cos 5tt? sin 5ttx--h • • •
7. —2 I (V2 — 1) cos 77? sin ttx + — cos 2tt? sin 2ttx
+ — (V2 + 1) cos 3tt? sin 3t7x - ■ • •
A34
App. 2   Answers to Odd-Numbered Problems
(2 — V2) cos 77? sin ttx — —- (2 + V2) cos 3irt sin 3 ttx
— (2 + v2) cos 577? sin 5ttx 25
17. m
8L2
tt
COS
	/     \2 ~1
	
c	— *
	U/ _
ttx I
I     + —g COS
377
377X
sin - +
L
L 3d
19. (a) w(0, t) = 0, (b) u{L, t) = 0, (c) 1^(0, r) = 0, (d) ux(L, t) = 0. C = -A, D = —B from (a), (b). Insert this. The coefficient determinant resulting from (c) (d) must be zero to have a nontrivial solution. This gives (22).
Problem Set 12.4, page 552
3. c2 = 300/[0.9/(2 • 9.80)] = 80.832 [m2/sec2] 11. Hyperbolic, u = /i(x) + f2(x + y) 13. Elliptic, u = J\(y + 3ix) + f2(y — 3ix) 15. Parabolic, u = x/j(x — y) + f2(x — y) 17. Parabolic, « = xf^lx + y) + f2(2x + y) 19. Hyperbolic, u = (l/y)Mxy) + My)
Problem Set 12.5, page 560
5. u = Sin 0.477X g-l-752-16^/100
7. ii = - (- sin 0.1 TTx e-o-ovna^t + I sin 02wc e-o.oiraaa.*
9. u
tt   \ tt
20V2
sin 0.1 TTx e-0-01752^ + - sin 0.3ttx e-o.mvat»^t _ 9
11. u = mt + mh, where wn = u — mt satisfies the boundary conditions of the text, so
^             mrx       .      „             2   rL                           ft TTX that wn = 2j Rn sin —— e     ""^> l^Bn= —      [f(x) — ux(x)] sin -dx
n=i L L   0 L
13. F = A cos px + B sin px, F'(0) = Bp = 0, B = 0, F'(L) = -Ap sin /?L = 0,
p = rnrlL, etc.
15. m = 1
2tt2
17. u = —— + 41 cos x e
19. w
tt
~12
1
cos 2x e 4t + — cos 3x <?"
1
+
+ cos 2x e     H--cos 4x e      h--cos 6;
23. — - 2
n—1
2„     - n ..." - —\T„-*x/,.2
27. ut - cAvxx = 0, w" = -Ne-axlc\ w =
so that w(0) = w(L) = 0. 29. u = (sin jttx sinh g7ry)/sinh 77 80 ^ 1
31. //
25. w = «T*
2a2
(1 - e~aL)x + 1
7t     , (2/7 — 1) sinh (2m — lW
71=]   v '
(2n — 1)ttx (2m — \)iry sin -—- sinh
24
24
App. 2   Answers to Odd-Numbered Problems
A35
^       sinh (mtoc/24) niry
33. u = A0x + X An -;- cos —— ,
sinn «7T 24
n—1
1    r24 1   r24 nrr>-
Ao = 24« l  f(y) dy'   An=nl  f(y) C0S "2T *
H77X «77(Ä — v) 2 r niTX
35.      An sin - sinh -— , An = —;- f(x) sin -dx
a a a sinh (mrb/a) J0 a
Problem Set 12.6, page 568
2 sin ap 2   rx sin ap
l sin ap ic   sin ap 2 ,
1. A = -, ß = 0, w = — - cos    e   p dp
Tip TT J0 p
3. A = <rp, B = 0, H =     cos pjc e-P-c2P2t afp
5. Set Trf = s\ A = 1 if 0 < p/n < 1, ß = 0, u = cos px <Tc2p2' a>
•'o
7. A = 2[cos /? + p sin p - l)/(77p2)|, ß = 0, « = A cos px e~cVt dp
-Vi
Problem Set 12.8, page 578
1. (a), (b) It is multiplied by V2. (c) Half 3. Bmn = 16/(mrc772) if m, n odd, 0 otherwise 5. Bmn = (-l)n 1 Hl(mmr2) if m odd, 0 if m even 7. fi„m = (-l)m+n4/(mmr2) 11. A cos V29 f sin 2x sin 5y
00 oo
13. —X X —cos (rV m2 + n2) sin mx sin «v
m, n odd
17. C77V260 (corresponding cigenfunctions F418 and F1614), etc.
19. Bmn = 0 (m or n even), fimn = \6kl{rnmr2) (rn, n odd)
21. Bmra = (-l)m+n144fl3fo3/(m3rt3-rr6)
/      [~9~    16 \       3tta 4ttv
23. cos I 7TT    — + — I sin - sin
\    V a
Problem Set 12.9, page 585
7. 3(V cos 0 + 1(V3 cos 30
220 /              1   „ 1 _
9. 55 + - \ r cos 0--r3 cos 30 + - r5 cos 50 - + ■ ■
7T    \ 3 5
^ 4      / 1 1 e
11. —--r cos 0 + - r3 cos 30 + — r5 cos 50 + ■ ■ ■
2      tt \ 9 25
15. Solve the problem in the disk r < a subject to u0 (given) on the upper semicircle and — u0 on the lower semicircle.
4«o / r 1     „ 1     „ \
u = —-   — sin 0 + —- r3 sin 30 + —F r5 sin 50 + ■ ■ • tt   \a 3as 5a° )
17. Increase by a factor V2 19. T = 6A2bpR2f2
App. 2   Answers to Odd-Numbered Problems 21. No
23. Differentiation brings in a factor 1/Am = Rl(cam).
Problem Set 12.10, page 593
11. v = F(r)G(t), F" + k2F = 0, G + c2k2G = 0, Fn = sin {nrrrlR),
2   rR n Trr
G„ = B.„ exp (-cWW), B„ = -      /-/(/-) sin — dr
R   o «
13. u = 100 15. U = |f3P3(cos <t>) - frP^cos (h)
17. 64r4P4(cos 0)
21. Analog of Example 1 in the text with 55 replaced by 50 23. v = r(cos 6)lr2 = xl(x2 + y2), v = xy/(x2 + y2)2
Problem Set 12.11, page 596
c(s) x
5. W = —^r + -5:-- , W(0, s) = 0, c(s) = 0. vv(x. f) = x(t - 1 + O
a"       5 (s + i)
7. w = f(x)g{f), xf'g + fg = xt, take /(x) = x to get g = ce~'' + f - 1 and
c = 1 from w(x, 0) = x(c - 1) = 0. 9. Set x2/(4c2t) = z2. Use z as a new variable of integration. Use erf(o°) = 1.
Chapter 12 Review Questions and Problems, page 597
19.	u =	cÁy)e* + c2{y)e-2*	21. K =	g(x)(\ -		
23.	u —	cos t sin x — \ cos 2t sin 2x	25. u =	I cos / sin x		— j cos 3t sin 3x
27.	u	sin (0.0277x) e-n004!S72t				
29.	■'/ =	™   Ln JE ,-0.004572, _ tt2   v 50	1 3t7X sin    „ e 9 50	-0.04115t	+ •	
31.	II =	100 cos4xe"16t				
33.	u = + •	— — — 1 — cos 2x e -2       77 \4	1 1- — cos 6x e 36	-36f 1	1 Too	cos lOxe"100*
37.	u =	h(y) + hix + >') 39. u =	f\(y - 2/x) +	fz(y +	2/x)	
41.	u =	xfi(y - x) + f2(y - x)				
49. u = (% — M0)(ln r)/ln (r^lr^) + (u0 In rx — «j In r0)/ln (r^r^f)
Problem Set 13.1, page 606
5. x - iy = -(x + iy), x = 0 7. 484
9. -5/169 11. -7/13 -(22/13)/ 13. -273 + 136/
15. -7/17 - (11/17)/ 17. x/(x2 + y2) 19. (x2 - >'2)/(x2 + y2)2
Problem Set 13.2, page 611
1. 3V2(cos (-í77) + / sin (-fir))
3. 5 (cos 77 + / sin 77) = 5 cos 77 5. cos §77 + i sin 577
App. 2   Answers to Odd-Numbered Problems
A37
7.1 Vol (cos arctan | + i sin arctan |) 9. -3tt/4
11. arctan (±3/4) 13. ±77/4 15. 3 77/4
17. 2.94020 + 0.59601/      19. 0.54030 - 0.84147/ 21. cos (—377) + / sin (— jj.77), cos §77 + i sin |77
23. ±(1 ±/)/V2 25.-1, cos ^77 ± / sin 7A77, cos §77 ± i sin §77
27. 4 + 3/, 4 - 8/ 29. § - /, 2 + 4/
35. |zi + z2|2 = (74 + z2)(Z! + z2) = (Zi + z2)(Xi + z2). Multiply out and use Re Z]z2    \zxz2\ (Prob. 32):
ZiZi + Zi22 + z2Zi + z2z2 = hi2 + 2 Re ZiZ2 + |z2j2 S y2 + 2|Zl||z2| +
= (kil + |z2|)2-
Take the square root to get (6).
Problem Set 13.3, page 617 1. Circle of radius §, center 3 + 2i
3. Set obtained from an open disk of radius 1 by omitting its center z = 1 5. Hyperbola xy =1 7. y-axis
9. The region above y = x
13. / = 1 - l/U + 1) = 1 - (x + 1 - iy)/[(x + l)2 + y2]; 0.9 - 0.1/ 15. (,v2 - y2 - 2/xy)/(A'2 + y2)2, -j/2      17. Yes since r2(sin 20)/r^ 0 19. Yes 21. 6z2(z3 + /)
23. 2/(1 - z)-3
Problem Set 13.4, page 623
1. Yes 3. No 5. Yes
7. No 9. Yes for z * 0
11. rx = xlr = cos 6, ry = sin ft ft; = -(sin 6>)/r, ft = (cos 0)/r,
(a) 0 = Mj. — (Jy = «r cos 0 + mh( —sin ft/r — vr sin // — t;fl(cos &)/r.
(b) 0 = ma + u,; = ur sin 0 + mw(cos 6)lr + vr cos ^ + v8( — sin f/)/r. Multiply (a) by cos ft (b) by sin ft and add. Etc.
13. z2/2 15. In \z\ + i Arg z 17. z3
19. No 21. No 23. c = 1, cos a- sinh v
27. Use (4), (5), and (1).
Problem Set 13.5, page 626
3. -1.13120 + 2.47173/, e = 2.71828      5. -/, 1
7. ea8(cos 5 - i sin 5), 2.22554 9. e~2x cos 2y, -e~2x sin 2y
11. exp (a2 - y2) cos 2xy, exp (x2 - y2) sin 2yv
13. e177'4, e57rili
15. Vr exp [i(0 + 2kir)/n\,   k = 0, • • % n - 1
17. 9em 19. z = hi 2 + m + 2nm (n = 0, ± 1, • ■ ■)
21. z = In 5 — arctan |/ ± 2»77/ (n = 0, 1, • ■ ■)
Problem Set 13.6, page 629
3. Use (11), then (5) for ely, and simplify.   5. Use (11) and simplify. 7. cos 1 cosh 1 - i sin 1 sinh 1 = 0.83373 - 0.98890/
AJ8
App. 2   Answers to Odd-Numbered Problems
9. 74.203, 74.210 11. -3.7245 - 0.51182/
13.-1 15. cosh 4 = 27.308
17. z = ±(2n + l)7rt'/2 19. z = |(2n + 1)tt - (-lfl.4436/
21. z — ±nm
25. Insert the definitions on the left, multiply out, simplify.
Problem Set 13.7, page 633
1. In 10 + Tri 3.1 In 8 - \ifi
5. In 5 + (arctan f - ff)i = 1.609 - 2.214/
7. 0.9273/ 9. \ In 2 - \m
11. ±(2n + 1)77/,   n = 0, 1, ■ ■ • 13. In 6 ± {In + Y)m,   n = 0, 1, • • ■
15. (tt - 1 ± 2mr)i,   n = 0, 1, • • •
17. In (/2) = (±2n + 1)tt/, 2 In / = ±(4n + l)7ri,   n = 0, 1, • ■ • 19. e03(cos 0.7 + / sin 0.7) = 1.032 + 0.870/
21. e2{\ + /)/V2 23. 64(cos (In 4) + / sin (In 4))
25. 2.8079 + 1.3179/ 27. (1 + /)/V2
Chapter 13 Review Questions and Problems, page 634
17. -32 - 24/ 19. -| - y 21. 5 - 3/
23. 6V2e37r'/4 25. 12«"* 27. ±(2 + 2/)
29. (±1 ± /)/V2 31. f(z) = \lz 33. f(z) = (1 + i)z2
35. f(£) = e*1 37. (-X2 + y2)l2 39. No
41.0 43.0.6435/ 45.-1.5431
Problem Set 14.1, page 645 1. Straight segment from 1 + 3/ to 4 + 12/
3. Circle of radius 3, center 4 4-/ 5. Semicircle, radius 1, center 0
7. Ellipse, half-axes 6 and 5 9. Parabola y = ^x3 from -1 - |i to 2 + 4/ 11. e-if (0g(§ 2tt)
13. t + i/t (1SfS4) 15. f + (4 - 4f2)/ (-IS(SI)
17. -a - ib + re~u (0    f ^ 2tt) 19. | + 1/
21. 0 23. 77/ + \i sinh 277
25. ill 27. -1 + / tanh |t7 = -1 + 0.6558/
29. 2 sinh \
Problem Set 14.2, page 653
1. 77/, no 3. 0, yes 5. 0, yes
7. 0, yes 9. 0, no 11. 0, yes
15. Yes, by the deformation principle 19. 77/ by path deformation
21. 77/ 23. 277/ 25. 0
27. (a) 0, (b) 77 29. 0
App. 2   Answers to Odd-Numbered Problems
Problem Set 14.3, page 657
1. -47T 3. 4tt 5. 8777
7.0 9.-77/ 11. m
13.77 15. 2Tri Ln 4 = 8.710/
17. 77/cosh2(l + /) = 77(-0.2828 + 1.6489/)
Problem Set 14.4, page 661
1.277ri/4 3. -2mevl2 5. 77/a3/3
7. 277/ if \a\ < 2, 0 if \a\ > 2 9. tt/(cos \ - sin |)
11. 2tt2/ 13. 77/<?a/2/24 if la - 2 - i| < 3, 0 if la - 2 - i\ > 3
Chapter 14 Review Questions and Problems, page 662
17.-677/ 19.0 21.-77/
23. \i sin 8 25. 0 27. 77       29. 0
Problem Set 15.1, page 672
1. Bounded, divergent, ±1 3. Bounded, convergent, 0
5. Unbounded
7. Bounded, divergent, ±1/V2 ± i, 0, 1, -2 9. Convergent, 0
13. \zn ~ l\< |e, \z*n ~ l*\ < \e {n > Me)), hence \zn + z*n - (I + <\e + \e 17. Convergent 19. Divergent
21. Conditionally convergent 23. Divergent by Theorem 3
27. n = 1100 + 75f| = 125 (why?); 1100 + 75/|125/125! = 125125/LV250t7 (125/e)125] = e,25/V'25077 = 6.91 ■ 1052
Problem Set 15.2, page 677
1. 2 anz2'" = 2 an(z2)n, \z2\ < r = Um \ajan+1\, hence \z\ <Vr. 3. -/, 1 5. -1, e by (6) and (1 + \ln)n -4 e.
7. 0, \bla\ 9. 0, I 11. 0, 1
13. 3 - 2i, 1 15. /, l/V'2 17. 0, V2
Problem Set 15.3, page 682
1.3 3. V2 5. V5/3
7. 1/V7 9. 1
Problem Set 15.4, page 690
1. 1 - 2z + 2z2 - |z3 + |z4 - + • • •,   R = 00
3. e~2i([ + (z + 2i) + \{z + 2/)2 + \{z + 2/)3 + ^(z + 2/)4 + • • •,   R = co
5. 1 - l(z - |t7)2 + Uz ~ W - jkö(z - |77)6 +----,   R = oc
7.1 + |i + |i(z - i) + (-J + \i){z - if2 - \{z -if +----,   R = V2
App. 2   Answers to Odd-Numbered Problems
11. 4(z - 1) + 10(7 - l)2 + I6(z - l)3 + 14(-. - l)4 + 6(7 - l)5 + (z - l)e 13. (2/Vtt)(z - zs/3 + z5(2!5) - z7/(3!7) + •••)>   R = °° 15. z3/(l !3) - z7/(3!7) + z1l/(5!ll) - + •••,   R = so
I n  -  i   1 -3   i    2 .5  i   17 _7  i n _ 1 _
19. Z + 3z   + igZ   + gigZ   + R ~ zTT
Problem Set 15.5, page 697
1. Use Theorem 1. 5. \z"\ = 1 and X 1/n2 converges. 9. |z + 1 - 2i\ ^ r < /? = 4 13. Nowhere
3. R = l/\ 7t> 0.56
7. jtanh" |z|| g 1, l/(«2 + 1) < 1/n 11. \z\ ^ 2 - S (8 > 0) 15. |z| g V5 - 8 (S > 0)
Chapter 15 Review Questions and Problems, page 698
11. », c3j 13. I, 1 Ln[(l + z)/(l - z)]
15. 17. IA/it, |1 - tt(z - 2/)2r1
19. 1/3
21.-1 " (z - 77() - (z - 77/)2/2! - ■ ■ -,    /? = OO
23.1 + |(z + 1) + l(z + l)2 + ie(z + I)3 + • • •,   R = 2
25. 1 + 3z + 6z2 + 10z3 + ••%/?= 1. Differentiate the geometric series
27. i + (z + 0 - i(z + 02 - (z + 03 + • ■ %  R = 1
29. -(z - lir) + jj- (z - §tt)3 - ^ (z - ^tt)5 +----, /? = x
Problem Set 16.1, page 707
1111
1. +  —  +  -2+  —  +   1+z +
R = 1
11111 1
3'?~z¥+27~i+24z_t2ö
z2 +
R = 00
1 1
7.
1 1
-7    + -Q 4
2z7 6z9 °   (z - l)""1
l
— e
= e
n=0 n+1
R = 2
R = oo 1
1
+ 1
2!
+
t/2       1 /
- + - + - (z - 0
: - i     4 8
16
R  = OG
(z - o2 -
li. - 2 (z + o"'1 = -
z + i
(z + 0----,Ä = 1
13.
+ 2 + (z - 1)
App. 2   Answers to Odd-Numbered Problems
A4!
n=0 n=0 "
go co |
17.2 z4n+2.m< i. -2753. h>i
TC-0 n-=0
z 1
19.---2 + -r + i + (z- i)
(z - 0      z - 1
21. (i - 4z) 2 z*\ kl < 1. (4- - -t) 2 4r. W > 1
n=0 ' v' 4    ' n-()
23. 2, — , |z + 1^1 > 0
n=0 \-">-
Problem Set 16.2, page 711
1. ±5, ±|, ■ • • (poles of 2nd order), 00 (essential singularity) 3. 0, ±Vtt, ±V2~77, • • ■ (simple poles), 00 (essential singularity) 5. °c (essential singularity)
7. ±1, ±1 (fourth-order poles), 30 (essential singularity) 9. ±i (essential singularities) 13. — 16z (fourth order)
15. ±1, ±2, • • • (third order) 17. + //V3 (simple)
19. ±2i (simple), 0, ±27rt, ±4m, • • ■ (second order) 21. 0, ±2tt, ±4tt, • • • (fourth order)
23. f(z) = (z- ZoTgd), g(zo) * 0, hence f(z) = (z - Zo)2ng2(z).
Problem Set 16.3, page 717 I. i, 4i
5. 1/5! (at z = 0)
9. -{(atz = 1), 3 (at z = -1) 15. eUz = 1 + 1/z + • • •, Ans. 2m 19. —477/ sinh \tj 23. 0
3. -\i (at z = 2z), \i (at -2;)
7. 1 (at ±«77) 11. -1 (at z = ±§77, ±f 77, • ■ •) 17. Simple poles at ±\. Ans. —4i 21. —4i sinh 5
25.1 (at z = 1), 2 (at z = §). Ans. 5m
Problem Set 16.4, page 725
1. 2tt/V13                      3. 27t-/35 5. 2tt/3
7. 0                                               9. 77 11. 277/3
13. 77/16                         15. 0 17. 77/2
19. 0                             21. 0 23. 77
25. 0 27. - 77/2
Chapter 16 Review Questions and Problems, page 726
17. 2777.73 19. 577 21. \tt cos 10
23. 77(74 25. 0 (n even), (-l)cn_1)/2 2m/(n - 1)! (n odd)
27. 677; 29. 2tt/7 31. 4t7/V3
33. 0 35. 77/2
A42
App. 2   Answers to Odd-Numbered Problems
Problem Set 17.1, page 733 3. Only the size
S. x = c, w = —y + ic, y = k, w = —k + ix
7. -3tt/4 < Arg w < 3tt/4, \w\ < 1/8 9. \w\ > 3
11. \w\ S 16, v g 0 13. Annulus 3 < |vv| < 5
15. In 2 = « = In 3, tt/4 = v = it/2 17. ±1, ±i
19. 0, ±1, ±2, - ■ ■ 21. -a/2
23. a and 0, V7a 25. M = ex = 1 when x = 0, / = e2:l
27.    = l/|z| = 1 on the unit circle,./ = l/|z|2
Problem Set 17.2, page 737
—iw 5i 5'Z ~ -2w + 3 7' Z ~ Aw - 2
9. - = 0 11. z = \ + i ± Vi + i
13. z = ±i 15. w = 4/z, etc.
17. a - d = 0, fe/c = 1 by (5) 19. vv = azld (a ŕO,<íŕO)
Problem Set 17.3, page 741
5. Apply the inverse g of f on both sides of n = f(Zi) to get        = g(/(Zi)) = Zi-
7. vv = (z + 2i)/(z - 20 9. w = z - 4
11. vv = l/z 13. w = (3iz + l)/z
15. w = (z + l)/(-3z + 1) 17. w = (2z - i)/(-iz - 2)
19. w = (z4 - 0/(-íz4 + O
Problem Set 17.4, page 745
1. Annulus 1 g |vv| š e2
3. 1/Vď < |vv| < Ve, 3rr/4 < arg vv < 5tt/4
5. 1 < |vv| < e, v > 0
7. vv-plane without 0
9. «2/cosh2 1 + y2/sinh2 1 ž 1. u š 0
11. Elliptic annulus bounded by «2/cosh2 1 + t;2/sinh2 1 = 1 and
M2/cosh2 5 + y2/sinh2 5 = 1 13. ±(2n + 1)77/2, n = 0, 1, • ■ ■
15. 0 < Im t < 77 is the image of R under t = z2. Ans. el = e~z 17. 0, ±L ±2i, ■ ■ ■
19. «2/cosh2 1 + i>2/sinh2 1 ž 1, v < 0 21. u < 0
23. -1 š u š 1, ľ = 0 (c = 0), M2/cosh2 c + t/2/sinh2 r = 1 (c ^ 0) 25. In 2 š m š ln 3, tt/4 š « i 77/2
Problem Set 17.5, page 747
1. w moves once around the unit circle.      5. —5/3, 2 sheets 7. -ill, 3 sheets 9. 0, 2 sheets
App. 2   Answers to Odd-Numbered Problems
Chapter 17 Review Questions and Problems, page 747
11. u = \v2 ~ 1, \v2 - 1     13. \w\ = 20.25, |arg w\ < tt/2
15. The domain between u = | — v2 and u = \ — \v2
17. |w + i| =| 19. w — 1 21. [arg vv| < tt/4
23. 0, (±1 ±_i)/V2 25. tt/8 ± nirll, n = 0,1, ■ • •
27. 0, ±(7V'2 29. w = iz 31. w = 1/z
33. vv = z/(z + 2) 35. ±V2~ 37. 2 ± V6
39. 1 + j ± VI + 2i 41. vv = e3z 43. z2/2/c
45. iz3 + 1
Problem Set 18.1, page 753
1. 20.v + 200. 20z + 200    3. 110 - 50xy, 110 + 25/'z2 5. F = (110/ln 2)Ln z        7. F = 200 - (100/ln 2) Ln z
13. Use Fig. 388 in Sec. 17.4 with the z- and vv-planes interchanged, and cos z = sin (z + \tt).
15. * = 220 - 11 Oxy
Problem Set 18.2, page 757
1. u2 - v2 = e2x(cos2 y - sin2 y), 4>.CT = 4e2x(cos2 y - sin2 y) = -<I>y?y, V2* = 0 3. Straightforward calculation, involving the chain rule and the Cauchy-Riemann equations
5. See Fig. 389 in Sec. 17.4. <J> = sin2 x cosh2 y - cos2 x sinh2 y. 9. (i) *=[/!(] - xy). (ii) w = /'z2 maps R onto -2 g u g 0, thus = t/iCl + |«) = f/x(l + i(-2xy)). 11. By Theorem 1 in Sec. 17.2
13. <!> = 10| 1 - (1/tt) Arg (z - 4)], F = 10[1 + (Hit) Ln (z - 4)] 15. Corresponding rays in the w-plane make equal angles, and the mapping is conformal.
Problem Set 18.3, page 760
3. (I00kl)y. Rotate through tt/2. 5. 100 - 2400/tt
7. Re F(z) = 100 + (200/tt) Re (arcsin z) 9. (240/77) Arg z 11. T0 + (2/7r)(F1 - T0) Arg z 13. 50 + (400/tt-) Arg z
Problem Set 18.4, page 766
1. V = iV2 = iK, W = -Kx = const, $ = Ky = const
3. F(z) = A"z (A' positive real)
5. V = (1 + 2i)K, F = (1 - 20#z
7. F(z) = z3
9. Hyperbolas (.v + 1 )y = const. Flow around a corner formed by x = — 1 and the x-axis.
11. y/(x2 + y2) = c or a'2 + (>• - k)2 = k2 13. F(z) = z/r0 + r0/z
App. 2   Answers to Odd-Numbered Problems
15. Use that w = arccos z is w = cos z with the roles of the z- and vt'-planes interchanged.
Problem Set 18.5, page 771 5. 1 - r2 cos 20
7. 2(r sin 6 - \r2 sin 26 + |r3 sin 30 - + ■ ■ ■) 9. |r2 sin 20 - \r6 sin 60 11. ^tt2 - 4(r cos 0 - Jr2 cos 20 + £r3 cos 30 - + • • ■)
4 / 1   , I    = \
13.       r sin 0--r3 sin 30 + — r5 sin 50 - + ■ • •
77 \ 9 25 /
Problem Set 18.6, page 774
1. No; \z\2 is not analytic.     3. Use (2). F(|) = f 5. 0>(4, -4) = -12
7. Use (3). 4>(1, 1) = -2   11. \F(ei&)\2 = 2 - 2 cos 20, 0 = 77/2, Max = 2
13. |F(z)| = [cos2 2x + sinh2 2y]1/2, z = ±(, Max = [1 + sinh2 2]1/2 = cosh 2 = 3.7622
15. No
Chapter 18 Review Questions and Problems, page 775
11. 4> = 10(1 - x + v), F = 10 - 10(1 + i)z
13. (20/ln 10) Ln z 15. (10/ln 10)(ln 100 - In r)
17. Arg z = const 19. {-Hit) Ln z
23. T(x, y) = x(2y + 1) = const 25. Circles (x - cf + y2 = c2
27. F(z) = ^- Ln (z - 5), Arg (2 - 5) = c
277
80 / 1   , 1 ,
29. 20 + — [r sin 0 + - r3 sin 30 + - rJ sin 50 + ■
77 \ 3 5
Problem Set 19.1, page 786
1. 0.9817- 102, -0.1010- 103, 0.5787- I0-2, -0.1360- 105 3.0.36443/(17.862 - 17.798) = 0.36443/0.064 = 5.6942,0.3644/(17.86 - 17.80) 0.3644/0.06 = 6.073, 0.364/(17.9 - 17.8) = 3.64, impossible
0.36443(17.862 +17.798)      0.36443-35.660 12.996
5. -—n-5—- = - = - = 5.7000,
17.8622 - 17.7982 319.05 - 316.77 2.28
13.00 13.0 13 10
5.702. —— = 5.70, — = 5.7, — = 5
2.28 2.28 2.3 2
7. 19.95, 0.049, 0.05013; 20, 0, 0.05
9. In the present calculation, (b) is more accurate than (a). 11. -0.126- 10-2, -0.402- 10"3; -0.267- HP6, -0.847- 10-7 13. Add first, then round.
. Oi = gi + e1 = ax + eY i  _ e2 + e22 _ +
a2      a2 + e2 a2      \      a2 )     a2      a2      a2 a2
App. 2   Answers to Odd-Numbered Problems
A45
		Ol			*2
\Ö2		«2			«2
hence
19. (a) 19/21 = 0.904761905, echop = erm
er,ehop      er,round = 0.2106" 10  5, etc.
= krll + k2| = &1 + ßr-Z
0.1905 • IO-f>,
Problem Set 19.2, page 796
1. g = 1.4 sin a", 1.37263 (= xs) 5. g = a4 + 0.2, 0.20165 (= x3)
7. 2.403 (= x5, exact to 3S) 9. 0.904557 (= x3)
11. 1.834243 (= .v4) 13. jc0 = 4.5, x4 = 4.73004 (6S exact)
15. (a) 0.5. 0.375, 0.377968, 0.377964; (b) 1/V7 = 0.377964 473
I7.xn+i = (2xn + 7/jen2)/3, 1.912931 (= x3)
19. (a) Algorithm Bisect (/, a0, b0, N) Bisection Method
This algorithm computes an interval [an, /?„] containing a solution of fix) = 0 (/ continuous) or it computes a solution cn, given an initial interval [a0, b0] such that /(ao)/(fc0) < 0. Here N is determined by (b - a)llN g B, )S the required accuracy.
INPUT: Initial interval \a0, b0], maximum number of iterations N. OUTPUT: Interval [aN, bN] containing a solution, or a solution cn. For n = 0, 1, ■ • •, N - I do: Compute c.„ = \(an + bn).
If f(cn) = 0 then OUTPUT cn. Stop. [Procedure completed] Else continue.
If f(an)f(cn) < 0 then an+1 = an and bn+1 = cn.
and b.
n + 1
Else set an+1 = c End
OUTPUT [aN, bN]. Stop. [Procedure completed] End BISECT
Note that [aN, bN] gives (aN + bN)/2 as an approximation of the zero and (bN as a corresponding error bound.
(b) 0.739085; (c) 1.30980, 0.429494 21.1.834243 23.0.904557
aN)l2
Problem Set 19.3, page 808
1. L0(x) = -2x + 19,   Lx(x) = 2a - 18,   Plix) = 0. 1082a + 1.2234, Pi(9.4) = 2.2405
3. 0.9971, 0.9943, 0.9915 (0.9916 4D), 0.9861 (0.9862 4D), 0.9835, 0.9809
5. p2(x) = -0.44304a2 + 1.30906a - 0.02322,   /?2(0.75) = 0.70929
7.p2(x) = -0.1434a2 + 1.0895a,  p2(0.5) = 0.5089,  p2(l.5) = 1.3116
9. L0 = -i(x - l)(.v - 2)(a - 3), Lx = \x(x - 2)(x - 3), L2 = -\x(x - 1)(a - 3),
L3 = \x(x - l)(x - 2);   p3(x) = 1 4- 0.039740a- - 0.335187x2 + 0.060645a3;
p3(0.5) = 0.943654 (6S-exact 0.938470),   p3(L5) = 0.510116 (0.511828),
p3(2.5) = -0.047993 (-0.048384)
A46
App. 2   Answers to Odd-Numbered Problems
Í3.p2(x) = 0.9461a - 0.2868x(x - l)/2 = -0.1434x2 + 1.0895a; 15. 0.722, 0.786
17. S/1/2 = 0.057 839, 8fm = 0.069 704, etc. Problem Set 19.4, page 815
9. [-1.39(x - 5)2 + Q.58(;c - 5)3]" = 0.004 at x = 5.8 (due to roundoff; should be 0). 11. 1 - fx2 + |x4
13. 4 - 12x2 - 8x3, 4 - 12x2 + 8x3. Yes
15. 1 - x2, -2(x - 1) - (x - l)2 + 2(x " l)3,
-1 + 2(x - 2) + 5(x - 2)2 - 6(x " 2)3 17. Curvature /7(1 + /'2)3/2 » f" if |/'| is small
19. Use that the third derivative of a cubic polynomial is constant, so that g" is
piecewise constant, hence constant throughout under the present assumption. Now integrate three times.
Problem Set 19.5, page 828
1. 0.747131 3. 0.69377 (5S-exact 0.69315)
5. 1.566 (4S-exact 1.557) 7. 0.894 (3S-exact 0.908)
9. Jh/2 + em = 1.55963 - 0.00221 = 1.55742 (6S-exact 1.55741)
11. Jm + em = 0.90491 + 0.00349 = 0.90840 (5S-exact 0.90842)
13. 0.94508, 0.94583 (5S-exact 0.94608)
15. 0.94614588, 0.94608693 (8S-exact 0.94608307)
17. 0.946083 (6S-exact)
19. 0.9774586 (7S-exact 0.9774377)
21.x - 2 = t, 1.098609 (7S-exact 1.098612)
23. x = Jif + 1), 0.7468241268 (10S-exact 0.7468241330)
25. (a) M2 = 2, M2* = \, \KM2\ = 2/(1 In2), n = 183, (b) /(iv) = 24/x5, 2m = 14 27. 0.08, 0.32, 0.176, 0.256 (exact)
29. 5(0.1040 - \ ■ 0.1760 + | • 0.1344 - \ ■ 0.0384) = 0.256
Chapter 19 Review Questions and Problems, page 830
17. 4.266, 4.38, 6.0, impossible 19. 49.980, 0.020; 49.980, 0.020008
21. 17.5565 g s ě 17.5675 23. The same as that of a.
25. -0.2, -0.20032, -0.200323
27. 3, 2.822785, 2.801665, 2.801386, 2.801386
29. 2.95647, 2.96087
31. 0.26, M2 = 6, M2* = 0, -0.02 šešO, 0.24 š a Š 0.26 33. 1.001005, -0.001476 šešO
Problem Set 20.1, page 839
1. .v, = —2.4, x2 = 5.3 5. Xi = 2, x2 = 1
3. No solution
l.x1 = 6.78, x2 = -11.3,% = 15.82
App. 2   Answers to Odd-Numbered Problems
9. xx - 0, x2 = tx arbitraiy, x3 = 5tx + 10 11. Xx — ti, x2 = ?2, both arbitrary, x3 = 1.25^ — 2.25?2 13. xx = 1.5, x2 = -3.5, x3 = 4.5, x4 = -2.5
Problem Set 20.2, page 844
1.
9.
11.
13.
I L a
0"
1_
0
1
6
0
1
9
0
3
1
0 5
8
0
1 i
9
xx = 4.2 x2 = 1.3
1 0		r4 -6^	xx = -4
_2 1_		.0 5_	x2 = 3
6
0
2
3
0 3
5
0 '2
0
0
L0
Xx = 0.5
, x2 = -2.5 x3 = 3.0 Xx = -1/30 ,x2= 2/15
x3 = 1/5
Xx = 3
, x2 = 6
x3 = 2 xx = 4
. x2 = 0
x3 = -2
x"! = 6
x„ = -2
3 x4
0 14
	"-2	4	-i1		341 6	-10
17.	-2	3	0	19.	15 2	4 3
	7	-12	2		-17	3
1
3 J
Problem Set 20.3, page 850
7. Exact 0.5, 0.5, 0.5
3. Exact 21.5, 0, -13.8       5. Exact 2, 1,4
9. (a) x(3)T = [0.49982   0.50001   0.50002], (b) x(3)T = [0.50333   0.49985   0.499681 11. 6, 15, 46, 96 steps; spectral radius 0.09, 0.35, 0.72, 0.85, approximately 13. [1.99934   1.00043   3.99684]T (Jacobi, step 5); [2.00004   0.998059 4.00072]1"
(Gauss-Seidel) _
17. V306 = 17.49, 12, 12   19. Vl8/c2 = 4.24|fc|, 4|fe|, 4|/c|
App. 2   Answers to Odd-Numbered Problems
Problem Set 20.4, page 858
1. 12, V62 = 7.87, 6, [|   -1 g] 5. 1.9, VT35 = 1.16, 1, |0.3 -0.1 7. 6, \ 6. 1, [1    1    1    1    1 1]
13. k = 100 • 100
17. 46 g 6 - 17 or 7- 17
21. [-0.6 2.8]T
3. 14, V5Ö = 7.07, 4, [-1    1   § -f] 0.5   1.0]
11. II A II ! = 17, (J A"1 II j = 17, K = 289
15. K — 1.2 ■ ^ = 1.469
19. [0   1]T, [1   -0.4]T, 289
23. 27, 748, 28375, 943656, 29070279
Problem Set 20.5, page 862
1. -11.4 + 5Ax 3. 8.95 - 0.388x
5. s = -675 + 90/, vm = 90 km/h 9. 4 - 0,75x - 0.125x2
11. 5.248 + 1.543a-, 3.900 + 0.532Ix + 2.021x2 13. -2.448 + 16.23a, -9.114 + 13.73x + 2.500x2, -2.270 + 1.466* -1.778X2 + 2.852xB
Problem Set 20.7, page 871
1. 5 s A s 9 3. 5, 0, 7; radii 4, 6, 6
5. |A - 4i\ g V2 + 0.1, |A| g 0.1, |A - 9i\ g V2 7. tu = 100, t22 = f33 = 1
9. They lie in the intervals with endpoints     ± (« — 1)10-6. (Why?) 11. 0 lies in no Gerschgorin disk, by (3) with >; hence det A = A2 • ■ • Xn + 0.
13. p(A) g Row sum norm || A\\x = max 2 lajfcl = f"1^* (lajjl + Gerschgorin radius) 15. VT53 = 12.37 17. Vl22 = 11.05 19. 6 g A g 10, 8 g A g 8
Problem Set 20.8, page 875
1. q = 4, 4.493, 4.4999; |e| g 1.5, 0.1849, 0.0206 3. g = 8, 8.1846, 8.2252; |e| g 1, 0.4769, 0.2200 5. q = 4, 4.786, 4.917; |e| g 1.63, 0.619, 0.399
l.q = 5.5, 5.5738, 5.6018; |e| § 0.5, 0.3115, 0.1899; eigenvalues (4S) 1.697, 3.382,
5.303, 5.618 9. y = Ax = Ax, yTx = AxTx, yTy = A2xTx,
e2 g yTy/xTx - (yTx/xTx)2 = A2 - A2 = 0 11. q = —2.8993 approximates —3 (0 of the given matrix),
1e| g 1-633, ■ • •, 0.7024 (Step 8)
Problem Set 20.9, page 882
3.500000   -1.802776 0 1.   -1.802776      6.730769 1.846154 .  0 1.846154 1.769230
0.980000 -0.441814 0
-0.441814 0.870164 0.371803
0"
0.371803 0.489836
App. 2   Answers to Odd-Numbered Problems
A49
5. Eigenvalues 8, 3, 1
5.64516 -2.50867
7.
-2.50867 0
7.91494 -0.646602 0
"18.3171
0.881767 . 0
'18.3910
0.177669 - 0
'7.00224
0.0571287
.0
'7.00322
0.0186419
,0
5.307219
0.374953 -0.646602
3.08458
0
0.374953 1.04762
0
0.0312469
7.45139 -1.56325 0
0.0312469 1.000482
0.881767 8.29042 0.360275 0.177669
8.23540
0.0100214 0.0571287
4.00088
0.0249333
0.0186419
4.00011
0.00154782
0
0.360275 1.39250 J 0
0.0100214 1.37363
18.3786 0.39651
0
0.0249333 0.996875 . 0
0.00154782 0.996669
-1.56325 3.544142 0.0983071
0'
0.0983071 1.00446.
'7.00298
0.0326363
.0
0.396511
8.24727
0.0600924
0.0326363 4.00034 0.00621221
0
0.0600924 1.37414 .
0
0.00621221 0.996681
Chapter 20 Review Questions and Problems, page 883
17. [4   -1   2]T 19. [6   -3 1|T
21. All nonzero entries of the factors are 1. 2.8193   -1.5904 -0.0482"
23.   -1.5904      1.2048   - 0.0241 (4D-values)
.-0.0482   - 0.0241 0.1205. 27. 15, V89, 8 29. 7, V2T, 4
25. Exact |-2   1 2]T
33. 6 35. 9
39. y = 1.98 + 0.9&X
41. Centers 1, 1, 1, radii 2.5, 1, 2.5 (A = 2.944, 0.028 ± 0.290;'. 3D) 43. Centers 5, 6, 8; radii 2, 1, 1, (A = 4.1864, 6.4707, 8.3429, 5S)
31. 14, V78, 7 37. 11.5-4.4578 = 51.2651
45.
1.5 -2.23607     0 '
-2.23607 5.8 -3.1 0 -3.1 6.7
, Step 3:
9.44973 -1.06216 0
-1.06216
4.28682 -0.00308
0
0.00308 0.26345.
App. 2   Answers to Odd-Numbered Problems
Problem Set 21.1, page 897
1. y = ex, 0.0382, 0.1245 (error of *5, x10)
3. y = x - tanh x (set y - x = u), 0.009292, 0.0188465 (error of x5, x10) 5. y = e1, 0.001275, 0.004200 (error of x5, xw) 7. y = 1/(1 - x2/2), 0.00029, 0.01187 (error of x5, xw) 9. y = 1/(1 - x2/2), 0.03547, 0.28715 (error of x5, x1Q) 11. y = 1/(1 - a-2/2); error -10~8, -4- 10"H, 10-7, +10"5;
about 1.3 • 10"5 by (10) 13. y = xe*\ error - 105 (for x = 1, • • •, 3) 19, 46, 85, 139, 213, 315, 454, 640, 889, 1219 15. y = 3 cos x — 2 cos2 x; error - 107: 0.18, 0.74, 1.73, 3.28, 5.59. 9.04, 14.33, 22.77, 36.80, 61.42
17. v = l/(x5 + 1), 0.000307, -0.000259 (error of x5, x10)
19. The errors are for E.-C. 0.02000, 0.06287, 0.05076, for Improved E.-C. -0.000455. 0.012086. 0.009601, for RK 0.0000012. 0.000016, 0.000536.
Problem Set 21.2, page 901 3. y = e-01x2; errors 10"6 to 6 • 10"6
5. y = tan a; y4, ■ • •, y10 (error - 105): 0.422798 (-0.48), 0.546315 (-1.2), 0.684161
(-2.4), 0.842332 (-4.4), 1.029714 (-7.5), 1.260288 (-13), 1.557626 (-22) 7. RK-error smaller, error • 105 = 0.4, 0.3, 0.2, 5.6 (for x = 0.4, 0.6, 0.8, 1.0) 9. y4 = 4.229 690, y5 = 4.556 859, y6 = 5.360 657, y7 = 8.082 563 11. Errors between -6 ■ 10"7 and +3 • 10"7. Solution ex - x - 1 13. Errors ■ 105 from x = 0.3 to 0.7: —5, —11, —19, —31, —47 15. (a) 0, 0.02, 0.0884, 0.215 848, y4 = 0.417 818, y5 = 0.708 887 (poor), (b) By 30-50%
Problem Set 21.3, page 908
3. Vj = ex, y2 = —ex. errors range from ±0.02 to ±0.23, monotone. 5. y\ = y2, y'2 = -4ylt y = yJ = 1, 0.84, 0.52, 0.0656, -0.4720; y = cos 2x 7. yx = 4e-' sin x, y2 = 4e~x cos x; errors from 0 to about ±0.1 9. Errors smaller by about a factor 104 11. y = 0.198669, 0.389494, 0.565220, 0.719632, 0.847790;
y' = 0.980132, 0.922062, 0.830020, 0.709991, 0.568572 13. y, = e-** - e-5x, y2 = e 'Ax + e'5*; y\ = 0.1341, 0.1807, 0.1832, 0.1657,
0.1409; y2 = 1.348, 0.9170, 0.6300, 0.4368, 0.3054 17. You get the exact solution, except for a roundoff error [e.g., yx = 2.761 608.
y(0.2) = 2.7616 (exact), etc.]. Why? 19. y = 0.198669, 0.389494, 0.565220, 0.719631, 0.847789; y' = 0.980132, 0.922061, 0.830019, 0.709988, 0.568568
Problem Set 21.4, page 916
3. 105, 155, 105, 115; Step 5: 104.94, 154.97, 104.97, 114.98 5. 0.108253, 0.108253, 0.324760, 0.324760; Step 10: 0.108538, 0.108396, 0.324902, 0.324831
App. 2   Answers to Odd-Numbered Problems
A51
7. 0, 0, 0, 0. All equipotential lines meet at the corners (why?). Step 5: 0.29298,
0.14649, 0.14649, 0.073245 9. -3«u + m,2 = -200, uu - 3m12 = -100 11. «12 = u-32 ' 31.25, h21 = m23 = 18.75, «;7c = 25 at the others 13. m21 = m23 = 0.25, «12 = h32 = -0.25, ujk = 0 else 15. (a) uu = — h]2 = —66. (b) Reduce to 4 equations by symmetry. Mn — "31 = ~Mis = ~u35 = —92.92, «21 = —m2s = —87.45, l'i2 = "s2 = ~~Mi4 = ~~"34 = —64.22, m22 = — m24 = —53.98,
«13 = «23 = «33 = 0
17. V3, wn = m21 = 0.0849, «12 = m22 = 0.3170. (0.1083, 0.3248 are 4S-values of the solution of the linear system of the problem.)
Problem Set 21.5, page 921
5. un = 0.766. «21 = 1.109, «12 = 1.957, m22 = 3.293
7. A as in Example 1, right sides -2, -2, -2, -2. Solution uu = m21 = 1.14286,
"i2 = "22 = 1.42857 11. — 4«n + h21 + h12 = —3, mu — 4m21 + h22 = —12, uXi - 4«12 + m22 = 0,
2m21 + 2h12 - 12h22 = -14, mu =      = 2, «21 = 4, m12 = 1. Here
-14/3 = -|(1 + 2.5) with 4/3 from the stencil. 13. b = [-380   -190,   -190,   Of; uu = 140, u21 = «12 = 90, «22 = 30
Problem Set 21.6, page 927
5. 0.1636, 0.2545 (t = 0.04, x = 0.2, 0.4), 0.1074, 0.1752 (f = 0.08), 0.0735, 0.1187 (r = 0.12), 0.0498, 0.0807 (t = 0.16), 0.0339, 0.0548 (t = 0.2; exact 0.0331, 0.0535) 7. Substantially less accurate, 0.15, 0.25 (t = 0.04), 0.100, 0.163 (t = 0.08) 9. Step 5 gives 0, 0.06279, 0.09336, 0.08364, 0.04707, 0.
11. Step 2: 0 (exact 0), 0.0453 (0.0422), 0.0672 (0.0658), 0.0671 (0.0628), 0.0394 (0.0373), 0 (0)
13. 0.1018, 0.1673, 0.1673, 0.1018 (/ = 0.04), 0.0219, 0.0355, •■•(? = 0.20) 15. 0.3301, 0.5706, 0.4522, 0.2380 (r = 0.04), 0.06538, 0.10604, 0.10565, 0.6543 (t = 0.20)
Problem Set 21.7, page 930
1. For* = 0.2, 0.4 we obtain 0.012, 0.02 (t = 0.2), 0.004, 0.008 (t = 0.4), -0.004,
-0.008 (t = 0.6), etc. 3. u{x, 1) = 0, -0.05, -0.10, -0.15, -0.075, 0 5. 0.190, 0.308, 0.308, 0.190 (0.178, 0.288, 0.288, 0.178 exact to 3D) 7. 0, 0.354, 0.766, 1.271, 1.679, 1.834, -••(/ = 0.1); 0, 0.575, 0.935, 1.135, 1.296,
1.357, •••(?= 0.2)
Chapter 21 Review Questions and Problems, page 930
17. y = tan x; 0 (0), 0.10050 (-0.00017), 0.20304 (-0.00033), 0.30981 (-0.00047), 0.42341 (-0.00062), 0.54702 (-0.00072)
A52
App. 2   Answers to Odd-Numbered Problems
19. 0.1003349 (0.8 • 10"7) 0.2027099 (1.6 • 10"7), 0.3093360 (2.1 ■ 10"7), 0.4227930
(2.3 • 10-7), 0.5463023 (1.8 • 10"7) 25. y(0.4) = 1.822798, y(0.5) = 2.046315,y(0.6) = 2.284161, y(0.7) = 2.542332,
y(0.8) = 2.829714, y(0.9) = 3.160288, y(1.0) = 3.557626 27. ya = 3e~9x, y2 = -5e'9x, [1.23251    -2.05419], [0.506362   -0.843937], • • %
[0.035113 -0.058522] 29. 1.96, 7.86, 29.46
31. h(Pn) = u(P31) = 270, u(P21) = u(P13) = u(P23) = m(P33) = 30,
u(P12) = u(P32) = 90, u(P22) = 60 35. 0.06279, 0.09336, 0.08364, 0.04707
37. 0, -0.352, -0.153, 0.153, 0.352, 0 if t = 0.12 and 0, 0.344, 0.166, -0.166,
-0.344, 0 if / = 0.24 39. 0.010956, 0.017720, 0.017747, 0.010964 if t = 0.2
Problem Set 22.1, page 939
3. / = 3(xa - 2)2 + 2(x2 + 4)2 - 44. Step 3: [2.0055 -3.9975]T 5. / = 0.50cj - l)2 + 0.7(x2 + 3)2 - 5.8, Step 3: [0.99406 -3.0015]T 7. / = 0.2(x1 - 0.2)2 + x22 - 0.008. Step 3: [0.493   -0.011]T, Step 6: [0.203 0.004]T
Problem Set 22.2, page 943
1. x3, x4 unused time on Mx, M2, respectively 3. No
U. /max = f(0, 5) = 10 13. /max = /(9, 6) = 36
15. /mi„ = /(3.5, 2.5) = -30
17. xx/3 + x2/2    100, Xj/3 + x2/6 g 80, / = 150x, + 100x2,
/max = /(210, 60) = 37500 19. 0.5xt + 0.75x2 S 45 (copper), O.Sjcj + 0.25x2 S 30, / = 120x, + 100x2,
/max = /(45, 30) = 8400
Problem Set 22.3, page 946
/ 2100    200 \
1. /(120/11, 60/11) = 480/11 3. /(—p : -^-J = 78000
5. Matrices with Rows 2 and 3 and Columns 4 and 5 interchanged 7./(0,^) = -10 9. ,f(5, 4, 6) = 478
Problem Set 22.4, page 952
1. ,f(4, 4) = 72 3. .f(20, 30) = 50 5. /(10, 5) = 5500
7. /(I, 1, 0) = 12 9. f(i 0,|) = 3
Chapter 22 Review Questions and Problems, page 952
11. Step 5: [0.353   -0.028]7 Slower 13. Of course! Step 5: [-1.003 1.897|T
21. f(2, 4) = 100 23. f(3, 6) = -54 25. f(50, 100) = 150
App. 2   Answers to Odd-Numbered Problems
A53
Problem Set 23.1, page 958
9.
"0	0	1	r		"0	1 1"		"0	1	1	1
0	0	0	i	11.	0	0 1	13.	0	0	0	0
1	0	0	0		_1	1 o.		1	0	0	0
J	1	0	0.					.0	0	0	0
15.©-
®-
		ßi
	1	" 1
	2	0
		
<ä	3	
>		1
	4	. 0
Edge l
0
0
1
0
Vertex	Incident Edges
1	-elt ~e2, e3, -e4
2	ei
3	e2- -e-i
4	?4
23.
1
X
S 2
>
3
L 0
1 -1"
0 0
1 1.
Problem Set 23.2, page 962
1. 4 3. 5 5. 4
9. The idea is to go backward. There is a vk_1 adjacent to vk and labeled k — 1, etc. Now the only vertex labeled 0 is ,v. Hence A(i>0) = 0 implies v0 = s, so that v0 — vx - • • ■ — ufe_x — vk is a path s     vk that has length jfc.
15. No; there is no way of traveling along (3, 4) only once.
21. From m to 100m, 10m, 2.5m, m + 4.6
Problem Set 23.3, page 966
1. (1, 2), (2, 4), (4, 3); L2 = 6, L3 = 18, L4 = 14
3. (1, 2), (1, 4), (2, 3); L2 = 2, L3 = 5, L4 = 5
5. (1, 4), (2, 4), (3, 4), (3, 5); L2 = 4, L3 = 3, L4 = 2, L5 = 8
7. (1, 5), (2, 3), (2, 6), (3, 4), (3, 5); L2 = 9, L3 = 7, L4 = 8, L5 = 4, L6 = 14
Problem Set 23.4, page 969
2 1 8
1. 1 (     3 L = 12 3. 4 f 2 L = 10     5. 1 - 2 ;   5 L = 28
4( 3-5 3^6-4
5 7
App. 2   Answers to Odd-Numbered Problems
2
9. 1 - 3 - 4 ( L = 38 11. Yes
5 — 6,
15. G is connected. If G were not a tree, it would have a cycle, but this cycle would provide two paths between any pair of its vertices, contradicting the uniqueness.
19. If we add an edge (u, v) to T, then since T is connected, there is a path u —> v in T which, together with (u, v), forms a cycle.
Problem Set 23.5, page 972
1. (1, 2), (1, 4), (3, 4), (4, 5), L = 12 3. (1, 2), (2, 8), (8, 7), (8, 6), (6, 5), (2, 4), (4, 3), L = 40 5. (1, 4), (3, 4), (2, 4), (3, 5), L = 20 7. (1, 2), (1, 3), (1, 4), (2, 6), (3, 5), L = 32 11. If G is a tree
13. A shortest spanning tree of the largest connected graph that contains vertex 1
Problem Set 23.6, page 978
1. 1 - 2 - 5, A/ = 2; 1 - 4 - 2 - 5, A/ = 2, etc.
3. 1 - 2 - 4 - 6, Af = 2; 1 - 2 - 3 - 5 - 6, A/ = 1, etc.
5. /la = 4, f13 = 1, /14 = 4, f42 = 4, /43 = 0, f25 = 8. f35 = 1, /
7. f12 = 4, f13 = 3, f24 = 4, /35 = 3, f54 = 2, f46 = 6, ,f56 = 1, /
9. {4, 5, 6}, 28 11. {2, 4, 6}, 50
13. 1-2-3-7, 4f = 2; 1-4-5-6 -7, A/=1;
1-2-3-6-7, A/= l;/max= 14 15. {3, 5, 7}, 22 17. 5 = {1, 4}, cap (S, T)
19. If/,, < ,u as well as 0
Problem Set 23.7, page 982 3. (2, 3) and (5, 6)
5. 1 - 2 - 5, At = 2;   1 - 4 - 2 - 5, At = 1;   / = 6 + 2+l=9 7. 1-2-4-6, At = 2;   I - 3 - 5 - 6, At = 1;   f = 4 + 2+1=7 9. By considering only edges with one labeled end and one unlabeled end 17. S = {1, 2, 4, 5}, T = {3, 6), cap (S, T) = 14
Problem Set 23.8, page 986
1. No 3. No 5. Yes, S = {1,4, 5, 8}
7. Yes; a graph is not bipartite if it has a nonbipartite subgraph. 9. 1 - 2 - 3 - 5 11. (1, 5), (2, 3) by inspection. The augmenting path 1—2 — 3 — 5
gives 1-2-3-5, that is, (1, 2), (3, 5). 13. (1, 4), (2, 3), (5, 7) by inspection. Or (1, 2), (3, 4), (5, 7) by the use of the path 1-2-3-4.
15. 3 19. 3 23. No: K5 is not planar.
25. K3
= 6 + 8 = 14
App. 2   Answers to Odd-Numbered Problems
Chapter 23 Review Questions and Problems, page 987
13.
A55
17.
"0	1	tr	
0	0	l	
J	0	0_	
"0	1	0	r
0	0	1	0
1	0	f)	i
-0	0	0	0-
	'0	1	0	r
	1	0	1	0
15.				
	0	1	0	1
	-1	0	1	0-
21.	Vertex	Incident Edges	23. 4
	1		
	2	-By, e3	
	3	'ei, -ca	
25.4			27. L
29. 1	- 4 - 3	- 2, L = 16	33./
15, L4
13
Problem Set 24.1, page 996
1. qL = 19, qM = 20, qu = 20.5 5. qL = 69.7, qM = 70.5, qv = 71.2 9. qL = 399, qM = 401, qv = 401
13. x = 70.49, s = 1.047, IQR = 1.5
17.0   0 300
3. qL = 38, qM = 44, qv = 54 7. to = 2.3, qM = 2.4, <f/r; = 2.45 11. x = 19.875, s = 0.835, IQR = 1.5 15. x = 400.4, s = 1.618, IQR = 2 19. 3.54. 1.29
Problem Set 24.2, page 999
1. 4 outcomes: HH, HT. TH. TT (H = Head, T = Tail) 3. 62 = 36 outcomes (1, 1), (1, 2), • • •, (6, 6) 5. Infinitely many outcomes S, ScSy SCS°S, ■ ■ ■ (S = "Six") 7. The space of ordered triples of nonnegative numbers 9. The space of ordered pairs of numbers 11. Yes
13. E = {5, SCS, SCSCS}, Ec = {SCSCSCS, SCSCSCSCS, •••}(£ = "Six")
Problem Set 24.3, page 1005 1. (a) 0.9a = 72.9%, (b)     • § ■ § = 72.65%
-l 490 . 489 . 488 . 487 . 486 _ Qrv t^0/ J- 500    499    498    497    49fi       W.JJ k
5. I - 25 = 0.96 7. 1 - 0.752 = 0.4375 < 0.5
9. P(MMM) + P(MMFM) + P(MFMM) + P(FMMM) = i + 3 • i = 4
A56
App. 2   Answers to Odd-Numbered Problems
11- 5% + §1 ~ 3% = le °y Theorem 3, or by counting outcomes 13. 0.08 + 0.04 - 0.08 • 0.04 = 11.68%
15. 0.954 = 81.5% 17. 1 - 0.974 = 11.5%
Problem Set 24.4, page 1010
3. In 40320 ways
7. 210, 70, 112, 28 11. (?f) = 635 013 559 600 15. 676000
Problem Set 24.5, page 1015
1. k = 1/55 by (6) 5. No because of (6)
9. P(X > 1200) = \   6[0.25 - (x ■
11. k = 2.5; 50% 17. X > b, X g b, X < c, X S c, etc.
Problem Set 24.6, page 1019
1. 2/3, 1/18 5. 4, 16/3
9. fi = I/O = 25; P = 20.2% 13. 750, 1, 0.002
Problem Set 24.7, page 1025
1. 0.0625, 0.25, 0.9375, 0.9375 5. 0.265
7. f(x) = 0.5xe-°3/x\, /(0) + f(l) 9. 1 - e'0'2 = 18%
it   120 13s
1j- 286- 28«' 286' 286
Problem Set 24.8, page 1031
1. 0.1587, 0.6306, 0.5, 0.4950 5. 16%. 9. About 23 13. t= 1084 hours
5. (23°) = 1140
9. 9!/(2!3!4!) = 1260. Ans. 1/1260 13. 1/84, 5/21
3. k = 1/8 by (10)
7. 1 - PCX S 3) = 0.5
= 0.896. Ans. 0.8963 = 72% = 1.1565; 26.9%
3. 3.5, 2.917 7. $643.50
11. i JL, (X - |)V20
15. 15c - 500c3 = 0.97, c = 0.0855
3. 64%
c_0-5(1.0 + 0.5) = 0.91. Ans. 9% 11. 0.99100 = 36.6%
3. 17.29, 10.71, 19.152 7. 31.1%, 95.5%. 11. About 58%
- 1.5)2] dx 13. k
Problem Set 24.9, page 1040
1.1/8,3/16,3/8 3.2/9,2/9,1/2
5- fz(y) = l/(/32 ~ a2) if «2 < >' < fh. anu 0 elsewhere
7. 27.45 mm, 0.38 mm 9. 25.26 cm, 0.0078 cm
App. 2   Answers to Odd-Numbered Problems
A57
13. Independent, fx(x) = 0.1<Taix if x > 0, f2(y) = 0.le~0-ly if v > 0, 36.8% 15. 50% 17. No
Chapter 24 Review Questions and Problems, page 1041
21. QL = 22.3, QM = 23.3, Q0 = 23.5      23. x = 22.89, s = 1.028, s2 = 1.056 25. H, TH, TTH, etc.
27. /(0) = 0.80816, f(l) = 0.18367, ,f(2) = 0.00816
29. Always B C A U fi. If also ACJ), then fi = AUB, etc.
31. 7/3, 8/9 33. 118.019, 1.98, 1.65%
35. 0, 2 37. p = 100/30
39. 16%, 2.3% (see Fig. 520 in Sec. 24.8)
Problem Set 25.2, page 1048
3. / = pk(\ — pj"~k, p = kin, k = number of successes in n trials 5. 11/20
7. / = f(x), d(ln /)% = L/p - (a- - 1)/(1 -p) = 0, p = 1/a-
9. fl = x 11. 0 = w/2 a- = l/.v
13. 0 = 1 15. Variability larger than perhaps expected
Problem Set 25.3, page 1057
1. CONF0.95 {37.47 g /a S 43.87} 3. Shorter by a factor V2
5. 4, 16 7. Cf. Example 2. n = 166
9. CONF0.99{ 20.07 20.33} 11. CONF0.99{ 63.71 g ^ g 66.29}
13. c = 1.96, x = 87, ^2 = 87 • 413/500 = 71.86, k - cslVn = 0.743, CONFa9g {86 S m = 881- CONF095{0.17 0.18}
15. CONF095{ 0.00045 £(r2s 0.00131}   17. CONF095{0.73    o-2 g 5.19}
19. CONF()95{23 g a2 ^ 548}. Hence a larger sample would be desirable.
21. Normal distributions, means —27, 81, 133, variances 16, 144, 400
23. Z = X + Y is normal with mean 105 and variance 1.25. Am. P(104 g Z ^ 106) = 63%
Problem Set 25.4, page 1067
1. t = V7(0.286 - 0)/4.31 = 0.18 < c = 1.94; do not reject the hypothesis. 3. c = 6090 > 6019; do not reject the hypothesis. 5. cP'ln = I, c — 28.36; do not reject the hypothesis. 7. fi < 28.76 or ft > 31.24
9. Alternative p * 1000, t = V20(996 - 1000)/5 = -3.58 < c = - 2.09 (Table A9, 19 degrees of freedom). Reject the hypothesis p = 1000 g.
11. Test p. = 0 against p. + 0. t = 2.11 < c = 2.36 (7 degrees of freedom). Do not reject the hypothesis.
13. a = 5%, c = 16.92 > 9 • 0.52/0.42 = 14.06; do not reject hypothesis.
15. l0 = VlO-9 - 17/19 (21.8 - 20.2)/V'9 ■ 0.62 + 8 • 0.52 = 6.3 > c = 1.74 (17 degrees of freedom). Reject the hypothesis and assert that B is better.
App. 2   Answers to Odd-Numbered Problems
17. v0 = 50/30 = 1.67 < c = 2.59 [(9. 15) degrees of freedom]; do not reject the hypothesis.
Problem Set 25.5, page 1071
1. LCL = 1 - 2.58 • 0.03/V6 = 0.968, UCL = 1.032 3. n = 10
5. Choose 4 times the original sample size (why?). 7. 2.58V0.024/V2 = 0.283, UCL = 27.783, LCL = 27.217 11. In 30% (5%) of the cases, approximately
13. UCL = np + 3Vnp{\ - p), CL = np, LCL = np - 3\'np(l - p) 15. CL = fi = 2.5, UCL = fi + 3\-> = 7.2, LCL = p. - 3V> is negative in (b) and we set LCL = 0.
Problem Set 25.6, page 1076 1. 0.9825, 0.9384, 0.4060 3. 0.8187, 0.6703, 0.1353
Problem Set 25.7, page 1079
1. xo2 = (30 - 50)2/5() + (70 - 50)2/50 = 16 > c = 3.84; no 3. 41
5. xoZ = 2.33 < c = 11.07. Yes
7. ej = npj = 370/5 = 74, x02 = 984/74 = 13.3, c = 9.49. Reject the hypothesis. 9- Xo2 = 1 < 3.84; yes 13. Combining the results for x = 10, 11, 12, we have K — r — 1 = 9 (r = 1 since we
estimated the mean, 1^ == 3.87). Xo2 = 12-98 < c = 16.92. Do not reject. 15.      = 49/20 + 49/60 = 3.27 < c = 3.84 (1 degree of freedom, a = 5%), which
supports the claim. 17. 42 even digits, accept.
Problem Set 25.8, page 1082 3. (|)18(1 + 18 + 153 + 816) = 0.0038
5. Hypothesis: A and B are equally good. Then the probability of at least 7 trials
favorable to A is |a + 8 • g8 = 3.5%. Reject the hypothesis. 7. Hypothesis p, = 0. Alternative p. > 0, x = 1.58,
t = V'10- 1.58/1.23 = 4.06 > c = 1.83 (a = 5%). Hypothesis rejected. 9. x = 9.67, s = 11.87, t0 = 9.67/(11.87/VT5) = 3.15 > c = 1.76 (a = 5%). Hypothesis rejected. 11. Consider Vj = Xj — pl0. 13. P(T S 2) = 0.1% from Table A12. Reject.
5. P(A\ 9) « e_30s(l + 300)
7. P(A; 0) « e-50fi 11. (1 - 0f, (1 - 0)5 + 500 - Of 15. 4>((9 - 12 + |)/V'12(1 - 0.12)) 17.(1 - if + 3-1(1 -|)2 =1
9. 19.5%, 14.7% 13. Because n is finite 0.22 (if c = 9)
App. 2   Answers to Odd-Numbered Problems
15. P(T g 15) = 10.8%. Do not reject. 17. P(T S2)= 2.8%. Reject.
Problem Set 25.9, page 1091
1. y = 1.9 + x
3. y = 6.7407 + 3.068 x
S.y = 4 + 4.8.y, 172 ft
7. y = -1146 + 4.32.1-       9.y = 0.5932 + 0.1138x, R = 1/0.1138 11. q0 = 76,K= 2.36V76/(7 • 944) = 0.253, CONF(),85{ -1.58 g ^ g -1.06) 13. 3.s.,2 = 500. 3sxy = 33.5, kx = 0.067, 3sv2 = 2.268, q0 = 0.023, K = 0.021 CÖNF0.9S{0.046 sKlg 0.088}
Chapter 25 Review Questions and Problems, page 1092
21. ß. = 5.33, a2 = 1722 23. lt will double.
25. CONF099{ 19.1 S fjL g 33.7) 27. CONF0 95{0.726 S/ii 0.751}
29. CONF0.95( 1.373 S/iS 1.451) 31. CONF099{0.05 § er2 g 10)
37. 30.14/3.8 = 7.93 < 8.25. Reject.
39. u0 = 2.5 < 6.0 [(9, 4) degrees of freedom); accept the hypothesis. 41. Decrease by a factor V2. By a factor 2.58/1.96 = 1.32.
33. c = 14.74 > 14.5; reject /jl0.
43. 0.9953, 0.9825. 0.9384, etc.
45. y = 1.70 + 0.55.v
APPENDIX
3
Auxiliary Material
A3.1 Formulas for Special Functions
For tables of numeric values, see Appendix 5. Exponential function ex (Fig. 544)
e = 2.71828 18284 59045 23536 02874 71353 (!) = ex+v, eJTev = ex~v, {exf = exlJ
Natural logarithm (Fig. 545)
(2) In (xy) = In x + In y, In (x/y) = In x — In y. In (xa) = a In x In x is the inverse of ex, and cln a; = x, e_ln x = e1" n/x) = 1/x.
Logarithm of base ten log10x or simply log x
(3) log x = M In x,      M = log e = 0.43429 44819 03251 82765 11289 18917
1 1
(4) lnx = — logx,      — = In 10 = 2.30258 50929 94045 68401 79914 54684
M M
logx is the inverse of 10x, and 10logx = x, 10-logx = 1/x.
Sine and cosine functions (Figs. 546, 547). In calculus, angles are measured in radians, so that sin x and cos x have period 2tt.
sin x is odd, sin (—x) = —sin x, and cos x is even, cos (—x) = cos x.
-2        0 2 x
Fig. 544.   Exponential function e*
Fig. 545.   Natural logarithm In x
A60
SEC. A3.1   Formulas for Special Functions
A61
15)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
y		l	Y 1	
/ ' -l		V     K ./	2k	
Fig. 546.   sin x
Fig. 547.   cos x
sin2 x + cos2 x
1° = 0.01745 32925 19943 radian 1 radian = 57° 17' 44.80625" = 57.29577 95131°
1
sin (x + y) = sin x cos y + cos x sin y sin (x — y) = sin x cos y — cos x sin y cos (x + y) = cos x cos y — sin a' sin y cos (a — y) - cos a cos y 4- sin x sin y
sin 2a = 2 sin x cos a, cos 2a = cos2 x — sin2 x
cos a
")		("
tJ	= cos	(t
		("
t)	= sin	2
sin (tt — a) = sin x,
,2 , _ 1,
cos (77 — a) = —cos a cosza = 5(1 + cos 2a), sin2 a = §(1 — cos 2a)
sin a sin)' = I [ — cos (a + y) + cos (a — y)] cos a cos y = |[cos (a + y) + cos (x — y)] sin a cos y = ||sin (a + y) + sin (a — y)|
« + v       u — v
sin h + sin v = 2 sin-cos-
2 2
cos // + cos v = 2 cos ■
v k — i,'
— cos-
2
u        k — V
cos (J — cos U
sin -
2
(13)     A Cosa + ß sinx = Va2 + B2 cos (a ± 5),
(14)      a cos a + B sin a = va2 + S2 sin (a ± 8),
tan 8 =
tan 8
sin 5 cos 5
sin 8 cos S
ß
A~
A R
A62
APP. 3   Auxiliary Material
Fig. 548.   tan x
Tangent, cotangent, secant, cosecant (Figs. 548, 549)
Fig. 549.   cot x
(15) tanx
sin x
cos x
cotx
sin x
sec x
cos x
(16)       tanix • v)
tanx + tan y
tan (x — y)
esc x
tanx — tan y
tan x tan y 1 + tan x tan y
Hyperbolic functions (hyperbolic sine sinhx, etc.; Figs. 550, 551)
(17)
(18)
(19)
(20)
(21)
sinhx = g (ť' - e x), cosh x = |(e* + e x)
tanh x
sinh x
coth x
cosh x
cosh x sinh x
cosh x + sinh x = ex, cosh x — sinh x = e~x
cosh2 x — sinh2 x = 1 sinh2x = |(cosh2x — 1), cosh2x = |(cosh2x + 1)
j_l
j_L
2 x
(dashed) and cosh x Fig. 551.   tanh x (dashed) and coth x
SEC. A3.1   Formulas for Special Functions
(22)
sinh (x ± y) = sinh x cosh y ± cosh x sinh y cosh (x ± y) = cosh x cosh y ± sinh x sinh y
tanh x ± tanh y
(23) tanh (x ± y) = —---~-
1 ± tanh x tanh y
Gamma function (Fig. 552 and Table A2 in App. 5). The gamma function T(a) is defined by the integral
rx
(24) T(a) =     e_tr_1 dr (a > 0)
which is meaningful only if a > 0 (or, if we consider complex a, for those a whose real part is positive). Integration by parts gives the important functional relation of the gamma function,
(25) T(a + 1) = aT(a).
From (24) we readily have F(l) = 1; hence if a is a positive integer, say k, then by repeated application of (25) we obtain
(26) T(k + 1) = k\ (k = 0, 1, • • •)■
This shows that the gamma function can be regarded as a generalization of the elementary factorial function. [Sometimes the notation (a — 1)1 is used for F(a), even for noninteger values of a, and the gamma function is also known as the factorial function.] By repeated application of (25) we obtain
F(a + 1)      T(a + 2) T(a + k + 1)
Y{a) = -
a(a + 1) a(a + ])(a + 2) ■ ■ ■ (a + k)
Via)
i			I)' !	i i i i i
-4		A	h	2 4a --2 --4
Fig. 552.   Gamma function
APP. 3   Auxiliary Material
and we may use this relation
T(a + k + 1)
(27) r(a) = -(tt*0, -1,-2, •••)
a(a + 1) • • ■ (a + k)
for defining the gamma function for negative a (# —1, —2, ■ ■ ■), choosing for k the smallest integer such that a + k + 1 > 0. Together with (24), rftis then gives a definition of T {a) for all a not equal to zero or a negative integer (Fig. 552).
It can be shown that the gamma function may also be represented as the limit of a product, namely, by the formula
(28) r(o) = lim   - -———     (a * 0, -1, > ■ ■).
n-^x a(a + \)(a + 2) • • • (a + n)
From (27) or (28) we see that, for complex a, the gamma function i(a) is a meromorphic function with simple poles at a = 0, — 1, — 2, ■ * * .
An approximation of the gamma function for large positive a is given by the Stirling formula
(29) T(a + 1) = Vlwa \ — where e is the base of the natural logarithm. We finally mention the special value
(30) r(|) = Vtt.
Incomplete gamma functions
X oo
(31) P(a,x) = J e-*?"-1 dt, Q(a, x) = J e'Y'-1 dt (a > 0)
0 x
(32) T(a) = P(a, x) + Q(a, x) Beta function
(33) B(x, y) = f ;x_1(l - tf~x dt (x > 0, y > 0) Representation in terms of gamma functions:
r(x)r(v)
(34) B(x, y) = V-
r(x + >■)
Error function (Fig. 553 and Table A4 in App. 5)
SEC. A3.1   Formulas for Special Functions
A65
Fig. 553.   Error function
erf (°°) = 1, complementary error function (37)
2         f _,2
erlc.T = 1 — err* = ——     e dt Vtt jx
Fresnel integrals1 (Fig. 554)
(38) C(*) = | cos (t2) dt, S(*) = f sin (f2) dt
C(°°) = Vtt/8, S(cc) = Vtt/8, complementary functions
CW = ,7 - C(*) = I cos (r2) J/
oo
) = [ sin (t2) dt
X
Sine integral (Fig. 555 and Tabic A4 in App. 5)
f sin t
(40) Si(*) = - dt
(39)
0.5
		
	c (x)                    .__. ^	
-		
-		//\"
-		Z~7 / ^
—	/         /          \    ~y  /      \_.	
	/         / -——	
-		
- /		
~A	I I a- 4~f I  I I 1 I  I i i' 1 I I  i I 1 ,  i i i  1 i i     i 1 i  [ i	1 1 1 1 1 1
1 2 3
Fig. 554.   Fresnel integrals
4 x
1 AUGUSTUS! FRESNEL (1788-1827), French physicist and mathematician. For tables see Ref. [GRIJ.
A66
APP. 3   Auxiliary Material
Site) 2
n/2 1
j_i_i_l
0 5
Fig. 555.   Sine integral
10 x
Si(oo) = 7j-/2, complementary function (41)
si(x) = — - Si(%
f   sin /
) = J   — *
Cosine integral (Tabic A4 in App. 5)
cos t
(42) ci(x) = - dt
Exponential integral
(43) Ei(x) = f —
dt
Logarithmic integral
rx dt
(44) \i(x) = —
Jo  In t
(x > 0)
(x > 0)
A3.2 Partial Derivatives
For differentiation formulas, see inside of front cover.
Let z = f(x, y) be a real function of two independent real variables, x and y. If we keep y constant, say, y = \\, and think of x as a variable, then f(x, yx) depends on x alone. If the derivative of f(x, \\) with respect to x for a value x = xx exists, then the value of this derivative is called the partial derivative of f(x, y) with respect to x at the point (xlt y-f) and is denoted by
dx
tel.yi)
Other notations are
or by
and
dz
dX
Czi,vO
zx(x\, v,):
these may be used when subscripts are not used for another purpose and there is no danger of confusion.
SEC. A3.2   Partial Derivatives
A67
We thus have, by the definition of the derivative,
(1)
01
dx
(a-i.Bi)
= lim
f JXX + Ax, \\) - f(xx, yx) Ax
The partial derivative of z, = ,f(x, y) with respect to y is defined similarly; we now keep x constant, say, equal to xlt and differentiate f(xx, y) with respect to v. Thus
(2)
3/
8y
(xi.yi)
dZ
; lim
Ay ^0
/(xx, yx + Ay) - f (xx, yx) Av
Other notations are fy(xx, yx) and Zj,^, >'i).
It is clear that the values of those two partial derivatives will in general depend on the point [xx, y{). Hence the partial derivatives dz/dx and c>z/dy at a variable point (x, y) are functions of x and y. The function dz/<)x is obtained as in ordinary calculus by differentiating z = fix, y) with respect to x, treatingy as a constant, and dz/dy is obtained by differentiating z with respect to y, treating x as a constant.
EXAMPLE 1 Let
fix, y) = .v y + x sin v. Then
2xy + sin y,
dy
x  + x cos v.
The partial derivatives dz/dx and dz/dy of a function z = fix, y) have a very simple geometric interpretation. The function z = fix, y) can be represented by a surface in space. The equation y = yx then represents a vertical plane intersecting the surface in a curve, and Ihe partial derivative dz/dx at a point (xx, \\) is the slope of the tangent (that is, tan a where a is the angle shown in Fig. 556) to the curve. Similarly, the partial derivative dz/dy at (Xj, yx) is the slope of the tangent to the curve x = xx on the surface z = fix,, y) at (xi, yj).
Fig. 556
A68
The partial derivatives dz/dx and dzldy are called first partial derivatives or partial derivatives of first order. By differentiating these derivatives once more, we obtain the four second partial derivatives (or partial derivatives of second order)1
J XX fyx fxy fyy
(3)
a2/	8	(df_
dx2 '	dx	\ dx
d2f	d	
dx dy	dx	\ <lv
d2f	d	(*L
dy dx	dy	\ dx
	d	/V
dy2	dy	
It can be shown that if all the derivatives concerned are continuous, then the two mixed partial derivatives are equal, so that the order of differentiation does not matter (see Ref. [GR4| in App. 1), that is,
d2Z d2Z
(4) - = -.
dx dy      dy dx
EXAMPLE 2    For the function in Example 1.
JXX =  2?, fXy   =   2.V  +  COS.V =   fyX, fyy   =   ~X SUi >>.
By differentiating the second partial derivatives again with respect to x and y, respectively, we obtain the third partial derivatives or partial derivatives of the third order of /, etc.
If we consider a function fix, y, z) of three independent variables, then we have the three first partial derivatives fx(x, y, z), fy(x, y, z), and fz(x, y, z). Here fx is obtained by differentiating f with respect to x, treating both y and z as constants. Thus, analogous to (1), we now have
df_ dx
= lim
(.ti,öi,zi) Ax—>0
f(Xi  + AX, >'!, ZX) ~ f(Xj_, >'!, Zl)
Ax
etc. By differentiating fy, fz again in this fashion we obtain the second partial derivatives of /, etc.
EXAMPLE 3    Let fix, y, z) = x2 + y2 + 2 + xy ez. Then
fx = 2x + y f', fy    2y i x e\ fz = 2z + xy ez.
fxx — 2. fxy ~ fyx ~~ ^ » fxz ~ fzx ~ X ^ >
fyy = 2. fyz = fzy = x ez, fzl = 2 + xy ez. ■
2 CAUTION! In the subscript notation the subscripts are written in the order in which we differentiate, whereas in the "d" notation the order is opposite.
SEC. A3.3   Sequences and Series
A69
A3.3
Sequences and Series
See also Chap. 15.
Monotone Real Sequences
We call a real sequence xv .v2, • • ■, xn, • • ■ a monotone sequence if it is either monotone increasing, that is,
We call xu x2, • ' • a bounded sequence if there is a positive constant K such that \xn\ < K for all n.
PROOF   Let xx, x2, ■ ■ • be a bounded monotone increasing sequence. Then its terms are smaller
than some number B and, since xv S xn for all n, they lie in the interval xx — xn § B, which will be denoted by I0. We bisect /0; that is. we subdivide it into two parts of equal length. If the right half (together with its endpoints) contains terms of the sequence, we denote it by IL. If it does not contain terms of the sequence, then the left half of/0 (together with its endpoints) is called Ix. This is the first step.
In the second step we bisect /l7 select one half by the same rule, and call it J2, and so on (see Fig. 557 on p. A70).
In this way we obtain shorter and shorter intervals /0, Ilt I2, • • • with the following properties. Each Im contains all /,„ for n > m. No term of the sequence lies to the right of I.m, and, since the sequence is monotone increasing, all xn with n greater than some number N lie in Im\ of course, N will depend on in, in general. The lengths of the im approach zero as m approaches infinity. Hence there is precisely one number, call it L, that lies in all those intervals,3 and we may now easily prove that the sequence is convergent with the limit L.
In fact, given an e > 0, we choose an m such that the length of /„, is less than e. Then L and all the x,„ with n > N(m) lie in Im, and, therefore, \xn — L\ < e for all those n. This completes the proof for an increasing sequence. For a decreasing sequence the proof is the same, except for a suitable interchange of "left" and "right" in the construction of those intervals. ■
3This statement seems to be obvious, but actually il is not; it may be regarded as an axiom of the real number system in the following form. Let yx, J2. ■ ■ ■ be closed intervals such that each Jm contains all Jn with n > m, and the lengths of the Jm approach zero as m approaches infinity. Then there is precisely one real number that is contained in all those intervals. This is the so-called Cantor-Dcdekind axiom, named after the German mathematicians GEORG CANTOR (1845-1918), the creator of set theory, and RICHARD DEDEKIND (1831-1916). known for his fundamental work in number theory. For further details sec Ref. [GR2] in App. 1. (An interval / is said lo be closed if its two endpoints are regarded as points belonging to /. It is said to be open if the endpoints are not regarded as points of /.)
or monotone decreasing, that is,
x2 g x3 s • • - .
THEOREM 1
If a real sequence is bounded and monotone, it converges.
A70
APP. 3   Auxiliary Material
x,       x„ x.,
i     i   i i i mini
B
j
h-h-
Fig. 557.   Proof of Theorem 1
THEOREM 2
Real Series
Leibniz Test for Real Series
Let X], x2, • ' " be real and monotone decreasing to zero, that is,
(1)
(a)   .q   ' x2  ' x3 "
(b)   lim xm — 0.
77?en ?/jc series with terms of alternating signs
X\      A*2 ~t~ X3      A"4 "T
converges, and for the remainder Kn after the nth term we have the estimate (2) \Rn\ g xn,!.
PROOF   Let s„ be the nth partial sum of the series. Then, because of (la),
S\ = X], S2 = Xi — x2 is sx,
SZ = s2 + x3 = s2> s3 = sl ~ (x2 ~ x3) — 's'l>
so that s2 '  s3 '   Si, Proceeding in this fashion, we conclude that (Fig. 558)
(3)
sl = s3 = s5
which shows that the odd partial sums form a bounded monotone sequence, and so do the even partial sums. Hence, by Theorem 1, both sequences converge, say,
lim s2r,
= s,
lim s2n
"2 "A *3 "I
Fig. 558.   Proof of the Leibniz test
SEC. A3.4   Grad, Div, Curl, V2 in Curvilinear Coordinates ATI
Now, since s2n + l — s2n = Xj^+j, we readily see that (lb) implies
s - s* = lim s2n+i ~ lim s2n = lim (s2n+i - s2n) = lim x2n+1 = 0.
Hence s* = 5, and the series converges with the sum s.
We prove the estimate (2) for the remainder. Since sn —* s, it follows from (3) that
•Sfcn+l = S§ 52n        and als0        *2n-l =    = s2n-
By subtracting j2n and s2n-i> respectively, we obtain
s2n l-l — s2n = s ~ s2n = 0 1? S — •'>'2ri-l = ^2»! ~~ *2n-l'
In these inequalities, the first expression is equal to x2n+1, the last is equal to — x2n, and the expressions between the inequality signs are the remainders R2n and R2n-i- Thus the inequalities may be written
-x2n+l — R2rl = 0, 0 i£ R2fl_1 = — X2„
and we see that they imply (2). This completes the proof. ■
A3.4 Grad, Div, Curl, V2
in Curvilinear Coordinates
To simplify formulas we write Cartesian coordinates x = xx, y = x2, z, = x3. We denote curvilinear coordinates by qlt q2, q3. Through each point P there pass three coordinate surfaces qx = const, q2 = const, q3 = const. They intersect along coordinate curves. We assume the three coordinate curves through P to be orthogonal (perpendicular to each other). We write coordinate transformations as
(1) % = Xifai, q2, q3), x2 = x2{qx, q2, q3), x3 = x3(qu q2, q3). Corresponding transformations of grad, div, curl, and V2 can all be written by using
(2) h^ii^2
k=l
'~'Cll
Next to Cartesian coordinates, most important are cylindrical coordinates c/, = r, q2 = 6. q3 = z. (Fig. 559a on p. A72) defined by
(3)   xx = qx cos q2 = r cos 6, x2 — qx sin q2 = r sin 6, x3 — q3 = z
and spherical coordinates qL = r, q2 = 0, q3 = <]) (Fig. 559b) defined by4
^    X] = qx cos q2 sin c/3 = r cos 6 sin <j>, x2 = qv sin q2 sin q3 = r sin 0 sin </>
x3 = qY cos q3 = r cos (p.
4This is the notation used in calculus and in many other books. It is logical since in it, 0 plays the same role as in polar coordinates. CAUTION! Some books interchange the roles of 0 and 4>.
APP. 3   Auxiliary Material
? (r, 0,z)
(a) Cylindrical coordinates (6) Spherical coordinates
Fig. 559.   Special curvilinear coordinates
In addition to the general formulas for any orthogonal coordinates qu q2, we shall give additional formulas for these important special cases.
Linear Element ds.   In Cartesian coordinates,
ds2 = dxf + dx22 + dxi
For the ^-coordinates,
(5) ds2 = lh2 dqx2 + h22 dq22 + h2 dq32.
i.Vi
(Sec. 9.5).
For polar coordinates set dz = 0. (5")
ds2 = dr2 + r2 sin2 $ dt? + r2 dd>2
(Cylindrical coordinates).
(Spherical coordinates).
Gradient, grad f = V/ = [fx, fZz, fxJ (partial derivatives; Sec. 9.7). In the (/-system, with u, v, w denoting unit vectors in the positive directions of the qlt q2, q3 coordinate curves, respectively,
(6)
(o\)
(6")
i   df      ia/      i of
grad / = Vf =---:— u + — -— v +---;— w
hx   aqx h2 dq2
df        1 df df grad /    V/ = — u + - — v + —
dr r do dz
Of 1      df        1 3/
grad / = V/ = — u + —— — v--— w
dr       r sin <b do       r dtp
h3 dq3
(Cylindrical coordinates) (Spherical coordinates).
Divergence div F = V-F = (F^ + (F2),;z + (F3)X3 (F = [Flt F2, F3], Sec. 9.8);
(7)  div F = V'F =
(7)    dlvF = V»F
1 8
r dr
— [h2hzFx) + — (h3lhF2) ,)q1 dq2
I dF2 dF3
W + ~ To" +
36 dz
(hih2F3)
(Cylindrical coordinates)
SEC. A3.4   Grad, Div, Curl, V2 in Curvilinear Coordinates
A73
1 d
I 3F2
(7 )    div F = V«F = _5" — (r^F-f) .
r   Sr r sin <p   ntl      r sin q> d<t>
(sin 4> Fs)       (Spherical coordinates).
Laplacian V2/ = V«Vf = div (grad /) = /.,. Xi t        + /     (Sec. 9.8)
(8) V2/
1
_       \ ftj    c'Jt/! /     dq2  \  h2 dq.
+
2      92/     1 0/      1   r*2/ a2/ (8') V2/- — + (-•
hyh2 df_ dq3 \ h3 8q3
(Cylindrical coordinates)
V2/
a,-2      r 9r      r2 sin2 </> SO2  '    2 d^2
1       a2/      1   d2f     cot </> a/
Curl (Sec. 9.9):
(Spherical coordinates).
(9)
curl F = V x F
	/ijU	h2\	h3w
1	8	d	d
h\h2h3			
		h2F2	h3F3
For cylindrical coordinates we have in (9) (as in the previous formulas)
h1 = hT = 1, h2 = h„ = qx — r, h3 = hz = 1
and for spherical coordinates we have
li1 = hr =1, hz = h„ = c/i sin q3 = r sin <\>, hs = h$ = q, = r.
» APPENDIX
4
Additional Proofs
Section 2.6, page 73
PROOF OF THEOREM 1 Uniqueness1
Assuming that the problem consisting of the ODE
(1) y" + p(x)y' + q(x)y = 0
and the two initial conditions
(3) y(x0) = K0, y'(x0) = Kx
has two solutions yt(x) and y2(x) on the interval / in the theorem, we show that their difference
y(x) = y,(x) - y2(x)
is identically zero on /; then yx = y2 on /, which implies uniqueness.
Since (1) is homogeneous and linear, y is a solution of that ODE on /, and since yx and y2 satisfy the same initial conditions, y satisfies the conditions
(10)
We consider the function and its derivative
From the ODE we have
y(.r0) = 0, y'(x0) = 0.
z(x) = y{xf + y'{xf z = 2yy + 2y y .
y = -py - qy-
By substituting this in the expression for z we obtain (11) z = 2yy' - 2py'2 - 2qyy .
Now, since y and y are real,
l.v ± y')2 - v2 • 2yv' • y'2 : ' 0.
'This proof was suggested by my colleague. Prof. A. D. Ziebur. In this proof we use formula numbers that have not yet been used in Sec. 2.6.
A74
APP. 4   Additional Proofs A75
From this and the definition of z we obtain the two inequalities
(12) (a)   2yy' g y2 + y'2 = z, (b)   -2yy' g y2 + y'2 = z.
From (12b) we have 2yy' § —z. Together, \2yy'] g z. For the last term in (11) we now obtain
-2qyy' g |-2^yy'| = |-?||2yy'| § \q\z. Using this result as well as — p :   ]/?| and applying (12a) to the term 2yy in (II), we find
Z g Z + 2|/?]y'2 + k|z. Since y'2 = y2 + y'2 = z, from this we obtain
z g (1 + 21/?] + |?|)z or, denoting the function in parentheses by h,
(13a) z = hz for all x on A
Similarly, from (11) and (12) it follows that
-z = -2yy' + 2py'2 + 2<?yy'
(13b)
§z + 2\p\z + \q\z = hz. The inequalities (13a) and (13b) arc equivalent to the inequalities (14) z ~ hz g 0, z + hz 0.
Integrating factors for the two expressions on the left arc
p   _ e~Sh(x) dx an£j p    _ eJh(x) dx
The integrals in the exponents exist because h is continuous. Since Fx and F2 arc positive, we thus have from (14)
I'\{z ~ hz) = (/'!?)' g 0 and         F2(z + hz) = (F2z)' § 0.
This means that F^Z is nonincreasing and F2z is nondecreasing on /. Since z(x0) = 0 by (10), when x g x0 we thus obtain
F\Z = (F\Z)xa = 0, F2z S (F2z)Io = 0
and similarly, when x =g jc0,
FlZgO, F27 g 0.
Dividing by Fx and F2 and noting that these functions are positive, we altogether have
Z g 0, z § 0 for all * on /.
This implies that z = y2 + y'2 = 0 on A Hence y = 0 or y2 = y2 on /. ■
A76 APP. 4   Additional Proofs
Section 5.4, pages 184
PROOF OF THEOREM 2   Frobenius Method. Basis of Solutions. Three Cases
The formula numbers in this proof are the same as in the text of Sec. 5.4. An additional formula not appearing in Sec. 5.4 will be called (A) (see below). The ODE in Theorem 2 is
„     b(x)   , c(x) (1) y" + y' + -V y = 0,
x x
where b(x) and c(x) are analytic functions. We can write it (1') x2y" + xb(x)y' + c(x)y = 0.
The indicial equation of (1) is
(4) r(r - 1) + b0r+ c0= 0.
The roots r,, r2 of this quadratic equation determine the general form of a basis of solutions of (1), and there are three possible cases as follows.
Case 1. Distinct Roots not Differing by an Integer.   A first solution of (1) is of the
form
(5) y^x) = xri(a0 + axx + a2x2 + • • •)
and can be determined as in the power series method. For a proof that in this case, the ODE (1) has a second independent solution of the form
(6) y2{x) = xrHA„ + AlX + A2x2 + ■ • •), see Ref. [All] listed in App. 1.
Case 2. Double Root. The indicial equation (4) has a double root r if and only if (b0 - l)2 - 4c0 = 0, and then r = |(1 - b0). A first solution
(7) y,(x) = xr(a0 + axx + a2x2 + •••), r = |(1 - b0),
can be determined as in Case 1. We show that a second independent solution is of the form
(8) y2(x) = y^x) In.v + xr(A1x + A2x2 + • • •) (x > 0).
We use the method of reduction of order (see Sec. 2.1), that is, we determine u{x) such that y2(x) = uixjy^ix) is a solution of (1). By inserting this and the derivatives
I i       .        r n li       i   ~   I   l   , II
y2 = u y, + m>'i, y2 = u >>! + 2« .vy + uy\
into the ODE (1') wc obtain
a-2(m"v1 + 2u'y[ + uy") + xb(u' \\ + uy[) + cuyx = 0.
APP. 4   Additional Proofs A77
Since yx is a solution of (1'), the sum of the terms involving u is zero, and this equation reduces to
2        It   I   m   2   '    '    i       / ' r\
x yxu  + 2x y\U + xbj\U = 0. By dividing by x2yx and insetting the power series for b we obtain
" 4. (n yi j. b° \   ' n
u   +12 — + — + •••!« =0.
Here and in the following the dots designate terms that are constant or involve positive powers of x. Now from (7) if follows that
y[ _ xr-x[ra0 + (r + l)ajx + • • ■] yx XT[a0 + a\x + ' ' 'J
1   / ra0 + (r + \)avx + ■ ■ • \ r
ein -t- a-iX + ■ ■ ■      I x
Hence the previous equation can be written
, 2r + b0
(A)
Since r = (1 — b(l)/2, the term (2r + b0)/x equals l/x, and by dividing by u we thus have
u
u' x
By integration we obtain In u - -In x + ■ • • , hence u = (l/x)e('' °. Expanding the exponential function in powers of x and integrating once more, we sec that u is of the form
u = Inx +      + kzx2 + ■ ■ ■ .
Inserting this into y2 = uylt we obtain for v2 a representation of the form (8).
Case 3. Roots Differing by an Integer. We write rx = r and r2 = r — p where p is a positive integer. A first solution
(9) yx(x) = xr*(a0 + axx + a2x2 + • • ■)
can be determined as in Cases 1 and 2. We show that a second independent solution is of the form
(10) y2(x) = kyx(x) Inx + xrHA0 + Axx + A2x2 + • - •)
where we may have k =h 0 or k = 0. As in Case 2 we set y2 = uyx. The first steps are literally as in Case 2 and give Eq. (A),
I 2> + b° + ■ ■
A78
APP. 4   Additional Proofs
Now by elementary algebra, the coefficient b0 — 1 of r in (4) equals minus the sum of the roots,
b0 - 1 = -fa + r2) = -(r + r - p) = -2r + p. Hence 2r + b0 = p + 1, and division by u gives
P + 1
The further steps are as in Case 2. Integrating, we find In u = —(p + 1) ln.v + • • • , thus
r -(P+l) (• • 0 u — x e
where dots stand for some scries of nonnegative integer powers of x. By expanding the exponential function as before we obtain a series of the form
I
In
A-P + 1   + xp
■ +
x x
We integrate once more. Writing the resulting logarithmic term first, we get
// = kp In x
kp-i
px'
kp+1x
Hence, by (9) we get for y2 = uy\ the formula
y2 = kpy\ In x + x*1
■ ■ ■ - fcp_jXP 1 + • • • 1 (a0 + aLx + ■ ■ ■)
But this is of the form (10) with k = kp since r± — p = r2 and the product of the two series involves nonnegative integer powers of x only. ■
Section 5.7, page 205
THEOREM
Reality of Eigenvalues
Ifp, q, r, and p in the Sturm-Li ouville equation (1) of Sec. 5.7 are real-valued and continuous on the interval a = x g b and r(x) > 0 throughout that interval (or r(x) < 0 throughout that interval), then all the eigenvalues of the Sturm—Liouville problem (1), (2), Sec. 5.1, are real.
PROOF   Let A = a + iB be an eigenvalue of the problem and let
y(x)  - u(x) + iv(x)
be a corresponding eigenfunction; here a, B, u, and u are real. Substituting this into (1), Sec. 5.7, we have
(pu + ipv')' + (q + ar + iBr)(u + iv) = 0.
APP. 4   Additional Proofs
A79
This complex equation is equivalent to the following pair of equations for the real and the imaginary parts:
(pu'Y + (q + ar)u — Brv = 0 (pv'Y + (q + ar)v + Bru = 0.
Multiplying the first equation by v, the second by —u and adding, we get
-B(u2 + v2)r = uipu')' - v(pu')' = [(pv')u - (pu')u]'.
The expression in brackets is continuous on a g x =s b, for reasons similar to those in the proof of Theorem 1, Sec. 5.7. Integrating over x from a to b, we thus obtain
B J (h2 + v2)r dx
p(uv  — u v)
Because of the boundary conditions the right side is zero; this is as in that proof. Since y is an eigenfunction, u2 + v2 # 0. Since y and r are continuous and r > 0 (or r < 0) on the interval a g x "s b, the integral on the left is not zero. Hence, B = 0, which means that A = a is real. This completes the proof. ■
Section 7.7, page 308
THEOREM
Determinants			
The definition of a determinant			
	«n	«12 a1n	
	«21	«22              • - • «2n.	
(7)                  D = det A =			
	«n1		
as given in Sec. 7.7 is unambiguous, that is, it yields the same value of D no matter			
which rows or columns we choose in		developings.	
PROOF    In this proof we shall use formula numbers not yet used in Sec. 7.7.
We shall prove first that the same value is obtained no matter which row is chosen.
The proof is by induction. The statement is true for a second-order determinant, for which the developments by the first row ana22 + «i2(~«2i) and by the second row «2i( — «12) + «22«n Sive tne same value a11a22 — a12a21. Assuming the statement to be true for an (n — l)st-order determinant, we prove that it is true for an nth-order determinant.
APP. 4   Additional Proofs
For this purpose we expand D in terms of each of two arbitrary rows, say, the /th and the 7'th, and compare the results. Without loss of generality let us assume i < j.
First Expansion.   We expand D by the /th row. A typical term in this expansion is
(19) alkClk = alk-(-\fkMlk.
The minor Mik of aik in D is an (n — l)st-order determinant. By the induction hypothesis we may expand it by any row. We expand it by the row corresponding to the /th row of D. This row contains the entries ajt (/ ¥= k). It is the (j — l)st row of Mik, because Mik does not contain entries of the /th row of D, and / < j. We have to distinguish between two cases as follows.
Case I. If / < k, then the entry aj: belongs to the /th column of Mik (see Fig. 560). Hence the term involving aiX in this expansion is
(20) afl ■ (cofactor of ajL in Mik) = a.}l ■ (- 1 )°~1'1 lMikjl
where Mi(y7 is the minor of a^ in Mik. Since this minor is obtained from Mlk by deleting the row and column of Ujh it is obtained from D by deleting the /th and j th rows and the kth and /th columns of D. We insert the expansions of the Mik into that of D. Then it follows from (19) and (20) that the terms of the resulting representation of D are of the form
(21 a) alkan ■ ( -1 )bMikj, (1 < k)
where
b = i + k + j + I - 1.
Case II. If / > k, the only difference is that then aj7 belongs to the (/ — I )st column of Mik, because Mik does not contain entries of the kth column of D, and k < I. This causes an additional minus sign in (20), and, instead of (21a), we therefore obtain
(21b) -aikajr(-\)bMikjl (/ > k)
where b is the same as before.
Fig. 560.   Cases I and II of the two expansions of D
APP. 4   Additional Proofs
Second Expansion. We now expand D at first by the /th row. A typical term in this expansion is
(22) = ajr(-\y+lM:jl.
By the induction hypothesis we may expand the minor of in D by its /th row, which corresponds to the /th row of D, since j > i.
Case I. If /c > /, the entry aik in that row belongs to the (A — 1 )st column of Mj(, because Mji does not contain entries of the /th column of D, and / < A (see Fig. 560). Hence the term involving aik in this expansion is
(23) alk ■ (cofactor of aik in Mfl) = aih • ( -1)« Kfc-1)MjkiJ,
where the minor of aik in M^j is obtained by deleting the /th and 7 th rows and the Ath and /th columns of Z) [and is, therefore, identical with MikjL in (20), so that our notation is consistent|. We insert the expansions of the into that of D. It follows from (22) and (23) that this yields a representation whose terms are identical with those given by (21a) when / < k.
Case II. If k < I, then aik belongs to the Ath column of MjL, we obtain an additional minus sign, and the result agrees with that characterized by (21b).
We have shown that the two expansions of D consist of the same terms, and this proves our statement concerning rows.
The proof of the statement concerning columns is quite similar; if we expand D in terms of two arbitrary columns, say, the Ath and the /th, we find that the general term involving Ojiaik is exactly the same as before. This proves that not only all column expansions of D yield the same value, but also that their common value is equal to the common value of the row expansions of D.
This completes the proof and shows that our definition of an nth-order determinant is unambiguous. ■
Section 9.3, page 377
PROOF OF FORMULA [2]
We prove that in right-handed Cartesian coordinates, the vector product v = a X b = [alt  a2,   a3] x [/;,,   b2, b3]
has the components
(2) vt = a2bd - aab2,      v2 = a3bx - axb%,      v3 = axb2 - a2bv
We need only consider the case v 0. Since v is perpendicular to both a and b, Theorem 1 in Sec. 9.2 gives a • v = 0 and b • v = 0: in components [see (2), Sec. 9.2],
a1v1 + a2v2 + a3v3 = 0
(3)
bxvx + b2v2 + b3v3 = 0.
APP. 4   Additional Proofs
Multiplying the first equation by b3, the last by a3. and subtracting, we obtain
(«3/?! - fli/?3)i'i = (a2b3 - a3b2)v2. Multiplying the first equation by bx, the last by ax, and subtracting, we obtain
{ayb2 - a%bx)v2 = (a3bL - axb3)v3.
We can easily verify that these two equations arc satisfied by
(4)     Vi = c(a2b3 - a3b2),      v2 = c(a3b1 - axb3),      v3 = c{axb2 - a2bx)
where c is a constant. The reader may verify by inserting that (4) also satisfies (3). Now each of the equations in (3) represents a plane through the origin in L'^Dg-space. The vectors a and b are normal vectors of these planes (see Example 6 in Sec. 9.2). Since v # 0, these vectors are not parallel and the two planes do not coincide. Hence their intersection is a straight line L through the origin. Since (4) is a solution of (3) and, for varying c, represents a straight line, we conclude that (4) represents L, and every solution of (3) must be of the form (4). In particular, the components of v must be of this form, where c is to be determined. From (4) we obtain
|v|2 = v* + v.2 + v* = c2[(a2b3 - a3b2f + (a^ - axb3f + (axb2 - a2bx)2}.
This can be written
|v|2 = c2[{ax2 + a22 + a32)(b2 + b2 + b2) - (a,b, + a2b2 + a3b3f],
as can be verified by performing the indicated multiplications in both formulas and comparing. Using (2) in Sec. 9.2, we thus have
|v|2 = c2[(a • a)(b • b) - (a • b)2].
By comparing this with formula (12) in Team Project 24 of Problem Set 9.3 we conclude that c = ± 1.
We show that c = +1. This can be done as follows.
If we change the lengths and directions of a and b continuously and so that at the end a = i and b = j (Fig. 186a in Sec. 9.3), then v will change its length and direction continuously, and at the end, v = i x j = k. Obviously we may effect the change so that both a and b remain different from the zero vector and are not parallel at any instant. Then v is never equal to the zero vector, and since the change is continuous and c can only assume the values +1 or —1, it follows that at the end c must have the same value as before. Now at the end a = i, b = j, v = k and, therefore, ax = 1, b2 = 1, v3 = 1, and the other components in (4) are zero. Hence from (4) we see that v3 = C = +1. This proves Theorem 1.
For a left-handed coordinate system, i X j = —k (see Fig. 186b in Sec. 9.3), resulting in c = — 1. This proves the statement right after formula (2). ■
APP. 4   Additional Proofs
A83
Section 9.9, page 416
PROOF OF THE INVARIANCE OF THE CURL
This proof will follow from two theorems (A and B), which we prove first.
THEOREM A
Transformation Law for Vector Components
For any vector v the components vx, v2, v3 and v2*, v3* in any two systems of Cartesian coordinates xt, x3, x3 and jc,*, x2*. x3", respectively, are related by
(1)
and conversely
(2)
with coefficients
(3)
satisfying
(4)
V* = CllVl + cL2v2 + cl3v3 V2* = c21v1 + c22v2 + c23v3
v3    = c31vl      c32v2 + c33v3,
Vl = C'llUl* + c21v2* + c3lv3*
CísPl c22v2* 4 C32U3
Ci3fl* + C23V2* + <;;;í'':í:í:
cn = i*«i
c2\ = j*'Í
c31 = k*«i
c22 =
C32 = k*«j
2 ckjc»
c13 = i*«k C23 = j**k c33 = k*k
(/c, m = 1,2, 3),
where the Kroneckcr delta2 is given by
8u
km
(k + m) (A = m)
and i, j, k and i*, j*, k* denote the unit vectors in the positive xx-, x2-, x3- and Xi*-, x2*-, x3*'-directions, respectively.
2LEOPOLD KRONECKKR (1823-1891), German mathematician at Berlin, who made important contributions to algebra, group theory, and number theory.
We shall keep our discussion completely independent of Chap. 7, but readers familiar with matrices should recognise that we are dealing with orthogonal transformations and matrices and that our present theorem follows from Theorem 2 in Sec. 8.3.
A84
APP. 4   Additional Proofs
PROOF   The representation of v in the two systems are
(5) (a)   x = Vl\ + v2\ + v3k        (b)  v = u^i* + u2*j* + v3*k*.
Since i* • i* = 1, i* • j* = 0, i* • k:I= = 0, we get from (5b) simply i* • v = vx* and from this and (5a)
Vl* = j* • y = i* • uyi + i* • u2j + i* • v3k = vxi* • i + v2\* • j + v3i* • k.
Because of (3), this is the first formula in (1), and the other two formulas are obtained similarly, by considering j* • v and then k* • v. Formula (2) follows by the same idea, taking i • v = vx from (5a) and then from (5b) and (3)
Uj = i ■ v = vt*i ' i* + u2*i • j* + rj3*i • k* = CuBj* + c2Xv2* + ^3^3*,
and similarly for the other two components.
We prove (4). We can write (1) and (2) briefly as
3 3
(6) (a)   vj = 2 cmjVm*, (b)   L>fc* = 2 c^Vj.
m = l j=l
Substituting Vj into vk*, we get
33 3/3 \
vk* = 2 ckj 2 cmjV„* = 2 "m*    2    .•'"••V I '
j — l       m—l m—1 v=l /
where k = 1, 2, 3. Taking k = 1, we have
Vr* = vx* 12 ''•''•j + is" 12 ri''> j + y3* 12 ri,--3,J •
For this to hold for every vector v, the first sum must be 1 and the other two sums 0. This proves (4) with k = 1 for m = 1, 2, 3. Taking k: = 2 and then A: = 3, we obtain (4) with k = 2 and 3, form = 1, 2, 3. ■
THEOREM B
Transformation Law for Cartesian Coordinates
The transformation of any Cartesian x1x2x3-coordinale system into any other Cartesian xx*x2*x3*-coordinate system is of the form
(7)
tvz'f/7 coefficients (3) aW constants bx, b2, b3; conversely,
(8)
k = 1, 2, 3.
APP. 4   Additional Proofs A85
Theorem B follows from Theorem A by noting that the most general transformation of a Cartesian coordinate system into another such system may be decomposed into a transformation of the type just considered and a translation; and under a translation, corresponding coordinates differ merely by a constant.
PROOF OF THE INVARIANCE OF THE CURL
We write again x,, x2, x3 instead of x, y, z, and similarly xt*, x2*, .v3* for other Cartesian coordinates, assuming that both systems are right-handed. Let al5 a2, a3 denote the components of curl v in the x^^-coordinates, as given by (1), Sec. 9.9, with
x = x-i,         y — xz, z = x3.
Similarly, let ax*, a2*, a3* denote the components of curl v in the x1*x2*x3*-coordinate system. We prove that the length and direction of curl v are independent of the particular choice of Cartesian coordinates, as asserted. We do this by showing that the components of curl v satisfy the transformation law (2), which is characteristic of vector components. We consider av We use (6a), and then the chain rule for functions of several variables (Sec. 9.6). This gives
dv3      dv2 /      dvin" dvrn*
«i = —---—- = 2 \L ™3 ~^z— L™2
dx?        5x3 ,  \ "x2 ax3
^ m. — 1
*   »  /      dvm*   dx* _       dvm* dx*
m=1 j= 1
From this and (7) we obtain
dXj*    dx2 dXj* dx3
3
dv,
*
m=lj—1 3
dVi* dv-r*
J2
~ (c33c22 ~ C3zC23)ai^ + (t'l3c'32 — c12c33)a2    + (c23c'l2 — C22c13)a3*-
Note what we did. The double sum had 3X3 = 9 terms, 3 of which were zero (when m — j), and the remaining 6 terms we combined in pairs as we needed them in getting a2*, a3*.
We now use (3), Lagrange's identity (see Team Project 24 in Problem Set 9.3) and k* x j* = -i* and k X j = -i. Then
C33C22 ~ c32<-'23 = (k* * k)(j* • j) - (k* • j)(j* • k) = (k* x j*) • (k x j) = i* • i = cn,
etc.
A86 APP. 4   Additional Proofs
Hence a, = cna,* + c2la2* + c3ia3*- This is of the form of the first formula in (2) in Theorem A, and the other two formulas of the form (2) are obtained similarly. This proves the theorem for right-handed systems. If the x^Xy-coordinates are left-handed, then k X j = +i, but then there is a minus sign in front of the determinant in (1), Sec. 9.9. ■
Section 10.2, pages 426-427
PROOF OF THEOREM 1, PART (b)   We prove that if
(1) f F(r) • dr = \ (Fx dx + F2 dy + F3 dz)
Jc Jc
with continuous Fx, F2, F3 in a domain D is independent of path in D, then F = grad / in D for some f; in components
(2') Fi = ~ , F2 = ^ ■ F3 = ~ .
dx dy dz
We choose any fixed A: (x0, y0, Zq) in D and any B: (x, y, z) in D and define / by
(3) f(x, y, z) = f0 + f (F, dx* + F2 dy* + F3 dz*)
JA
with any constant f0 and any path from A to in D. Since A is fixed and we have independence of path, the integral depends only on the coordinates x, y, z, so that (3) defines a function f(x, y, z) in D. We show that F = grad / with this /, beginning with the first of the three relations (2r). Because of independence of path we may integrate from A to Bv: (xL, y, z) and then parallel to the x-axis along the segment BXB in Fig. 561 with Bx chosen so that the whole segment lies in D. Then
B, B
f(x, y, z) = f0 +      (Fx dx* + F2 dy* + F3 dz*) +     (Ft dx* + F2 dy* + F3 dz*).
JA JB1
We now take the partial derivative with respect to x on both sides. On the left we gel dfldx. We show that on the right we get Fx. The derivative of the first integral is zero because A: (x0, y0, z0) and (-^l; y, z) do not depend on x. We consider the second integral. Since on the segment B^B, both y and z are constant, the terms F2dy* and
Fig. 561.   Proof of Theorem 1
APP. 4   Additional Proofs
A87
F3 dz* do not contribute to the derivative of the integral. The remaining part can be written as a definite integral,
f /•, dx* = | / jf.vv, z) dx*.
Hence its partial derivative with respect to x is Fx(x, y, z), and the first of the relations (2') is proved. The other two formulas in (2') follow by the same argument.
Section 13.4, page 620
PROOF OF THEOREM 1   Cauchy-Riemann Equations
We prove that Cauchy-Riemann equations
(1) It,    =    Vy, Uy    = ~VX
are sufficient for a complex function f(z) = u(x, y) + iv{x, y) to be analytic; precisely, if the real part u and Ihe imaginary part v of f(z) satisfy (1) in a domain D in the complex plane and if the partial derivatives in (1) are continuous in D, then f(z) is analytic in D.
In this proof we write Az = Ax + /Ay and A/ = f(z + Az) — f(z). The idea of proof is as follows.
(a) We express Af in terms of first partial derivatives of u and v, by applying the mean value theorem of Sec. 9.6.
(b) We get rid of partial derivatives with respect to y by applying the Cauchy-Riemann equations.
(c) We let Az approach zero and show that then A//Az as obtained approaches a limit, which is equal to ux + ivx, the right side of (4) in Sec. 13.4, regardless of the way of approach to zero.
(a) Let P: (x, y) be any fixed point in D. Since D is a domain, it contains a neighborhood of P. We can choose a point Q: (x + Ax, y + Ay) in this neighborhood such that the straight-line segment PQ is in D. Because of our continuity assumptions we may apply the mean value theorem in Sec. 9.6. This yields
u(x + Ax, y + Ay) - u(x, y) = (Ax^Mj) + (Ay)uy{Mx) v(x + Ax, y + Ay) - v(x, y) = (Ax)vJM2) + (Ay)vy(M2)
where Mx and M2 (# M1 in general!) are suitable points on that segment. The first line is Re Af and the second is Im Af, so that
Af = {Ax)ux(M1) + (Ay)Uy(M,) + i[(Ax)vx(M2) + (Ay)vy(M2)].
(b) uy = — vx and vy = ux by the Cauchy-Riemann equations, so that
Af = (Ax)Mt(Mx) - (Ay)vx(M-d + i[(Ax)vx(M2) + {Ay)ux{M2)].
A88 APP. 4   Additional Proofs
Also Az = Ax + /Ay, so that we can write Ax = Az - I Ay in the first term and Ay = (Az — Ax)/i — —/(Az — Ax) in the second term. This gives
Af = (Az - iAy)uT(Mx) + '(Az - Ax)vx(M1) + i[(Ax)vx(M2) + i A v)//,(,!/,) ].
By performing the multiplications and reordering we obtain
Af = (Aztn,(Mi) - /Ay !„,.<>/,) ~ uT(M2)}
+ i[(Az)vx(Mi) ~ A.vj r (V/,) - r.(A/,i]|.
Division by Az now yields
Af iAy iAx
(A) — = ux(Mi) + iuJMJ ~ -j- (n,(iWi) - «,(M2)} - — [vJMJ ~ vx(M2)}.
(c) We finally let Az approach zero and note that |Av/Az| = 1 and |Ajc/Az| g 1 in (A). Then Q: (x + A.v, y + Ay) approaches P: (x, v), so that M, and M2 must approach P. Also, since the partial derivatives in (A) arc assumed to be continuous, they approach their value at P. In particular, the differences in the braces {• • ■} in (A) approach zero. Hence the limit of the right side of (A) exists and is independent of the path along which Az 0. We see that this limit equals the right side of (4) in Sec. 13.4. This means that f(z) is analytic at every point z in D, and the proof is complete. ■
Section 14.2, pages 647-648
GOURSAT'S PROOF OF CAUCHY'S INTEGRAL THEOREM   Goursal proved Cauchy's integral theorem without assuming that /'(z) is continuous, as follows.
We start with the case when C is the boundary of a triangle. We orient C counterclockwise. By joining the midpoints of the sides we subdivide the triangle into four congruent triangles (Fig. 562). Let CIs Cn, Cm, CIV denote their boundaries. We claim that (sec Fig. 562).
Indeed, on the right we integrate along each of the three segments of subdivision in both possible directions (Fig. 562), so that the corresponding integrals cancel out in pairs, and the sum of the integrals on the right equals the integral on the left. We now pick an integral on the right that is biggest in absolute value and call its path Cj. Then, by the triangle inequality (Sec. 13.2),
<f fdz		<f fdz	+	$ fdz	+	f fdz	+	<f fdz	■ 4	$ fdz
						Cm				Jc1
Fig. 562.   Proof of Cauchy's integral theorem
APP. 4   Additional Proofs
A89
We now subdivide the triangle bounded by C, as before and select a triangle of subdivision with boundary C2 for which
Cj
/ dz	Then	$ fdz	i42	4 fdz
		Jc		Jc2
Continuing in this fashion, we obtain a sequence of triangles 7\, T2, • • ■ with boundaries Ci, C2, ■ • • that are similar and such that Tn lies in Tm when n > m, and
(2)
dp fdz ^4"  j> fdz
n = 1, 2,
Let Zq be the point that belongs to all these triangles. Since / is differentiable at z the derivative / (z0) exists. Let
(3)
Kz)
./'■) - f(Z0)
f'(Zo).
Solving this algebraically for f(z) we have
f(z) = /(Z0) + Cz - 2o)/'(zo) + Kz)(z - Zo). Integrating this over the boundary Cn of the triangle Tn gives
<f  /(z) & = <f  f(zo) dz + $  (z- z0)/'(Zo) & + <f  A(«)(z - zo) dz.
c„ ■ c„ Jc„ Jc„
Since /(z0) and /'(z0) are constants and Cn is a closed path, the first two integrals on the right are zero, as follows from Cauchy's proof, which is applicable because the integrands do have continuous derivatives (0 and const, respectively). We thus have
f(z)dz
h(z)(z - zo) dz.
Since f'(zo) is the limit of the difference quotient in (3), for given e > 0 we can find a 8 > 0 such that
(4)
\h(z)\ < e when
< 8.
We may now take n so large that the triangle Tn lies in the disk |z - z0| < 8. Let Ln be the length of Cn. Then |z - z.ol < Ln for all z on Cn and z0 in Tn, From this and (4) we have \h(z)(z - z0)| < eLn. The A/L-iinequality in Sec. 14.1 now gives
(5)
/(z) dz
h(z)(z ~ Zo) *
Now denote the length of C by L. Then the path Cx has the length Lx = L/2, the path C2 has the length L2 = Lx/2 = LI A, etc., and C,, has the length Ln = L/2". Hence Lr2 = L2/4n. From (2) and (5) we thus obtain
fdz
4n
4
A90
By choosing e (> 0) sufficiently small we can make the expression on the right as small as we please, while the expression on the left is the definite value of an integral. Consequently, this value must be zero, and the proof is complete.
The proof for the case in which C is the boundary of a polygon follows from the previous proof by subdividing the polygon into triangles (Fig. 563). The integral corresponding to each such triangle is zero. The sum of these integrals is equal to the integral over C, because we integrate along each segment of subdivision in both directions, the corresponding integrals cancel out in pairs, and we are left with the integral over C.
The case of a general simple closed path C can be reduced to the preceding one by inscribing in C a closed polygon P of chords, which approximates C "sufficiently accurately," and it can be shown that there is a polygon P such that the integral over P differs from that over C by less than any preassigned positive real number e, no matter how small. The details of this proof are somewhat involved and can be found in Ref. [D6| listed in App. 1. ■
Fig. 563.   Proof of Cauchy's integral theorem for a polygon
Section 15.1, page 667
PROOF OF THEOREM 4   Cauchy's Convergence Principle for Series
(a) In this proof we need two concepts and a theorem, which we list first.
1. A bounded sequence sy, s2, • • • is a sequence whose terms all lie in a disk of (sufficiently large, finite) radius K with center at the origin; thus |sn| < K for all n.
2. A limit point a of a sequence Sy, s2, • • • is a point such that, given an e > 0, there are infinitely many terms satisfying \sn — a\ < e. (Note that this does not imply convergence, since there may still be infinitely many terms that do not lie within that circle of radius e and center a.)
Example: |, |, §, a> Tfi' it- ' ' ' nas me limit points 0 and 1 and diverges.
3. A bounded sequence in the complex plane has at least one limit point. (Bolzano-Weierstrass theorem; proof below. Recall that "sequence" always mean infinite sequence.)
(b) We now turn to the actual proof that Zy + z% + " ■ ■ converges if and only if for every e > 0 we can find an N such that
(1) \Zn+i + ' ' ' + Zn t pi < e      ror every n > N and p = 1, 2, • • • .
Here, by the definition of partial sums,
$n l p      ->n      <-n+1 "f       ' "f Zn+p'
APP. 4   Additional Proofs
A91
Writing n + p = r, we see from this that (1) is equivalent to
(1*) \sr - sn\ < e for all r > N and n > N.
Suppose that sx, .v2, ■ ■ ■ converges. Denote its limit by s. Then for a given e > 0 we can find an N such that
€
\sn ~ s\ < y f°r every n > N.
Hence, if r > N and a > N, then by the triangle inequality (Sec. 13.2),
i i i     i        i     i e e
K ~ an\ = \(sr - s) - (sn - s)\ '■' \sr - s\ + \sn - s\ < — + — = e,
that is, (1*) holds.
(c) Conversely, assume that Aj, s2, • • ■ satisfies (1*). We first prove that then the sequence must be bounded. Indeed, choose a fixed e and a fixed n = n0 > N in (1*). Then (1 *) implies that all sr with r > N lie in the disk of radius e and center s„o and only finitely many terms slt • ■ • , sN may not lie in this disk. Clearly, we can now find a circle so large that this disk and these finitely many terms all lie within this new circle. Hence the sequence is bounded. By the Bolzano-Weierstrass theorem, it has at least one limit point, call it s.
We now show that the sequence is convergent with the limit s. Let e > 0 be given. Then there is an N* such that \sT - sj < ell for all r > N* and n > N*, by (1*). Also, by the definition of a limit point, \sn — s\ < ell for infinitely many n, so that we can find and fix an n > N* such that |s„ — s\ < ell. Together, for every r > A/*,
\sr - s\ = \(sr - sn) + [sn - 5)| '   |sr - sn\ + \sn - s\ < — + — = e;
that is, the sequence sh s2, • ■ ■ is convergent with the limit s. ■
THEOREM	Bolzano-Weierstrass Theorem3	
	A bounded infinite sequence z-y, Z2, z3, ■	■ ■ in the complex plane has at least one
	limit point.	
PROOF It is obvious that we need both conditions: a finite sequence cannot have a limit point, and the sequence 1, 2, 3, ■ ■ ■ , which is infinite but not bounded, has no limit point. To prove the theorem, consider a bounded infinite sequence z±, z2, • • • and let K be such that \zn\ < K for all «. If only finitely many values of the zn are different, then, since the sequence is infinite, some number z must occur infinitely many times in the sequence, and, by definition, this number is a limit point of the sequence.
We may now turn to the case when the sequence contains infinitely many different terms. We draw a large square Q0 that contains all z„. We subdivide Q0 into four congruent squares, which we number 1, 2, 3, 4. Clearly, at least one of these squares (each taken
3BKRNARD BOLZANO (1781-1848), Austrian mathematician and professor of religious studies, was a pioneer in the study of point sets, the foundation of analysis, and mathematical logic. For W'eierstrass, see Sec. 15.5.
A92
APP. 4   Additional Proofs
with its complete boundary) must contain infinitely many terms of the sequence. The square of this type with the lowest number (1, 2, 3, or 4) will be denoted by <2t- This is the first step. In the next step we subdivide (2i into four congruent squares and select a square Q2 by the same rule, and so on. This yields an infinite sequence of squares Q0, Qi> Q-2i' ' ' > Qm ' ' ' wrm the property that the side of Qn approaches zero as n approaches infinity, and Qm contains all Q„ with n > m. It is not difficult to see that the number which belongs to all these squares,4 call it z = a, is a limit point of the sequence. In fact, given an e > 0, we can choose an N so large that the side of the square QN is less than e and, since QN contains infinitely many zn, we have \zn — a\ < e for infinitely many n. This completes the proof. ■
Section 15.3, pages 681-682
PART (b) OF THE PROOF OF THEOREM 5
We have to show that
2 ««.
(Z + At/
A;
nz
thus,
2 an Az[(z + Az)""2 + 2z(z + A;)"" (z + Azf - z
+ ••• + («- 1)Z"-2],
n— 1
If we set
(7a)
Az
= Az[(z + Az)""2 + 2z(z + Az)"-3 + ••• + («- l)zre"2l - Az = b and z = a, thus Az = b — a, this becomes simply
bn - an
= (b - a)Ar
b — a
where An is the expression in the brackets on the right,
(7b) An = b'1-'2 + 2abn~3 + ?>a2bn-4 +••• + («
(n = 2, 3,
IK
thus, A2 = 1, A3 since then
b + 2a, etc. We prove (7) by induction. When n — 2, then (7) holds, (b + a)(b - a)
2a
2a
(h - a)A2
Assuming that (7) holds for n = k, we show that it holds for n = k + 1. By adding and subtracting a term in the numerator and then dividing we first obtain
uk i 1
baK
bak - a
k+l
= b
bk - ak b — a
+ ak.
4 The fact that such a unique number z = a exists seems to be obvious, but it actually follows from an axiom of the real number system, the so-called Cantor-Dedekind axiom: see footnote 3 in App. A3.3.
APP. 4   Additional Proofs
A93
By the induction hypothesis, the right side equals b[(b — a)Ak + kak 1] + ak. Direct calculation shows that this is equal to
(b - a){bAk + lea*'1} + cika*'1 + ak.
From (7b) with n — k we see that the expression in the braces {• * •} equals
b*'1 + 2abk~2 + ■ ■ • + (k - \)bak~2 + kak~l = Akl.
Hence our result is
bk+l - ak+1
--- =(b- a)Ak,! + (k + l)ak.
b — a
Taking the last term to the left, we obtain (7) with n = k + 1. This proves (7) for any integer n g 2 and completes the proof. ■
Section 18.2, page 754
ANOTHER PROOF OF THEOREM 1   without the use of a harmonic conjugate
We show that if w = U + iv = f(z) is analytic and maps a domain D conformally onto a domain D* and $>*(u, v) is harmonic in D*, then
(1) ®(x, y) = ®*(u(x, y), v(x, y))
is harmonic in 73, that is. V2<t> = 0 in D. We make no use of a harmonic conjugate of <£*, but use straightforward differentiation. By the chain rule,
Wc apply the chain rule again, underscoring the terms that will drop out when we form V24):
<f)   = cj> *„   + (cb* u  + <v>* v )u
^XX x U    1/1XX    '    \^UUvlX Uu WX' llx
+ '!», -r,,,. + ($*,«,, + «»'*. rx)v,.
<$yy is the same with each x replaced by y. We form the sum V2<1>. In it, 4>*„ = <I>*„ is multiplied by
UXVX   + UyVy
which is 0 by the Cauchy-Riemann equations. Also V2u = 0 and V2i> = 0. There remains
Y24> = QZJm* + u2) + $Uvx2 + v2). By the Cauchy-Riemann equations this becomes
V2<£ = (0>*u + <I>*.)(h,2 + v2)
and is 0 since <t>* is harmonic.
APPEN
Tables
D I X 5
For Tables of Laplace transforms see Sees. 6.8 and 6.9. For Tables of Fourier transforms see Sec. 11.10.
If you have a Computer Algebra System (CAS), you may not need the present tables, but you may still find them convenient from time to time.
Table Al   Bessel Functions
For more extensive tables see Ref. [GR1] in App. 1.
X	J0(x)		X		Mx)	X	4C&)	
0.0	1.0000	0.0000	3.0	-0.2601	0.3391	6.0	0.1506	-0.2767
0.1	0.9975	0.0499	3.1	-0.2921	0.3009	6.1	0.1773	-0.2559
0.2	0.9900	0.0995	3.2	-0.3202	0.2613	6.2	0.2017	-0.2329
0.3	0.9776	0.1483	3.3	-0.3443	0.2207	6.3	0.2238	-0.2081
0.4	0.9604	0.1960	3.4	-0.3643	0.1792	6.4	0.2433	-0.1816
0.5	0.9385	0.2423	3.5	-0.3801	0.1374	6.5	0.2601	-0.1538
0.6	0.9120	0.2867	3.6	-0.3918	0.0955	6.6	0.2740	-0.1250
0.7	0.8812	0.3290	3.7	-0.3992	0.0538	6.7	0.2851	-0.0953
0.8	0.8463	0.3688	3.8	-0.4026	0.0128	6.8	0.2931	-0.0652
0.9	0.8075	0.4059	3.9	-0.4018	-0.0272	6.9	0.2981	-0.0349
1,0	0.7652	0.4401	4.0	-0.3971	-0.0660	7.0	0.3001	-0.0047
1.1	0.7196	0.4709	4.1	-0.3887	0.1033	7.1	0.2991	0.0252
1.2	0.6711	0.4983	4.2	-0.3766	-0.1386	7.2	0.2951	0.0543
1.3	0.6201	0.5220	4.3	-0.3610	-0.1719	7.3	0.2882	0.0826
1.4	0.5669	0.5419	4.4	-0.3423	-0.2028	7.4	0.2786	0.1096
1.5	0.5118	0.5579	4.5	-0.3205	-0.2311	7.5	0.2663	0.1352
1.6	0.4554	0.5699	4.6	-0.2961	-0.2566	7.6	0.2516	0.1592
1.7	0.3980	0.5778	4.7	-0.2693	-0.2791	7.7	0.2346	0.1813
1.8	0.3400	0.5815	4.8	-0.2404	-0.2985	7.8	0.2154	0.2014
1.9	0.2818	0.5812	4.9	-0.2097	-0.3147	7.9	0.1944	0.2192
2.0	0.2239	0.5767	5.0	-0.1776	0.3276	8.0	0.1717	0.2346
2.1	0.1666	0.5683	5.1	-0.1443	-0.3371	8.1	0.1475	0.2476
2.2	0.1104	0.5560	5.2	-0.1103	-0.3432	8.2	0.1222	0.2580
2.3	0.0555	0.5399	5.3	-0.0758	-0.3460	8.3	0.0960	0.2657
2.4	0.0025	0.5202	5.4	-0.0412	-0.3453	8.4	0.0692	0.2708
2.5	-0.0484	0.4971	5.5	-0.0068	-0.3414	8.5	0.0419	0.2731
2.6	-0.0968	0.4708	5.6	0.0270	-0.3343	8.6	0.0146	0.2728
2.7	-0.1424	0.4416	5.7	0.0599	-0.3241	8.7	-0.0125	0.2697
2.8	-0.1850	0.4097	5.8	0.0917	-0.3110	8.8	-0.0392	0.2641
2.9	-0.2243	0.3754	5.9	0.1220	-0.2951	8.9	-0.0653	0.2559
J0(x) = 0 for J! = 2.40483, 5.52008, 8.65373, 11.7915, 14.9309, 18.0711, 21.2116, 24.3525, 27.4935, 30.6346 Jt(x) = 0 for* = 3.83171, 7.01559, 10.1735, 13.3237. 16.4706, 19.6159, 22.7601, 25.9037, 29.0468, 32.1897
A94
Table Al (continued)
X		Yt(x)	X	Y0(x)	Yi(x)	X	Y0(x)	Yiix)
0.0	(-co)	(-*)	2.5	0.498	0.146	5.0	-0.309	0.148
0.5	-0.445	-1.471	3.0	0.377	0.325	5.5	-0.339	-0.024
1.0	0.088	-0.781	3.5	0.189	0.410	6.0	-0.288	-0.175
1.5	0.382	0.412	4.0	- 0.017	0.398	6.5	-0.173	-0.274
2.0	0.510	-0.107	4.5	-0.195	0.301	7.0	-0.026	-0.303
Table A2   Gamma Function [see (24) in App. A3.1J
a	T(a)	a	T(a)	a	r»	a	I»	a	i»
1.00	1.000 000	1.20	0.918 169	1.40	0.887 264	1.60	0.893 515	1.80	0.931 384
1.02	0.988 844	1.22	0.913 106	1.42	0.886 356	1.62	0.895 924	1.82	0.936 845
1.04	0.978 438	1.24	0.908 521	1.44	0.885 805	1.64	0.898 642	1.84	0.942 612
1.06	0.968 744	1.26	0.904 397	1.46	0.885 604	1.66	0.901 668	1.86	0.948 687
1.08	0.959 725	1.28	0.900 718	1.48	0.885 747	1.68	0.905 001	1.88	0.955 071
1.10	0.951 351	1.30	0.897 471	1.50	0.886 227	1.70	0.908 639	1.90	0.961 766
1.12	0.943 590	1.32	0.894 640	1.52	0.887 039	1.72	0.912 581	1.92	0.968 774
1.14	0.936 416	1.34	0.892 216	1.54	0.888 178	1.74	0.916 826	1.94	0.976 099
1.16	0.929 803	1.36	0.890 185	1.56	0.889 639	1.76	0.921 375	1.96	0.983 743
1.18	0.923 728	1.38	0.888 537	1.58	0.891 420	1.78	0.926 227	1.98	0.991 708
1.20	0.918 169	1.40	0.887 264	1.60	0.893 515	1.80	0.931 384	2.00	1.000 000
Table A3   Factorial Function and Its Logarithm with Base 10
n	n\	log («!)	n	n\	log (n!)	n	n\	log («!)
1	1	0.000 000	6	720	2.857 332	11	39 916 800	7.601 156
2	2	0.301 030	7	5 040	3.702 431	12	479 001 600	8.680 337
3	6	0.778 151	8	40 320	4.605 521	13	6 227 020 800	9.794 280
4	24	1.380 211	9	362 880	5.559 763	14	87 178 291 200	10.940 408
5	120	2.079 181	10	3 628 800	6.559 763	15	1 307 674 368 000	12.116 500
Table A4   Error Function, Sine and Cosine Integrals [see (35), (40), (42) in App. A3.11
X	erf x	Si(x)	ci(.r)	X	erf x	Si(*)	ciOi)
0.0	0.0000	0.0000	00	2.0	0.9953	1.6054	-0.4230
0.2	0.2227	0.1996	1.0422	2.2	0.9981	1.6876	-0.3751
0.4	0.4284	0.3965	0.3788	2.4	0.9993	1.7525	-0.3173
0.6	0.6039	0.5881	0.0223	2.6	0.9998	1.8004	-0.2533
0.8	0.7421	0.7721	-0.1983	2.8	0.9999	1.8321	0.1865
1.0	0.8427	0.9461	-0.3374	3.0	1.0000	1.8487	-0.1196
1.2	0.9103	1.1080	-0.4205	3.2	1.0000	1.8514	-0.0553
1.4	0.9523	1.2562	- 0.4620	3.4	1.0000	1.8419	0.0045
1.6	0.9763	1.3892	-0.4717	3.6	1.0000	1.8219	0.0580
1.8	0.9891	1.5058	-0.4568	3.8	1.0000	1.7934	0.1038
2.0	0.9953	1.6054	-0.4230	4.0	1.0000	1.7582	0.1410
A96
APP. 5 Tables
Table A5   Binomial Distribution
Probability function f(x) [see (2), Sec. 24.7] and distribution function F(x)
		P =	= 0.1	P =	= 0.2	P '■	= 0.3	P	= 0.4	P =	= 0.5
n	,t	/w	F(x)	fix)	Fix)	m	m	m	F(x)	m	Fix)
		0.		0.		0.		ü.		0.	
1	0	9000	0.9000	8000	0.8000	7000	0.7000	6000	0.6000	5000	0.5000
	1	1000	1.0000	2000	1.0000	3000	1.0000	4000	1.0000	5000	1.0000
	0	8100	0.8100	6400	0.6400	4900	0.4900	3600	0.3600	2500	0.2500
2	1	1800	0.9900	3200	0.9600	4200	0.9100	4800	0.8400	5000	0.7500
	2	0100	1.0000	0400	1.0000	0900	1.0000	1600	1.0000	2500	1.0000
	0	7290	0.7290	5120	0.5120	3430	0.3430	2160	0.2160	1250	0.1250
	1	2430	0.9720	3840	0.8960	4410	0.7840	4320	0.6480	3750	0.5000
j	?	0270	0.9990	0960	0.9920	1890	0.9730	2880	0.9360	3750	0.8750
	3	0010	1.0000	0080	1.0000	0270	1.0000	0640	1.0000	1250	1.0000
	0	6561	0.6561	4096	0.4096	2401	0.2401	1296	0.1296	0625	0.0625
	1	2916	0.9477	4096	0.8192	4116	0.6517	3456	0.4752	2500	0.3125
4	2	0486	0.9963	1536	0.9728	2646	0.9163	3456	0.8208	3750	0.6875
	3	0036	0.9999	0256	0.9984	0756	0.9919	1536	0.9744	2500	0.9375
	4	0001	1.0000	0016	1.0000	0081	1.0000	0256	1.0000	0625	1.0000
	0	5905	0.5905	3277	0.3277	1681	0.1681	0778	0.0778	0313	0.0313
	1	3281	0.9185	4096	0.7373	3602	0.5282	2592	0.3370	1563	0.1875
	2	0729	0.9914	2048	0.9421	3087	0.8369	3456	0.6826	3125	0.5000
J	3	0081	0.9995	0512	0.9933	1323	0.9692	2304	0.9130	3125	0.8125
	4	0005	1.0000	0064	0.9997	0284	0.9976	0768	0.9898	1563	0.9688
	5	0000	1.0000	0003	1.0000	0024	1.0000	0102	1.0000	0313	1.0000
	0	5314	0.5314	2621	0.2621	1176	0.1176	0467	0.0467	0156	0.0156
	1	3543	0.8857	3932	0.6554	3025	0.4202	1866	0.2333	0938	0.1094
	2	0984	0.9841	2458	0.9011	3241	0.7443	3110	0.5443	2344	0.3438
6	3	0146	0.9987	0819	0.9830	1852	0.9295	2765	0.8208	3125	0.6563
	4	0012	0.9999	0154	0.9984	0595	0.9891	1382	0.9590	2344	0.8906
	5	0001	1.0000	0015	0.9999	0102	0.9993	0369	0.9959	0938	0.9844
	6	0000	1.0000	0001	1.0000	0007	1.0000	0041	1.0000	0156	1.0000
	0	4783	0.4783	2097	0.2097	0824	0.0824	0280	0.0280	0078	0.0078
	1	3720	0.8503	3670	0.5767	2471	0.3294	1306	0.1586	0547	0.0625
	2	1240	0.9743	2753	0.8520	3177	0.6471	2613	0.4199	1641	0.2266
7	3	0230	0.9973	1147	0.9667	2269	0.8740	2903	0.7102	2734	0.5000
/	4	0026	0.9998	0287	0.9953	0972	0.9712	1935	0.9037	2734	0.7734
	5	0002	1.0000	0043	0.9996	0250	0.9962	0774	0.9812	1641	0.9375
	6	0000	1.0000	0004	1.0000	0036	0.9998	0172	0.9984	0547	0.9922
	7	0000	1.0000	0000	1.0000	0002	1.0000	0016	1.0000	0078	1.0000
	0	4305	0.4305	1678	0.1678	0576	0.0576	0168	0.0168	0039	0.0039
	1	3826	0.8131	3355	0.5033	1977	0.2553	0896	0.1064	0313	0.0352
	2	1488	0.9619	2936	0.7969	2965	0.5518	2090	0.3154	1094	0.1445
	3	0331	0.9950	1468	0.9437	2541	0.8059	2787	0.5941	2188	0.3633
8	4	0046	0.9996	0459	0.9896	1361	0.9420	2322	0.8263	2734	0.6367
	5	0004	1.0000	0092	0.9988	0467	0.9887	1239	0.9502	2188	0.8555
	6	0000	1.0000	0011	0.9999	0100	0.9987	0413	0.9915	1094	0.9648
	7	0000	1.0000	0001	1.0000	0012	0.9999	0079	0.9993	0313	0.9961
	8	0000	1.0000	0000	1.0000	0001	1.0000	0007	1.0000	0039	1.0000
Table A6   Poisson Distribution
Probability function f(x) [see (5), Sec. 24.7] and distribution function Fix)
X	P /«	= 0.1 Fix)	fix)	= 0.2 Fix)	P = fix)	= 0.3 Fix)	P = fix)	= 0.4 F(x)	P fix)	= 0.5 Fix)
	0.		0.		0.		0.		0.	
0	9048	0.9048	8187	0.8187	7408	0.7408	6703	0.6703	6065	0.6065
1	0905	0.9953	1637	0.9825	2222	0.9631	2681	0.9384	3033	0.9098
2	0045	0.9998	0164	0.9989	0333	0.9964	0536	0.9921	0758	0.9856
3	0002	1.0000	0011	0.9999	0033	0.9997	0072	0.9992	0126	0.9982
4	0000	1.0000	0001	1.0000	0003	1.0000	0007	0.9999	0016	0.9998
5							0001	1.0000	0002	1.0000
	P '■	= 0.6	p'	= 0.7	P	= 0.8	P '	0.9	P	= 1
X	fix)	Fix)	fix)	Fix)	fix)	Fix)	fix)	Fix)	fix)	Fix)
	0.		0.		0.		0.		0.	
0	5488	0.5488	4966	0.4966	4493	0.4493	4066	0.4066	3679	0.3679
1	3293	0.8781	3476	0.8442	3595	0.8088	3659	0.7725	3679	0.7358
2	0988	0.9769	1217	0.9659	1438	0.9526	1647	0.9371	1839	0.9197
3	0198	0.9966	0284	0.9942	0383	0.9909	0494	0.9865	0613	0.9810
4	0030	0.9996	0050	0.9992	0077	0.9986	0111	0.9977	0153	0.9963
5	0004	1.0000	0007	0.9999	0012	0.9998	0020	0.9997	0031	0.9994
6			0001	1.0000	0002	1.0000	0003	1.0000	0005	0.9999
7									0001	1.0000
	P m	= 1.5 Fix)	P fix)	= 2 Fix)	P fix)	= 3 Fix)	P fix)	= 4 Fix)	P fix)	= 5 Fix)
	0.		0.		0.		0.		0.	
0	2231	0.2231	1353	0.1353	0498	0.0498	0183	0.0183	0067	0.0067
1	3347	0.5578	2707	0.4060	1494	0.1991	0733	0.0916	0337	0.0404
2	2510	0.8088	2707	0.6767	2240	0.4232	1465	0.2381	0842	0.1247
3	1255	0.9344	1804	0.8571	2240	0.6472	1954	0.4335	1404	0.2650
4	0471	0.9814	0902	0.9473	1680	0.8153	1954	0.6288	1755	0.4405
5	0141	0.9955	0361	0.9834	1008	0.9161	1563	0.7851	1755	0.6160
6	0035	0.9991	0120	0.9955	0504	0.9665	1042	0.8893	1462	0.7622
7	0008	0.9998	0034	0.9989	0216	0.9881	0595	0.9489	1044	0.8666
8	0001	1.0000	0009	0.9998	0081	0.9962	0298	0.9786	0653	0.9319
9			0002	1.0000	0027	0.9989	0132	0.9919	0363	0.9682
10					0008	0.9997	0053	0.9972	0181	0.9863
11					0002	0.9999	0019	0.9991	0082	0.9945
12					0001	1.0000	0006	0.9997	0034	0.9980
13							0002	0.9999	0013	0.9993
14							0001	1.0000	0005	0.9998
15									0002	0.9999
16									0000	1.0000
A98
APP. 5 Tables
Table A7   Normal Distribution
Values of the distribution function <J>(z) [see (3), Sec. 24.8]. <t>(-z) = 1 - <3?(z)
z		z		z	<h;)	z	4>(z)	z		z.	
	0.		0.		0.		0.		0.		0.
0.01	5040	0.51	6950	1.01	8438	1.51	9345	2.01	9778	2.51	9940
0.02	5080	0.52	6985	1.02	8461	1.52	9357	2.02	9783	2.52	9941
0.03	5120	0.53	7019	1.03	8485	1.53	9370	2.03	9788	2.53	9943
0.04	5160	0.54	7054	1.04	8508	1.54	9382	2.04	9793	2.54	9945
0.05	5199	0.55	7088	1.05	8531	1.55	9394	2.05	9798	2.55	9946
0.06	5239	0.56	7123	1.06	8554	1.56	9406	2.06	9803	2.56	9948
0.07	5279	0.57	7157	1.07	8577	1.57	9418	2.07	9808	2.57	9949
0.08	5319	0.58	7190	1.08	8599	1.58	9429	2.08	9812	2.58	9951
0.09	5359	0.59	7224	1.09	8621	1.59	9441	2.09	9817	2.59	9952
0.10	5398	0.60	7257	1.10	8643	1.60	9452	2.10	9821	2.60	9953
0.11	5438	0.61	7291	1.11	8665	1.61	9463	2.11	9826	2.61	9955
0.12	5478	0.62	7324	1.12	8686	1.62	9474	2.12	9830	2.62	9956
0.13	5517	0.63	7357	1.13	8708	1.63	9484	2.13	9834	2.63	9957
0.14	5557	0.64	7389	1.14	8729	1.64	9495	2.14	9838	2.64	9959
0.15	5596	0.65	7422	1.15	8749	1.65	9505	2.15	9842	2.65	9960
0.16	5636	0.66	7454	1.16	8770	1.66	9515	2.16	9846	2.66	9961
0.17	5675	0.67	7486	1.17	8790	1.67	9525	2.17	9850	2.67	9962
0.18	5714	0.68	7517	1.18	8810	1.68	9535	2.18	9854	2.68	9963
0.19	5753	0.69	7549	1.19	8830	1.69	9545	2.19	9857	2.69	9964
0.20	5793	0.70	7580	1.20	8849	1.70	9554	2.20	9861	2.70	9965
0.21	5832	0.71	7611	1.21	8869	1.71	9564	2.21	9864	2.71	9966
0.22	5871	0.72	7642	1.22	8888	1.72	9573	2.22	9868	2.72	9967
0.23	5910	0.73	7673	1.23	8907	1.73	9582	2.23	9871	2.73	9968
0.24	5948	0.74	7704	1.24	8925	1.74	9591	2.24	9875	2.74	9969
0.25	5987	0.75	7734	1.25	8944	1.75	9599	2.25	9878	2.75	9970
0.26	6026	0.76	7764	1.26	8962	1.76	9608	2.26	9881	2.76	9971
0.27	6064	0.77	7794	1.27	8980	1.77	9616	2.27	9884	2.77	9972
0.28	6103	0.78	7823	1.28	8997	1.78	9625	2.28	9887	2.78	9973
0.29	6141	0.79	7852	1.29	9015	1.79	9633	2.29	9890	2.79	9974
0.30	6179	0.80	7881	1.30	9032	1.80	9641	2.30	9893	2.80	9974
0.31	6217	0.81	7910	1.31	9049	1.81	9649	2.31	9896	2.81	9975
0.32	6255	0.82	7939	1.32	9066	1.82	9656	2.32	9898	2.82	9976
0.33	6293	0.83	7967	1.33	9082	1.83	9664	2.33	9901	2.83	9977
0.34	6331	0.84	7995	1.34	9099	1.84	9671	2.34	9904	2.84	9977
0.35	6368	0.85	8023	1.35	9115	1.85	9678	2.35	9906	2.85	9978
0.36	6406	0.86	8051	1.36	9131	1.86	9686	2.36	9909	2.86	9979
0.37	6443	0.87	8078	1.37	9147	1.87	9693	2.37	9911	2.87	9979
0.38	6480	0.88	8106	1.38	9162	1.88	9699	2.38	9913	2.88	9980
0.39	6517	0.89	8133	1.39	9177	1.89	9706	2.39	9916	2.89	9981
0.40	6554	0.90	8159	1.40	9192	1.90	9713	2.40	9918	2.90	9981
0.41	6591	0.91	8186	1.41	9207	1.91	9719	2.41	9920	2.91	9982
0.42	6628	0.92	8212	1.42	9222	1.92	9726	2.42	9922	2.92	9982
0.43	6664	0.93	8238	1.43	9236	1.93	9732	2.43	9925	2.93	9983
0.44	6700	0.94	8264	1.44	9251	1.94	9738	2.44	9927	2.94	9984
0.45	6736	0.95	8289	1.45	9265	1.95	9744	2.45	9929	2.95	9984
0.46	6772	0.96	8315	1.46	9279	1.96	9750	2.46	9931	2.96	9985
0.47	6808	0.97	8340	1.47	9292	1.97	9756	2.47	9932	2.97	9985
0.48	6844	0.98	8365	1.48	9306	1.98	9761	2.48	9934	2.98	9986
0.49	6879	0.99	8389	1.49	9319	1.99	9767	2.49	9936	2.99	9986
0.50	6915	1.00	8413	1.50	9332	2.00	9772	2.50	9938	3.00	9987
APP. 5   Tables A99
Table A8   Normal Distribution
Values of z for given values or Q>(z) [see (3), Sec. 24.8] and D(z) = 0%) - <5(-z) Example: z = 0.279 if O(z) = 61%; z = 0.860 if D(z) = 61%.
%	em	Z(O)	%		z(»)	%		Z(D)
1	-2.326	0.013	41	-0.228	0.539	81	0.878	1.311
2	-2.054	0.025	42	-0.202	0.553	82	0.915	1.341
3	-1.881	0.038	43	-0.176	0.568	83	0.954	1.372
4	-1.751	0.050	44	-0.151	0.583	84	0.994	1.405
5	-1.645	0.063	45	-0.126	0.598	85	1.036	1.440
6	-1.555	0.075	46	-0.100	0.613	86	1.080	1.476
7	-1.476	0.088	47	0.075	0.628	87	1.126	1.514
8	-1.405	0.100	48	-0.050	0.643	88	1.175	1.555
9	-1.341	0.113	49	-0.025	0.659	89	1.227	1.598
10	-1.282	0.126	50	0.000	0.674	90	1.282	1.645
11	-1.227	0.138	51	0.025	0.690	91	1.341	1.695
12	-1.175	0.151	52	0.050	0.706	92	1.405	1.751
13	-1.126	0.164	53	0.075	0.722	93	1.476	1.812
14	-1.080	0.176	54	0.100	0.739	94	1.555	1.881
15	-1.036	0.189	55	0.126	0.755	95	1.645	1.960
16	-0.994	0.202	56	0.151	0.772	96	1.751	2.054
17	-0.954	0.215	57	0.176	0.789	97	1.881	2.170
18	-0.915	0.228	58	0.202	0.806	97.5	1.960	2.241
19	-0.878	0.240	59	0.228	0.824	98	2.054	2.326
20	-0.842	0.253	60	0.253	0.842	99	2.326	2.576
21	-0.806	0.266	61	0.279	0.860	99.1	2.366	2.612
22	-0.772	0.279	62	0.305	0.878	99.2	2.409	2.652
23	-0.739	0.292	63	0.332	0.896	99.3	2.457	2.697
24	-0.706	0.305	64	0.358	0.915	99.4	2.512	2.748
25	-0.674	0.319	65	0.385	0.935	99.5	2.576	2.807
26	-0.643	0.332	66	0.412	0.954	99.6	2.652	2.878
27	-0.613	0.345	67	0.440	0.974	99.7	2.748	2.968
28	-0.583	0.358	68	0.468	0.994	99.8	2.878	3.090
29	-0.553	0.372	69	0.496	1.015	99.9	3.090	3.291
30	-0.524	0.385	70	0.524	1.036			
31	-0.496	0.399	71	0.553	1.058	99.91	3.121	3.320
32	-0.468	0.412	72	0.583	1.080	99.92	3.156	3.353
33	-0.440	0.426	73	0.613	1.103	99.93	3.195	3.390
34	-0.412	0.440	74	0.643	1.126	99.94	3.239	3.432
35	-0.385	0.454	75	0.674	1.150	99.95	3.291	3.481
36	-0.358	0.468	76	0.706	1.175	99.96	3.353	3.540
37	-0.332	0.482	77	0.739	1.200	99.97	3.432	3.615
38	-0.305	0.496	78	0.772	1.227	99.98	3.540	3.719
39	-0.279	0.510	79	0.806	1.254	99.99	3.719	3.891
40	-0.253	0.524	80	0.842	1.282			
A100 APP. 5 Tables
Table A9 t-Distribution
Values of z for given values of the distribution function F(z) (see (8) in Sec. 25.3) Example: For 9 degrees of freedom, z = 1-83 when F(z) = 0.95.
				Number of Degrees of Freedom						
										
	1	2	3	4	5	6	7	8	9	10
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.32	0.29	0.28	0.27	0.27	0.26	0.26	0.26	0.26	0.26
0.7	0.73	0.62	0.58	0.57	0.56	0.55	0.55	0.55	0.54	0.54
0.8	1.38	1.06	0.98	0.94	0.92	0.91	0.90	0.89	0.88	0.88
0.9	3.08	1.89	1.64	1.53	1.48	1.44	1.41	1.40	1.38	1.37
0.95	6.31	2.92	2.35	2.13	2.02	1.94	1.89	1.86	1.83	1.81
0.975	12.7	4.30	3.18	2.78	2.57	2.45	2.36	2.31	2.26	2.23
0.99	31.8	6.96	4.54	3.75	3.36	3.14	3.00	2.90	2.82	2.76
0.995	63.7	9.92	5.84	4.60	4.03	3.71	3.50	3.36	3.25	3.17
0.999	318.3	22.3	10.2	7.17	5.89	5.21	4.79	4.50	4.30	4.14
				Number of Degrees of Freedom						
F(z)										
	11	12	13	14	15	16	17	18	19	20
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.26	0.26	0.26	0.26	0.26	0.26	0.26	0.26	0.26	0.26
0.7	0.54	0.54	0.54	0.54	0.54	0.54	0.53	0.53	0.53	0.53
0.8	0.88	0.87	0.87	0.87	0.87	0.86	0.86	0.86	0.86	0.86
0.9	1.36	1.36	1.35	1.35	1.34	1.34	1.33	1.33	1.33	1.33
0.95	1.80	1.78	1.77	1.76	1.75	1.75	1.74	1.73	1.73	1.72
0.975	2.20	2.18	2.16	2.14	2.13	2.12	2.11	2.10	2.09	2.09
0.99	2.72	2.68	2.65	2.62	2.60	2.58	2.57	2.55	2.54	2.53
0.995	3.11	3.05	3.01	2.98	2.95	2.92	2.90	2.88	2.86	2.85
0.999	4.02	3.93	3.85	.3.79	3.73	3.69	3.65	3.61	3.58	3.55
				Number of Degrees of Freedom						
F(z)										
	22	24	26	28	30	40	50	100	200	00
0.5	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
0.6	0.26	0.26	0.26	0.26	0.26	0.26	0.25	0.25	0.25	0.25
0.7	0.53	0.53	0.53	0.53	0.53	0.53	0.53	0.53	0.53	0.52
0.8	0.86	0.86	0.86	0.85	0.85	0.85	0.85	0.85	0.84	0.84
0.9	1.32	1.32	1.31	1.31	1.31	1.30	1.30	1.29	1.29	1.28
0.95	1.72	1.71	1.71	1.70	1.70	1.68	1.68	1.66	1.65	1.65
0.975	2.07	2.06	2.06	2.05	2.04	2.02	2.01	1.98	1.97	1.96
0.99	2.51	2.49	2.48	2.47	2.46	2.42	2.40	2.36	2.35	2.33
0.995	2.82	2.80	2.78	2.76	2.75	2.70	2.68	2.63	2.60	2.58
0.999	3.50	3.47	3.43	3.41	3.39	3.31	3.26	3.17	3.13	3.09
APP. 5 Tables
A101
Table A10   Chi-square Distribution
Values of x for given values of the distribution function F(z) (see Sec. 25.3 before (17)). Example: For 3 degrees of freedom, z = 11.34 when F(z) = 0.99.
				Number of Decrees of Freedom						
m										
	1	2	3	4	5	6	7	8	9	10
0.005	0.00	0.01	0.07	0.21	0.41	0.68	0.99	1.34	1.73	2.16
0.01	0.00	0.02	0.11	0.30	0.55	0.87	1.24	1.65	2.09	2.56
0.025	0.00	0.05	0.22	0.48	0.83	1.24	1.69	2.18	2.70	3.25
0.05	0.00	0.10	0.35	0.71	1.15	1.64	2.17	2.73	3.33	3.94
0.95	3.84	5.99	7.81	9.49	1 1.07	12.59	14.07	15.51	16.92	18.31
0.975	5.02	7.38	9.35	11.14	12.83	14.45	16.01	17.53	19.02	20.48
0.99	6.63	9.21	11.34	13.28	15.09	16.81	18.48	20.09	21.67	23.21
0.995	7.88	10.60	12.84	14.86	16.75	18.55	20.28	21.95	23.59	25.19
				Number of Degrees of Freedom						
F(z)										
	11	12	13	14	15	16	17	18	19	20
0.005	2.60	3.07	.3.57	4.07	4.60	5.14	5.70	6.26	6.84	7.43
0.01	3.05	3.57	4.11	4.66	5.23	5.81	6.41	7.01	7.63	8.26
0.025	3.82	4.40	5.01	5.63	6.26	6.91	7.56	8.23	8.91	9.59
0.05	4.57	5.23	5.89	6.57	7.26	7.96	8.67	9.39	10.12	10.85
0.95	19.68	21.03	22.36	23.68	25.00	26.30	27.59	28.87	30.14	31.41
0.975	21.92	23.34	24.74	26.12	27.49	28.85	30.19	31.53	32.85	34.17
0.99	24.72	26.22	27.69	29.14	30.58	32.00	33.41	34.81	36.19	37.57
0.995	26.76	28.30	29.82	31.32	32.80	34.27	35.72	37.16	38.58	40.00
				Number of Degrees of Freedom						
F(z)										
	21	22	23	24	25	26	27	28	29	30
0.005	8.0	8.6	9.3	9.9	10.5	11.2	11.8	12.5	13.1	13.8
0.01	8.9	9.5	10.2	10.9	11.5	12.2	12.9	13.6	14.3	15.0
0.025	10.3	11.0	11.7	12.4	13.1	13.8	14.6	15.3	16.0	16.8
0.05	11.6	12.3	13.1	13.8	14.6	15.4	16.2	16.9	17.7	18.5
0.95	32.7	33.9	35.2	36.4	37.7	38.9	40.1	41.3	42.6	43.8
0.975	35.5	36.8	38.1	39.4	40.6	41.9	43.2	44.5	45.7	47.0
0.99	38.9	40.3	41.6	43.0	44.3	45.6	47.0	48.3	49.6	50.9
0.995	41.4	42.8	44.2	45.6	46.9	48.3	49.6	51.0	52.3	53.7
F(z)
Number of Degrees of Freedom
40
50
60
70
80
90
100
100 (Approximation)
\(h - 2.58)2 \(h - 2.33)2 - 1.96)2
0.005 0.01
0.025 0.05
0.95 0.975 0.99 0.995
20.7
22.2
24.4
26.5
55.8
59.3
63.7
66.8
28.0
29.7
32.4
34.8
67.5
71.4
76.2
79.5
35.5 37.5 40.5
43.2
79.1
83.3
88.4 92.0
43.3
45.4 48.8 51.7
90.5 95.0
100.4 104.2
51.2
53.5 57.2 60.4
101.9 106.6 112.3 116.3
59.2 61.8 65.6 69.1
113.1 118.1 124.1 128.3
67.3
70.1
74.2 77.9
124.3 129.6 135.8 140.2
I.64)2
5in t 1.64)2
\<h - 1.96)2
|(/i + 2.33)2
i(h + 2.58)2
In the last column, h = v2m - 1, where m is the number of degrees of freedom.
A102
APP. 5 Tables
Table All   F-Distribution with (m, n) Degrees of Freedom
Values of z for which the distribution function F(z) [see (13), Sec. 25.4] has the value 0.95 Example: For (7, 4) d.f., z = 6.09 if F(z) = 0.95.
n	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6	m = 1	m = 8	m = 9
1	161	200	216	225	230	234	237	239	241
2	18.5	19.0	19.2	19.2	19.3	19.3	19.4	19.4	19.4
3	10.1	9.55	9.28	9.12	9.01	8.94	8.89	8.85	8.81
4	7.71	6.94	6.59	6.39	6.26	6.16	6.09	6.04	6.00
5	6.61	5.79	5.41	5.19	5.05	4.95	4.88	4.82	4.77
6	5.99	5.14	4.76	4.53	4.39	4.28	4.21	4.15	4.10
7	5.59	4.74	4.35	4.12	3.97	3.87	3.79	3.73	3.68
8	5.32	4.46	4.07	3.84	3.69	3.58	3.50	3.44	3.39
9	5.12	4.26	3.86	3.63	3.48	3.37	3.29	3.23	3.18
10	4.96	4.10	3.71	3.48	3.33	3.22	3.14	3.07	3.02
11	4.84	3.98	3.59	3.36	3.20	3.09	3.01	2.95	2.90
12	4.75	3.89	3.49	3.26	3.11	3.00	2.91	2.85	2.80
13	4.67	3.81	3.41	3.18	3.03	2.92	2.83	2.77	2.71
14	4.60	3.74	3.34	3.11	2.96	2.85	2.76	2.70	2.65
15	4.54	3.68	3.29	3.06	2.90	2.79	2.71	2.64	2.59
16	4.49	3.63	3.24	3.01	2.85	2.74	2.66	2.59	2.54
17	4.45	3.59	3.20	2.96	2.81	2.70	2.61	2.55	2.49
18	4.41	3.55	3.16	2.93	2.77	2.66	2.58	2.51	2.46
19	4.38	3.52	3.13	2.90	2.74	2.63	2.54	2.48	2.42
20	4.35	3.49	3.10	2.87	2.71	2.60	2.51	2.45	2.39
22	4.30	3.44	3.05	2.82	2.66	2.55	2.46	2.40	2.34
24	4.26	3.40	3.01	2.78	2.62	2.51	2.42	2.36	2.30
26	4.23	3.37	2.98	2.74	2.59	2.47	2.39	2.32	2.27
28	4.20	3.34	2.95	2.71	2.56	2.45	2.36	2.29	2.24
30	4.17	3.32	2.92	2.69	2.53	2.42	2.33	2.27	2.21
32	4.15	3.29	2.90	2.67	2.51	2.40	2.31	2.24	2.19
34	4.13	3.28	2.88	2.65	2.49	2.38	2.29	2.23	2.17
36	4.11	3.26	2.87	2.63	2.48	2.36	2.28	2.21	2.15
38	4.10	3.24	2.85	2.62	2.46	2.35	2.26	2.19	2.14
40	4.08	3.23	2.84	2.61	2.45	2.34	2.25	2.18	2.12
50	4.03	3.18	2.79	2.56	2.40	2.29	2.20	2.13	2.07
60	4.00	3.15	2.76	2.53	2.37	2.25	2.17	2.10	2.04
70	3.98	3.13	2.74	2.50	2.35	2.23	2.14	2.07	2.02
80	3.96	3.11	2.72	2.49	2.33	2.21	2.13	2.06	2.00
90	3.95	3.10	2.71	2.47	2.32	2.20	2.11	2.04	1.99
100	3.94	3.09	2.70	2.46	2.31	2.19	2.10	2.03	1.97
150	3.90	3.06	2.66	2.43	2.27	2.16	2.07	2.00	1.94
200	3.89	3.04	2.65	2.42	2.26	2.14	2.06	1.98	1.93
1000	3.85	3.00	2.61	2.38	2.22	2.11	2.02	1.95	1.89
GO	3.84	3.00	2.60	2.37	2.21	2.10	2.01	1.94	1.88
APP. 5   Tables A103
Table All   F-Distribution with (m, n) Degrees of Freedom (continued)
Values of z for which the distribution function F(z) [see (13), Sec. 25.4] has the value 0.95
n	m = 10	m = 15	m = 20	m = 30	m = 40	m = 50	m = 100	GO
1	242	246	248	250	251	252	253	254
2	19.4	19.4	19.4	19.5	19.5	19.5	19.5	19.5
3	8.79	8.70	8.66	8.62	8.59	8.58	8.55	8.53
4	5.96	5.86	5.80	5.75	5.72	5.70	5.66	5.63
5	4.74	4.62	4.56	4.50	4.46	4.44	4.41	4.37
6	4.06	3.94	3.87	3.81	3.77	3.75	3.71	3.67
7	3.64	3.51	3.44	3.38	3.34	3.32	3.27	3.23
8	3.35	3.22	3.15	3.08	3.04	3.02	2.97	2.93
9	3.14	3.01	2.94	2.86	2.83	2.80	2.76	2.71
10	2.98	2.85	2.77	2.70	2.66	2.64	2.59	2.54
11	2.85	2.72	2.65	2.57	2.53	2.51	2.46	2.40
12	2.75	2.62	2.54	2.47	2.43	2.40	2.35	2.30
13	2.67	2.53	2.46	2.38	2.34	2.31	2.26	2.21
14	2.60	2.46	2.39	2.31	2.27	2.24	2.19	2.13
15	2.54	2.40	2.33	2.25	2.20	2.18	2.12	2.07
16	2.49	2.35	2.28	2.19	2.15	2.12	2.07	2.01
17	2.45	2.31	2.23	2.15	2.10	2.08	2.02	1.96
18	2.41	2.27	2.19	2.11	2.06	2.04	1.98	1.92
19	2.38	2.23	2.16	2.07	2.03	2.00	1.94	1.88
20	2.35	2.20	2.12	2.04	1.99	1.97	1.91	1.84
22	2.30	2.15	2.07	1.98	1.94	1.91	1.85	1.78
24	2.25	2.11	2.03	1.94	1.89	1.86	1.80	1.73
26	2.22	2.07	1.99	1.90	1.85	1.82	1.76	1.69
28	2.19	2.04	1.96	1.87	1.82	1.79	1.73	1.65
30	2.16	2.01	1.93	1.84	1.79	1.76	1.70	1.62
32	2.14	1.99	1.91	1.82	1.77	1.74	1.67	1.59
34	2.12	1.97	1.89	1.80	1.75	1.71	1.65	1.57
36	2.11	1.95	1.87	1.78	1.73	1.69	1.62	1.55
38	2.09	1.94	1.85	1.76	1.71	1.68	1.61	1.53
40	2.08	1.92	1.84	1.74	1.69	1.66	1.59	1.51
50	2.03	1.87	1.78	1.69	1.63	1.60	1.52	1.44
60	1.99	1.84	1.75	1.65	1.59	1.56	1.48	1.39
70	1.97	1.81	1.72	1.62	1.57	1.53	1.45	1.35
80	1.95	1.79	1.70	1.60	1.54	1.51	1.43	1.32
90	1.94	1.78	1.69	1.59	1.53	1.49	1.41	1.30
100	1.93	1.77	1.68	1.57	1.52	1.48	1.39	1.28
150	1.89	1.73	1.64	1.54	1.48	1.44	1.34	1.22
200	1.88	1.72	1.62	1.52	1.46	1.41	1.32	1.19
1000	1.84	1.68	1.58	1.47	1.41	1.36	1.26	1.08
cc	1.83	1.67	1.57	1.46	1.39	1.35	1.24	1.00
A104
APP. 5 Tables
Table All   F-Distribution with (m, n) Degrees of Freedom (continued)
Values of z for which the distribution function F(z) [see (13), Sec. 25.4] has the value 0.99
n	m = 1	m = 2	m = 3	m = 4	m = 5	m = 6	m = 7	m = 8	m = 9
1	4052	4999	5403	5625	5764	5859	5928	5981	6022
2	98.5	99.0	99.2	99.2	99.3	99.3	99.4	99.4	99.4
3	34.1	30.8	29.5	28.7	28.2	27.9	27.7	27.5	27.3
4	21.2	18.0	16.7	16.0	15.5	15.2	15.0	14.8	14.7
5	16.3	13.3	12.1	11.4	11.0	10.7	10.5	10.3	10.2
6	13.7	10.9	9.78	9.15	8.75	8.47	8.26	8.10	7.98
7	12.2	9.55	8.45	7.85	7.46	7.19	6.99	6.84	6.72
8	11.3	8.65	7.59	7.01	6.63	6.37	6.18	6.03	5.91
9	10.6	8.02	6.99	6.42	6.06	5.80	5.61	5.47	5.35
10	10.0	7.56	6.55	5.99	5.64	5.39	5.20	5.06	4.94
11	9.65	7.21	6.22	5.67	5.32	5.07	4.89	4.74	4.63
12	9.33	6.93	5.95	5.41	5.06	4.82	4.64	4.50	4.39
13	9.07	6.70	5.74	5.21	4.86	4.62	4.44	4.30	4.19
14	8.86	6.51	5.56	5.04	4.69	4.46	4.28	4.14	4.03
15	8.68	6.36	5.42	4.89	4.56	4.32	4.14	4.00	3.89
16	8.53	6.23	5.29	4.77	4.44	4.20	4.03	3.89	3.78
17	8.40	6.1 1	5.18	4.67	4.34	4.10	3.93	3.79	3.68
18	8.29	6.01	5.09	4.58	4.25	4.01	3.84	3.71	3.60
19	8.18	5.93	5.01	4.50	4.17	3.94	3.77	3.63	3.52
20	8.10	5.85	4.94	4.43	4.10	3.87	3.70	3.56	3.46
22	7.95	5.72	4.82	4.31	3.99	3.76	3.59	3.45	3.35
24	7.82	5.61	4.72	4.22	3.90	3.67	3.50	3.36	3.26
26	7.72	5.53	4.64	4.14	3.82	3.59	3.42	3.29	3.18
28	7.64	5.45	4.57	4.07	3.75	3.53	3.36	3.23	3.12
30	7.56	5.39	4.51	4.02	3.70	3.47	3.30	3.17	3.07
32	7.50	5.34	4.46	3.97	3.65	3.43	3.26	3.13	3.02
34	7.44	5.29	4.42	3.93	3.61	3.39	3.22	3.09	2.98
36	7.40	5.25	4.38	3.89	3.57	3.35	3.18	3.05	2.95
38	7.35	5.21	4.34	3.86	3.54	3.32	3.15	3.02	2.92
40	7.31	5.18	4.31	3.83	3.51	3.29	3.12	2.99	2.89
50	7.17	5.06	4.20	3.72	3.41	3.19	3.02	2.89	2.78
60	7.08	4.98	4.13	3.65	3.34	3.12	2.95	2.82	2.72
70	7.01	4.92	4.07	3.60	3.29	3.07	2.91	2.78	2.67
80	6.96	4.88	4.04	3.56	3.26	3.04	2.87	2.74	2.64
90	6.93	4.85	4.01	3.54	3.23	3.01	2.84	2.72	2.61
100	6.90	4.82	3.98	3.51	3.21	2.99	2.82	2.69	2.59
150	6.81	4.75	3.91	3.45	3.14	2.92	2.76	2.63	2.53
200	6.76	4.71	3.88	3.41	3.11	2.89	2.73	2.60	2.50
1000	6.66	4.63	3.80	3.34	3.04	2.82	2.66	2.53	2.43
	6.63	4.61	3.78	3.32	3.02	2.80	2.64	2.51	2.41
APP. 5 Tables
A105
Table All   F-Distribution with (m, n) Degrees of Freedom (continued)
Values of z for which the distribution function F(z) [sec (13), Sec. 25.4] has the value 0.99
11	m = 10	m = 15	m = 20	m = 30	m = 40	m = 50	m = 100	30
1	6056	6157	6209	6261	6287	6303	6334	6366
2	99.4	99.4	99.4	99.5	99.5	99.5	99.5	99.5
3	27.2	26.9	26.7	26.5	26.4	26.4	26.2	26.1
4	14.5	14.2	14.0	13.8	13.7	13.7	13.6	13.5
5	10.1	9.72	9.55	9.38	9.29	9.24	9.13	9.02
6	7.87	7.56	7.40	7.23	7.14	7.09	6.99	6.88
7	6.62	6.31	6.16	5.99	5.91	5.86	5.75	5.65
8	5.81	5.52	5.36	5.20	5.12	5.07	4.96	4.86
9	5.26	4.96	4.81	4.65	4.57	4.52	4.42	4.31
10	4.85	4.56	4.41	4.25	4.17	4.12	4.01	3.91
11	4.54	4.25	4.10	3.94	3.86	3.81	3.71	3.60
12	4.30	4.01	3.86	3.70	3.62	3.57	3.47	3.36
13	4.10	3.82	3.66	3.51	3.43	3.38	3.27	3.17
14	3.94	3.66	3.51	3.35	3.27	3.22	3.11	3.00
15	3.80	3.52	3.37	3.21	3.13	3.08	2.98	2.87
16	3.69	3.41	3.26	3.10	3.02	2.97	2.86	2.75
17	3.59	3.31	3.16	3.00	2.92	2.87	2.76	2.65
18	3.51	3.23	3.08	2.92	2.84	2.78	2.68	2.57
19	3.43	3.15	3.00	2.84	2.76	2.71	2.60	2.49
20	3.37	3.09	2.94	2.78	2.69	2.64	2.54	2.42
22	3.26	2.98	2.83	2.67	2.58	2.53	2.42	2.31
24	3.17	2.89	2.74	2.58	2.49	2.44	2.33	2.21
26	3.09	2.81	2.66	2.50	2.42	2.36	2.25	2.13
28	3.03	2.75	2.60	2.44	2.35	2.30	2.19	2.06
30	2.98	2.70	2.55	2.39	2.30	2.25	2.13	2.01
32	2.93	2.65	2.50	2.34	2.25	2.20	2.08	1.96
34	2.89	2.61	2.46	2.30	2.21	2.16	2.04	1.91
36	2.86	2.58	2.43	2.26	2.18	2.12	2.00	1.87
38	2.83	2.55	2.40	2.23	2.14	2.09	1.97	1.84
40	2.80	2.52	2.37	2.20	2.1 1	2.06	1.94	1.80
50	2.70	2.42	2.27	2.10	2.01	1.95	1.82	1.68
60	2.63	2.35	2.20	2.03	1.94	1.88	1.75	1.60
70	2.59	2.31	2.15	1.98	1.89	1.83	1.70	1.54
80	2.55	2 27	2 12	1.94	1.85	1.79	1.65	1.49
90	2.52	2.24	2.09	1.92	1.82	1.76	1.62	1.46
100	2.50	2.22	2.07	1.89	1.80	1.74	1.60	1.43
150	2.44	2.16	2.00	1.83	1.73	1.66	1.52	1.33
200	2.41	2.13	1.97	1.79	1.69	1.63	1.48	1.28
1000	2.34	2.06	1.90	1.72	1.61	1.54	1.38	1.11
cc	2.32	2.04	1.88	1.70	1.59	1.52	1.36	1.00
A106
APP. 5 Tables
Table A12 Distribution Function F[x) - P(T § x) of the Random Variable T in Section 25.8
	n
X	= 3
	0.
0	167
1	500
	n
X	= 20
	0.
50	001
51	002
52	002
53	003
54	004
55	005
56	006
57	007
58	008
59	010
60	012
61	014
62	017
63	020
64	023
65	027
66	032
67	037
68	043
69	049
70	056
71	064
72	073
73	082
74	093
75	104
76	117
77	130
78	144
79	159
80	176
81	193
82	211
83	230
84	250
85	271
86	293
87	315
88	339
89	362
90	387
91	411
92	436
93	462
94	487
	n
X	=4
	0.
0	042
1	167
2	375
	
	n
X	= 19
	0.
43	001
44	002
45	002
46	003
47	003
48	004
49	005
50	006
51	008
52	010
53	012
54	014
55	017
56	021
57	025
58	029
59	034
60	040
61	047
62	054
63	062
64	072
65	082
66	093
67	105
68	119
69	133
70	149
71	166
72	184
73	203
74	223
75	245
76	267
77	290
78	314
79	339
80	365
81	391
82	418
8?	445
84	473
85	500
	n
X	=5
	0.
0	008
1	042
2	117
3	242
4	408
	
	n
X	= 18
	0.
38	001
39	002
40	003
41	003
42	004
43	005
44	007
45	009
46	011
47	013
48	016
49	020
50	024
51	029
52	034
53	041
54	048
55	056
56	066
57	076
58	088
59	100
60	115
61	130
62	147
63	165
64	184
65	205
66	227
67	250
68	275
69	300
70	327
71	354
72	383
73	411
74	441
75	470
76	500
	n
X	=6
	0.
0	001
1	008
2	028
3	068
4	136
5	235
6	360
7	500
	
	n
X	= 17
	0.
32	001
33	002
34	002
35	003
36	004
37	005
38	007
39	009
40	011
41	014
42	017
43	021
44	026
45	032
46	038
47	046
48	054
49	064
50	076
51	088
52	102
53	118
54	135
55	154
56	174
57	196
58	220
59	245
60	271
61	299
62	328
63	358
64	388
65	420
66	452
67	484
	n
X	=7
	0.
1	001
2	005
3	015
4	035
5	068
6	119
7	191
8	281
9	386
10	500
	n
X	= 16
	0.
27	001
28	002
29	002
30	003
31	004
32	006
33	008
34	010
35	013
36	016
37	021
38	026
39	032
40	039
41	048
42	058
43	070
44	083
45	097
46	114
47	133
48	153
49	175
50	199
51	225
52	253
53	282
54	313
55	345
56	378
57	412
58	447
59	482
	n
X	= 8
	0.
2	001
3	003
4	007
5	016
6	031
7	054
8	089
9	138
10	199
11	274
12	360
13	452
	n
X	= 15
	0.
23	001
24	002
25	003
26	004
27	006
28	008
29	010
30	014
31	018
32	023
33	029
34	037
35	046
36	057
37	070
38	084
39	101
40	120
41	141
42	164
43	190
44	218
45	248
46	279
47	313
48	349
49	385
50	423
51	461
52	500
X	n = 9
	0.
4	001
5	003
6	006
7	012
8	022
9	038
10	060
11	090
12	130
13	179
14	23S
15	306
16	381
17	460
	n
X	= 14
	0.
18	001
19	002
20	002
21	003
22	005
23	007
24	010
25	013
26	018
27	024
28	031
29	040
30	051
31	063
32	079
33	096
34	117
35	140
36	165
37	194
38	225
39	259
40	295
41	334
42	374
43	415
44	457
45	500
	n
X	= 10
	0.
6	001
7	002
8	005
9	008
10	014
11	023
12	036
13	054
14	078
15	108
16	146
17	190
18	242
19	300
20	364
21	431
22	500
	n
X	= 13
	0.
14	001
15	001
16	002
17	003
18	005
19	007
20	011
21	015
22	021
23	029
24	038
25	050
26	064
27	082
28	102
29	126
30	153
31	184
32	218
33	255
34	295
35	338
36	383
37	429
38	476
	n
X	= 11
	0.
8	001
9	002
10	003
11	005
12	008
13	013
14	020
15	030
16	043
17	060
18	082
19	109
20	141
21	179
22	223
23	271
24	324
25	381
26	440
27	500
	n
X	= 12
	0.
11	001
12	002
13	003
14	004
15	007
16	010
17	016
IS	022
19	031
20	043
21	058
22	076
23	098
24	125
25	155
26	190
27	230
28	273
29	319
30	369
31	420
32	473
PHOTO CREDITS
Pan A Opener: John Sohm/Chromoshohm/Photo Researchers.
Part B Opener: Lester Lcfkowitz/Corbis Images.
Part C Opener: Science Photo Library/Photo Researchers.
Part D Opener: Lester Lefkowitz/Corbis Images.
Part E Opener: Walter Hodges/Stone/Getty Images.
Chapter 19, Figure 434: From Dennis Sharp, ARCHITECKTUR. ©1973 der
deutschsprachigen Ausgabe Edition Praeger GmbH, München. Part F Opener: Charles O'Rear/Corbis Images. Part G Opener: Greg Pease/Stone/Getty Images. Appendix 1 Opener: Patrick Bennett/Corbis Images. Appendix 2 Opener: Richard T. Nowitz/Corbis Images. Appendix 3 Opener: Chris Kapolka/Stone/Getty Images. Appendix 4 Opener: John Olson/Corbis Images.
PI
INDEX
Page numbers Al, A2, A3,      refer to App. 1 to App. 5 at the end of the book.
Abel 78 Absolute
convergence 667, 697
frequency 994, 1000
value 607 Absolutely integrable 508 Acceleration 395, 995 Acceptance sampling 1073 Adams-Bashforth methods 899 Adams-Moulton methods 900 Adaptive 824 Addition of
complex numbers 603
matrices 275
means 1038
normal random variables 1050
power series 174, 680
variances 1039
vectors 276, 324, 367 Addition rule 1002 Adiabatic 561 ADI method 915 Adjacency matrix 956 Adjacent vertices 955 Airfoil 732
Airy's equation 552, 904 Algebraic multiplicity 337, 865 Algorithm 777, 783
Dijkstra 964
efficient 962
Ford-Fulkerson 979
Gauss 837
Gauss-Seidel 848
Greedy 967
Kruskal 967
Moore 960
polynomially bounded 962 Prim 971
Runge-Kutta 892, 904 stable, unstable 783 Aliasing 526
Allowable number of defectives 1073 Alternating
direction implicit method 915
path 983 Alternative hypothesis 1058 Ampere 92 Amplification 89 Amplitude spectrum 506 Analytic function 175, 617, 681 Analytic at infinity 711 Angle
between curves 35
between vectors 372 Angular speed 381, 765 Annulus 613 Anticommutative 379 AOQ, AOQL 1075-1076 Approximate solution of
differential equations 9, 886-934
eigenvalue problems 863-882
equations 787-796
systems of equations 833-858 Approximation
least squares 860
polynomial 797
trigonometric 502 A priori estimate 794 AQL 1074 Arc of a curve 391 Archimedian principle 68 Arc length 393 Arctan 634 Area 435, 442, 454 Argand diagram 605 Argument 607 Artificial variable 949 Assignment problem 982 Associated Legendre functions 182 Asymptotically
equal 191, 1009
normal 1057
stable 148
12 Index
Attractive 148
Augmented matrix 288, 833
Augmenting path 975, 983
theorem 977, 984 Autonomous 31, 151 Average (see Mean value) Average outgoing quality 1075 Axioms of probability 1001
B
Back substitution 289, 834 Backward
differences 807
edge 974, 976
Euler method 896, 907 Band matrix 914 Bashforth method 899 Basic
feasible solution 942, 944
variables 945 Basis 49, 106, 113, 138, 300, 325, 360 Beam 120, 547 Beats 87
Bellman optimality principle 963 Bell-shaped curve 1026 Bernoulli 30
distribution 1020
equation 30
law of large numbers 1032
numbers 690 Bessel 189
equation 189, 204
functions 191, 198, 202. 207, A94
functions, tables A94
inequality 215, 504 Beta function A64 Bezier curves 816 BFS 960
Bijective mapping 729 Binary 782 Binomial
coefficients 1009
distribution 1020, A96
series 689
theorem 1010 Binormal (Fig. 210) 397 Bipartite graph 982, 985 Birthday problem 1010
Bisection method 796 Bolzano-Weierstrass theorem A91 Bonnet 181 Boundary 433
conditions 203, 540, 558, 571, 587
point 433, 613
value problem 203, 558 Bounded
domain 646
function 38
region 433
sequence A69 Boxplot 995 Branch
cut 632
point 746 Breadth first search 960 Buoyance force 68
C
Cable 52, 198, 593
CAD (Computer aided design) 810
Cancellation law 321
Cantor-Dedekind axiom A69
Capacitance 92
Capacitor 92
Capacity
of a cut set 976
of an edge 973 Cardano 602 Cardioid 443
Cartesian coordinates 366, 604
CAS (Computer algebra system) vii, 777
Catenary 399
Cauchy 69
convergence principle 667, A90
determinant 112
-Goursat theorem 647
-Hadamard formula 676
inequality 660
integral formula 654
integral theorem 647, 652
method of steepest descent 938
principal value 719, 722
product 680
-Riemann equations 37, 618, 621 -Schwarz inequality 326, 859 Cayley transformation 739
Index
Center 143
of a graph 973
of gravity 436, 457
of a power series 171 Central
differences 808
limit theorem 1057
moments 1019 Centrifugal, centripetal 396 Cgs system: Front cover Chain rule 401 Characteristic
determinant 336, 864
equation 59, 111, 336, 551, 864
function 542, 574
of a partial differential equation 551
polynomial 336, 864
value 324, 864
vector 324, 864 Chebyshev polynomials 209 Chinese postman problem 963 Chi-square 1055, 1077, A101 Cholesky's method 843 Chopping 782 Chromatic number 987 Circle 391
of convergence 675 Circuit 95 Circular
disk 613
helix 391, 394
membrane 580 Circulation 764 Cissoid 399 Clairaut equation 34 Class intervals 994 Closed
disk 613
integration formula 822, 827
interval A69
path 959
point set 613
region 433 Coefficient matrix 288, 833 Coefficients of a
differential equation 46
power series 171
system of equations 287, 833 Cofactor 309 Collatz's theorem 870
Column 275
space 300
sum norm 849
vector 275 Combination 1007 Combinatorial optimization 954-986 Comparison test 668 Complement 613, 988 Complementary
error function A64
Fresnel integrals A65
sine integral A66 Complementation rule 1002 Complete
bipartite graph 987
graph 958
matching 983
orthonormal set 214 Complex
conjugate numbers 605
exponential function 57, 623
Fourier integral 519
Fourier series 497
function 614
hyperbolic functions 628, 743 impedance 98 indefinite integral 637 integration 637-663, 712-725 line integral 633 logarithm 630, 688 matrices 356 number 602 plane 605
plane, extended 710 potential 761 sequence 664 series 666 sphere 710
trigonometric functions 626, 688 trigonometric polynomial 524 variable 614 vector space 324, 359
Complexity 961
Component 366, 374
Composite transformation 281
Compound interest 9
Compressible fluid 412
Computer
aided design (CAD) 810 algebra system (CAS) vii, 777
14
Index
Computer (Cant.)
graphics 287
software (see Software) Conchoid 399 Condition number 855 Conditionally convergent 667 Conditional probability 1003 Conduction of heat 465, 552, 757 Cone 406, 448 CONF 1049 Confidence
intervals 1049-1058
level 1049
limits 1049 Conformal mapping 730, 754 Conic sections 355 Conjugate
complex numbers 605
harmonic function 622 Connected
graph 960
set 613 Conservative 415, 428 Consistent equations 292, 303 Constraints 937 Consumer's risk 1075 Continuity
of a complex function 615
equation 413
of a vector function 387 Continuous
distribution 1011
random variable 1011, 1034 Contour integral 647 Contraction 789 Control
chart 1068
limit 1068
variables 936 Convergence
absolute 667
circle of 675
conditional 667
interval 172, 676
of an iterative process 793, 848
mean 214
mean-square 214
in norm 214
principle 667
radius 172
Convergence (Cont.)
of a sequence 386, 664
of a series 171, 666
superlinear 795
tests 667-672
uniform 691 Conversion of an ODE to a system 134 Convolution 248, 523 Cooling 14
Coordinate transformations A71, A84 Coordinates
Cartesian 366, 604
curvilinear A71
cylindrical 587, A71
polar 137, 443, 580, 607
spherical 588, A71 Coriolis acceleration 396 Corrector 890, 900 Correlation analysis 1089 Cosecant 627, A62 Cosine
of a complex variable 627, 688, 743
hyperbolic 688
integral A66, A95
of a real variable A60 Cotangent 627, A62 Coulomb 92
law 409 Covariance 1039, 1085 Cramer's rule 306-307, 312 Crank-Nicolson method 924 Critical
damping 65
point 31, 142, 730
region 1060 Cross product 377 Grout's method 841 Cubic spline 811 Cumulative
distribution function 1011
frequency 994 Curl 414, 430, 472. A71 Curvature 397 Curve 389
arc length of 393
fitting 859
orientation of 390
piecewise smooth 421
rectifiable 393
simple 391
Index
Curve (Cont.)
smooth 421, 638
twisted 391 Curvilinear coordinates A71 Cut set 976 Cycle 959 Cycloid 399 Cylinder 446
flow around 763, 767 Cylindrical coordinates 587, A71
D
D'AlemberTs solution 549, 551 Damping 64, 88 Dantzig 944 DATA DESK 991 Decay 5
Decreasing sequence A69 Decrement 69 Dedekind A69 Defect 337 Defective item 1073 Definite complex integral 639 Definiteness 356 Deformation of path 649 Degenerate feasible solution 947 Degree of
precision 822
a vertex 955 Degrees of freedom 1052, 1055, 1066 Deleted neighborhood 712 Delta
Dirac 242
Kronecker 210, A83 De Moivre 610
formula 610
limit theorem 1031 De Morgan's laws 999 Density 1014 Dependent
linearly 49, 74, 106, 297, 300, 325
random variables 1036 Depth first search 960 Derivative
of a complex function 616, 658
directional 404
left-hand 484
right-hand 484
of a vector function 387
DERIVE 778 Descartes 366 Determinant 306, 308
Cauchy 112
characteristic 336, 864
of a matrix 305
of a matrix product 321
Vandermondc 112 DFS 960
Diagonalization 351 Diagonal matrix 284 Diagonally dominant matrix 868 Diameter of a graph 973 Differences 802, 804, 807-808 Difference table 803 Differentiable complex function 616 Differential 19, 429
form 20, 429
geometry 389
operator 59 Differential equation (ODE and PDE)
Airy 552, 904
Bernoulli 30
Bessel 189, 204
Cauchy-Riemann 37, 618, 621
with constant coefficients 53, 111
elliptic 551, 909
Euler-Cauchy 69, 185
exact 20
homogeneous 27, 46, 105, 535 hyperbolic 551, 928 hypergeometric 188 Laguerre 257
Laplace 407, 465. 536, 579, 587, 910 Lcgendre 177, 204, 590 linear 26. 45, 105, 535 nonhomogeneous 27, 46, 105, 535 nonlinear 45, 151, 535 numeric methods for 886-934 ordinary 4
parabolic 551, 909, 922
partial 535
Poisson 910, 918
separable 12
Sturm-Liouville 203
of vibrating beam 547, 552
of vibrating mass 61, 86, 135, 150, 243, 261,
342, 499 of vibrating membrane 569-586 of vibrating string 538
16
Differentiation
analytic functions 691 complex functions 616 Laplace transforms 254 numeric 827 power scries 174, 680 series 696
vector functions 387 Diffusion equation 464, 552 Diffusivity 552 Digraph 955 Dijkstra's algorithm 964 Dimension of vector space 300, 325, 369 Diodes 399 Dirac's delta 242 Directed
graph 955
line segment 364
path 982 Directional derivative 404 Direction field 10 Direct method 845 Dirichlet 467
discontinuous factor 509
problem 467, 558, 587, 915 Discharge of a source 767 Discrete
Fourier transform 525
random variable 1011, 1033
spectrum 507, 524 Disjoint events 998 Disk 613 Dissipative 429 Distribution 1010
Bernoulli 1020
binomial 1020, A96
chi-square 1055, A101
continuous 1011
discrete 1011, 1033
Fisher's F- 1066, A102
-free test 1080
function 1011, 1032
Gauss 1026
hypergeometric 1024
marginal 1035
multinomial 1025
normal 1026, 1047-1057, 1062-1067, A98 Poisson 1022, 1073, A97 Student's t- 1053, A100 two-dimensional 1032
Distribution {Cant.)
uniform 1015, 1017. 1034 Divergence
theorem of Gauss 459
of vector fields 410, A72 Divergent
sequence 665
series 171, 667 Divided differences 802 Division of complex numbers 604, 609 Domain 401, 613, 646 Doolittle's method 841 Dot product 325, 371 Double
Fourier series 576
integral 433
labeling 968
precision 782 Driving force 84 Drumhead 569 Duffing equation 159 Duhamel's formula 597
E
Eccentricity of a vertex 973 Echelon form 294 Edge 955
coloring 987
incidence list 957 Efficient algorithm 962 Eigenbasis 349
Eigenfunction 204, 542, 559, 574
expansion 210 Eigenspace 336, 865 Eigenvalue problems for
matrices (Chap. 8) 333-363
matrices, numerics 863-882
ODEs (Sturm-Liouville problems) 203-216
PDEs 540-593
systems of ODEs 130-165 Eigenvector 334, 864 EISPACK 778 Elastic membrane 340 Electrical network {see Networks) Electric circuit {see Circuit) Electromechanical analogies 96 Electromotive force 91 Electrostatic
field 750
Index
Electrostatic (Cont.)
potential 588-592, 750 Element of matrix 273 Elementary
matrix 296
operations 292 Elimination of first derivative 197 Ellipse 390 Ellipsoid 449 Elliptic
cylinder 448
paraboloid 448
partial differential equation 551, 909 Empty set 998
Engineering system: Front cover Entire function 624, 661, 711 Entry 273, 309 Equality of
complex numbers 602
matrices 275
vectors 365 Equally likely 1000 Equipotential
lines 750, 762
surfaces 750 Equivalence relation 296 Equivalent linear systems 292 Erf 568, 690, A64, A95 Error 783
bound 784
estimation 785
function 568, 690, A64, A95
propagation 784
Type I, Type II 1060 Essential singularity 708 Estimation of parameters 1046-1057 Euclidean
norm 327
space 327 Euler 69
backward methods lor ODEs 896, 907 beta function A64
-Cauchy equation 69. 108, 116, 185, 589 -Cauchy method 887, 890, 903 constant 200
formula 58, 496, 624, 627, 687
formulas for Fourier coefficients 480, 487
graph 963
numbers 690
method for systems 903
Euler (Cont.)
trail 963 Evaporation 18 Even function 490 Event 997
Everett interpolation formula 809 Exact
differential equation 20 differential form 20, 429 Existence theorem
differential equations 37, 73, 107, 109, 137, 175
Fourier integral 508
Fourier series 484
Laplace transforms 226
Linear equations 302 Expectation 1016 Experiment 997 Explicit solution 4 Exponential
decay 5
function, complex 57, 623
function, real A60
growth 5,31
integral A66 Exposed vertex 983 Extended complex plane 710, 736 Extension, periodic 494 Extrapolation 797 Extremum 937
F
Factorial function 192, 1008, A95
Failure 1021
Fair die 1000
Falling body 8
False position 796
Family of curves 5, 35
Faraday 92
Fast Fourier transform 526 F-distribution 1066, A102 Feasible solution 942 Fehlberg 894 Fibonacci numbers 683 Field
conservative 415, 428 of force 385
gravitational 385, 407, 411, 587
18
Field (Cont.)
irrotational 415, 765
scalar 384
vector 384
velocity 385 Finite complex plane 710 First
fundamental form 457 Green's formula 466 shilling theorem 224
Fisher, R. A. 1047. 1066, 1077 F-distribution 1066, A102
Fixed
decimal point 781 point 736, 781, 787
Flat spring 68
Floating point 781
Flow augmenting path 975
Flows in networks 973-981
Fluid flow 412, 463, 761
Flux 412. 450 integral 450
Folium of Descartes 399
Forced oscillations 84, 499
Ford-Fulkerson algorithm 979
Forest 970
Form
Hermitian 361 quadratic 353 skew-Hermitian 361
Forward
differences 804 edge 974. 976
Four-color theorem 987
Fourier 477
-Bessel series 213, 583 coefficients 480, 487 coefficients, complex 497 constants 210 cosine integral 511 cosine series 491 cosine transform 514, 529 double series 576 half-range expansions 494 integral 508, 563 integral, complex 519 -Legendre series 212, 590 matrix 525 series 211, 480, 487 series, complex 497
Fourier {Cont.)
series, generalized 210 sine integral 51 1 sine series 491, 543 sine transform 514, 530 transform 519, 531, 565 transform, discrete 525 transform, fast 526
Fractional linear transformation 734
Fraction defective 1073
Fredholm 201
Free
fall 18
oscillations 61 Frenet formulas 400 Frequency 63
of values in samples 994 Fresncl integrals 690, A65 Friction 18-19 Frohenius 182
method 182
norm 849
theorem 869 Fulkcrson 979 Full-wave rectifier 248 Function
analytic 175, 617
Bessel 191, 198, 202, 207, A94
beta A64
bounded 38
characteristic 542, 574
complex 614
conjugate harmonic 622
entire 624, 661, 71 I
error 568, 690, A64, A95
even 490
exponential 57, 623, A60 factorial 192, 1008, A95 gamma 192, A95 Hankel 202
harmonic 465, 622, 772 holomorphic 617 hyperbolic 628, 743, A62 hypergeometric 188 inverse hyperbolic 634 inverse trigonometric 634 Legendre 177 logarithmic 630, A60 meromorphic 711 Neumann 201
Index
Function (Cont.) odd 490
orthogonal 205, 482 orthonormal 205, 210 periodic 478 probability 1012, 1033 rational 617 scalar 384 space 382 staircase 248 step 234
trigonometric 626, 688, A60
unit step 234
vector 384 Function space 327 Fundamental
form 457
matrix 139
mode 542
period 485
system 49, 106, 113, 138 theorem of algebra 662
G
Galilei 15
Gamma function 192, A95
GAMS 778
Gauss 188
distribution 1026 divergence theorem 459 elimination method 289, 834 hypergeometric equation 188 integration formula 826 -Jordan elimination 317, 844 least squares 860, 1084 quadrature 826 -Seidel iteration 846, 913
General
powers 632
solution 64, 106, 138, 159 Generalized
Fourier series 210
function 242
solution 545
triangle inequality 608 Generating function 181, 216, 258 Geometric
multiplicity 337
series 167, 668, 673, 687, 692
Gerschgorin's theorem 866 Gibbs phenomenon 490, 510 Global error 887 Golden Rule 15, 23 Goodness of fit 1076 Gösset 1066 Goursat 648, A88 Gradient 403, 415, 426, A72
method 938 Graph 955
bipartite 982
complete 958
Euler 963
planar 987
sparse 957
weighted 959 Gravitation 385, 407, 411, 587 Greedy algorithm 967 Greek alphabet: Back cover Green 439
formulas 466
theorem 439, 466 Gregory-Newton formulas 805 Growth restriction 225 Guldin's theorem 458
H
Hadamard's formula 676 Half-life time 9 Half-plane 613 Half-range Fourier series 494 Half-wave rectifier 248, 489 Halving 819, 824 Hamiltonian cycle 960 Hanging cable 198 Hankel functions 202 Hard spring 159 Harmonic
conjugate 622
function 465, 622, 772
oscillation 63
series 670 Heat
equation 464, 536, 553, 757, 923 potential 758 Heaviside 221
expansions 245 formulas 247
no
Index
Heavisidc (Cont.)
function 234 Helicoid 449 Helix 391, 394, 399 Heimholt/ equation 572 Henry 92 Hermitc
interpolation 816
polynomials 216 Hermitian 357, 361 Hertz 63
Hesse's normal form 375 Heun's method 890 High-frequency line equations 594 Hilbert 201, 326
matrix 858
space 326 Histogram 994 Holomorphic 617 Homogeneous
differential equation 27, 46, 105, 535
system of equations 288, 304, 833 Hooke's law 62
Householder's tridiagonalization 875
Hyperbolic-differential equation 551, 909, 928 functions, complex 628, 743 functions, real A62 paraboloid 448
partial differential equations 551, 928
spiral 399 Hypergeometric
differential equation 188
distribution 1024
functions 188
series 188 Hypocycloid 399 Hypothesis 1058
I
Idempotent matrix 286 Identity
of Lagrange 383
matrix (see Unit matrix)
theorem for power series 679
transformation 736
trick 351 Ill-conditioned 851 Image 327, 729
Imaginary
axis 604
part 602
unit 602 Impedance 94, 98 Implicit solution 20 Improper integral 222, 719, 722 Impulse 242 IMSL 778 Incidence
list 957
matrix 958 Inclusion theorem 868 Incomplete gamma function A64 Incompressible 765 Inconsistent equations 292, 303 Increasing sequence A69 Indefinite
integral 637, 650
integration 640 Independence of path 426, 648 Independent
events 1004
random variables 1036 Indicial equation 184 Indirect method 845 Inductance 92 Inductor 92 Inequality
Bessel 215, 504
Cauchy 660
ML- 644
Schur 869
triangle 326, 372, 608 Infinite
dimensional 325
population 1025, 1045
sequence 664
series (see Series) Infinity 710, 736 Initial
condition 6, 48, 137, 540
value problem 6, 38, 48, 107, 886, 902 Injective mapping 729 Inner product 325, 359, 371
space 326 Input 26, 84, 230 Instability (see Stability) Integral
contour 647
Index
Integral (Cant.) definite 639 double 433 equation 252 Fourier 508, 563 improper 719, 722 indefinite 637, 650 line 421, 633 surface 449
theorems, complex 647, 654
theorems, real 439, 453, 469
transform 221, 513
triple 458 Integrating factor 23 Integration
complex functions 637-663, 701-727
Laplace transforms255
numeric 817-827
power series 680
scries 695 Integro-differential equation 92 Interest 9, 33 Interlacing of zeros 197 Intermediate value theorem 796 Interpolation 797-815
Hermite 816
Lagrange 798
Newton 802, 805, 807
spline 811 Interquartile range 995 Intersection of events 998 Interval
closed A69
of convergence 172, 676
estimate 1046
open 4, A69 Invariant subspace 865 Inverse
hyperbolic functions 634
mapping principle 733
of a matrix 315, 844
trigonometric functions 634 Inversion 735 Investment 9, 33 Irreducible 869 Irregular boundary 919 Irrotational 415, 765 Isocline 10
Isolated singularity 707 Isotherms 758
Iteration
for eigenvalues 872 for equations 787-794 Gauss-Seidel 846, 913 Jacobi 850 Picard 41
J
Jacobian 436, 733
Jacobi iteration 850
Jerusalem, Shrine of the Book 814
Jordan 316
Joukowski airfoil 732
K
Kirchhoff s laws 92, 973 Kronecker delta 210, A83 Kmskal's algorithm 967 Kutta 892
L
k, k, L 853 L2 863 Labeling 968 Lagrange 50
identity of 383
interpolation 798 Laguerre polynomials 209, 257 Lambert's law 43 LAPACK 778 Laplace 221
equation 407, 465, 536, 579, 587, 910
integrals 512
limit theorem 1031
operator 408
transform 221, 594 Laplacian 443, A73 Latent root 324 Laurent series 701, 712 Law of
absorption 43
cooling 14
gravitation 385
large numbers 1032
mass action 43
the mean (see Mean value theorem) LC-circuit 97
112
Index
LCL 1068
Least squares 860, 1084 Lebesgue 863 Left-hand
derivative 484
limit 484 Left-handed 378 Legendre 177
differential equation 177, 204, 590
functions 177
polynomials 179. 207, 590, 826 Leibniz 14
convergence test A70 Length
of a curve 393
of a vector 365 Leonardo of Pisa 638 Leontief 344 Leslie model 341 Libby 13
Liebmann's method 913 Likelihood function 1047 Limit
of a complex function 615
cycle 157
left-hand 484
point A90
right-hand 484
of a sequence 664
vector 386
of a vector function 387 Lineal element 9 Linear
algebra 271-363
combination 106, 325
dependence 49, 74, 106, 108. 297, 325
differential equation 26, 45. 105, 535
element 394, A72
fractional transformation 734
independence 49. 74, 106, 108, 297, 325
interpolation 798
operator 60
optimization 939
programming 939
space (see Vector space)
system of equations 287, 833
transformation 281, 327 Linearization of systems of ODEs 151 Line integral 421, 633 Lines of force 751
UNPACK 779 Liouville 203
theorem 661 Lipschitz condition 40 List 957 Ljapunov 148 Local
error 887
minimum 937 Logarithm 630, 688, A60 Logarithmic
decrement 69
integral A66
spiral 399 Logistic population law 30 Longest path 959 Loss of significant digits 785 Lotka-Volterra population model 154 Lot tolerance per cent defective 1075 Lower
control limit 1068
triangular matrix 283 LTPD 1075 LU-factorization 841
M
Maclaurin 683
series 683
trisectrix 399 Magnitude of a vector (see Length) Main diagonal 274, 309 Malthus's law 5, 31 MAPLE 779 Mapping 327. 729 Marconi 63
Marginal distributions 1035
Markov process 285.341
Mass-spring system 61, 86, 135, 150, 243, 252,
261, 342, 499 Matching 985 MATHCAD 779 MATHEMATICA 779 Mathematical expectation 1019. 1038 MATLAB 779 Matrix
addition 275
augmented 288. 833
band 914
Index
Matrix (Cont.) diagonal 284
eigenvalue problem 333-363, 863-882
Hermitian 357
identity (see Unit matrix)
inverse 315
inversion 315, 844
multiplication 278, 321
nonsingular 315
norm 849. 854
normal 362, 869
null (see Zero matrix)
orthogonal 345
polynomial 865
scalar 284
singular 315
skew-Hermilian 357
skew-symmetric 283, 345
sparse 812, 912
square 274
stochastic 285
symmetric 283, 345
transpose 282
triangular 283
tridiagonal 812, 875, 914
unit 284
unitary 357
zero 276 Max-flow min-cut theorem 978 Maximum 937
flow 979
likelihood method 1047
matching 983
modulus theorem 772
principle 773 Mean convergence 214 Mean-square convergence 214 Mean value of a (an)
analytic function 771
distribution 1016
function 764
harmonic function 772
sample 996 Mean value theorem 402, 434, 454 Median 994, 1081 Membrane 569-586 Meromorphic function 711 Mesh incidence matrix 278 Method of
false position 796
113
Method of (Cont.)
least squares 860, 1084
moments 1046
steepest descent 938
undetermined coefficients 78, 117. 160
variation of parameters 98, 118, 160 Middle quartile 994 Minimum 937, 942, 946 MINITAB 991 Minor 309 Mixed
boundary value problem 558, 587, 759, 917
triple product 381 Mixing problem 13, 130, 146. 163, 259 Mks system: Front cover ML-inequality 644 Möbius 453
strip 453, 456
transformation 734 Mode 542. 582
Modeling 2, 6. 13, 61, 84, 130, 159, 222, 340, 499,
538, 569, 750-767 Modified Bessel functions 203 Modulus 607 Molecule 912 Moivre's formula 610 Moment
central 1019
of a distribution 1019
of a force 380
generating function 1026
of inertia 436, 455, 457
of a sample 1046
vector 380 Monotone sequence A69 Moore's shortest path algorithm 960 Morera's theorem 661 Moulton 900
Moving trihedron (see Trihedron) M-test for convergence 969 Multinomial distribution 1025 Multiple point 391 Multiplication of
complex numbers 603, 609
determinants 322
matrices 278, 321
means 1038
power series 174, 680
vectors 219, 371, 377 Multiplication rule for events 1003
114
Index
Multiplicity 337, 865 Multiply connected domain 646 Multistep method 898 "Multivalued function" 615 Mutually exclusive events 998
N
Nabla 403 NAG 779 Natural
frequency 63
logarithm 630, A60
spline 812 Neighborhood 387, 613 Nested form 786 NETLIB 779
Networks 132, 146, 162, 244, 260, 263, 277, 331
in graph theory 973 Neumann, C. 201
functions 201
problem 558, 587, 917 Newton 14
-Cotes formulas 822
interpolation formulas 802, 805, 807
law of cooling 14
law of gravitation 385
method 800
-Raphson method 800
second law 62 Neyman 1049, 1058 Nicolson 924 Nicomedes 399 Nilpotent matrix 286 NIST 779 Nodal
incidence matrix 277
line 574 Node 142, 797 Nonbasic variables 945 Nonconservative 428 Nonhomogcneous
differential equation 27, 78, 116. 159, 305, 535
system of equations 288, 304 Nonlinear differential equations 45, 151 Nonorientable surface 453 Nonparametric test 1080 Nonsingular matrix 315 Norm 205, 326, 346, 359, 365, 849
Normal
acceleration 395 asymptotically 1057 to a curve 398 derivative 444, 465
distribution 1026, 1047-1057. 1062-1067, A98
two-dimensional 1090 equations 860, 1086 form of aPDE 551 matrix 362, 869 mode 542, 582 plane 398 to a plane 375 random variable 1026 to a surface 447 vector 375, 447
Null
hypothesis 1058 matrix (see Zero matrix) space 301
vector (see Zero vector) Nullity 301
Numeric methods 777-934 differentiation 827 eigenvalues 863-882 equations 787-796 integration 817-827 interpolation 797-816 linear equations 833-858 matrix inversion 315, 844 optimization 936-953 ordinary differential equations (ODEs) 886-908
partial differential equations (PDEs) 909-930 Nystrom method 906
o
O 962
Objective function 936 OC curve 1062 Odd function 490
ODE 4 (see also Differential equations) Ohm's law 92 One-di men si onal
heat equation 553
wave equation 539 One-sided
derivative 484
test 1060
Index
One-step method 898 One-to-one mapping 729 Open
disk 613
integration formula 827
interval A69
point set 613 Operating characteristic 1062 Operational calculus 59, 220 Operation count 838 Operator 59. 327
Optimality principle, Bellman's 963 Optimal solution 942 Optimization 936-953, 959-990 Orbit 141 Order 887, 962
of a determinant 308
of a differential equation 4, 535
of an iteration process 793 Ordering 969
Ordinary differential equations 2-269, 886-908
(see also Differential equation) Orientable surface 452 Orientation of a
curve 638
surface 452 Orthogonal
coordinates A71
curves 35
eigenvectors 350
expansion 210
functions 205, 482
matrix 345
polynomials 209
series 210
trajectories 35
transformation 346
vectors 326, 371 Orthonormal functions 205, 210 Oscillations
of a beam 547,552
of a cable 198
in circuits 91
damped 64, 88
forced 84
free 61, 500, 547
harmonic 63
of a mass on a spring 61, 86, 135, 150, 243,
252, 261, 342, 499 of a membrane 569-586
Oscillations (Cont.) self-sustained 157 of a string 204, 538, 929 undamped 62,
Osculating plane 398
Outcome 997
Outer product 377
Outlier 995
Output 26, 230
Overdamping 65
Overdetermined system 292
Overflow 782
Overrelaxation 851
Overtone 542
Paired comparison 1065
Pappus's theorem 458
Parabolic differential equation 551, 922
Paraboloid 448
Parachutist 12
Parallelepiped 382
Parallel flow 766
Parallelogram
equality 326, 372, 612
law 367
Parameter of a distribution 1016 Parametric representation 389, 446 Parking problem 1023 Parseval's equality 215, 504 Partial
derivative 388, A66
differential equation 535, 909
fractions 231, 245
pivoting 291, 834
sum 171, 480, 666 Particular solution 6, 48, 106, 159 Pascal 399 Path
in a digraph 974
in a graph 959
of integration 421, 637 PDE 535, 909 (see also Differential equation) Peaceman-Rachford method 915 Pearson, E. S. 1058 Pearson, K. 1066 Pendulum 68, 152, 156 Period 478
116
Index
Periodic
extension 494
function 478 Permutation 1006
Perron-Frobenius theorem 344, 869
Pfaffian form 429
Phase
angle 88
of complex number (see Argument) lag 88
plane 141, 147
portrait 141, 147 Picard
iteration method 41
theorem 709 Piecewise
continuous 226
smooth 421, 448, 639 Pivoting 291, 834 Planar graph 987 Plane 315, 375 Plane curve 391 Poincare 216 Point
estimate 1046
at infinity 710, 736
set 613 '
source 765
spectrum 507, 524 Poisson 769
distribution 1022, 1073, A97
equation 910. 918
integral formula 769 Polar
coordinates 437, 443, 580, 607-608
form of complex numbers 607
moment on inertia 436 Pole 708 Polynomial
approximation 797
matrix 865 Polynomial!}' bounded 962 Polynomials 617
Chebyshev 209
Hermite 216
Laguerre 207, 257
Legendrc 179. 207, 590, 826
trigonometric 502 Population in statistics 1044 Population models 31, 154, 341
Position vector 366 Positive definite 326, 372 Possible values 1012 Postman problem 963 Potential 407, 427, 590. 750, 762
complex 763
theory 465. 749 Power
method for eigenvalues 872
of a test 1061
series 167, 673
series method 167 Precision 782 Predator-prey 154 Predictor-corrector 890, 900 Pre-Hilbert space 326 Prim's algorithm 971 Principal
axes theorem 354
branch 632
diagonal (see main diagonal) directions 340 normal (Fig. 210) 397 part 708
value 607, 630, 632, 719, 722 Prior estimate 794 Probability 1000, 1001
conditional 1003
density 1014, 1034
distribution 1010, 1032
function 1012, 1033 Producer's risk 1075 Product (see Multiplication) Projection of a vector 374 Pseudocode 783 Pure imaginary number 603
Q
QR-factorization method 879 Quadratic
equation 785
form 353
interpolation 799 Qualitative methods 124, 139-165 Quality control 1068 Quartile 995 Quasilinear 551, 909 Quotient of complex numbers 604
Index
R
Rachford method 915 Radiation 7, 561 Radiocarbon dating 13 Radius
of convergence 172, 675, 686
of a graph 973 Random
experiment 997
numbers 1045
variable 1010. 1032 Range of a
function 614
sample 994 Rank of a matrix 297, 299, 311 Raphson 790 Rational function 617 Ratio test 669 Rayleigh 159
equation 159
quotient 872 flC-circuit 97, 237, 240 Reactance 93 Real
axis 604 part 602 sequence 664 vector space 324, 369 Rectangular
membrane 571 pulse 238, 243 rale 817
wave 211, 480, 488, 492 Rectifiable curve 393 Rectification of a lot 1075 Rectifier 248, 489, 492 Rectifying plane 398 Reduction of order 50. 116 Region 433, 614 Regression 1083
coefficient 1085, 1088
line 1084 Regula falsi 796 Regular
point of an ODE 183
Sturm-Liouville problem 206 Rejectable quality level 1075 Rejection region 1060 Relative
class frequency 994
Relative (Cont.)
error 784
frequency 1000 Relaxation 850 Remainder 171, 666 Removable singularity 709 Representation 328 Residual 849, 852 Residue 713 Residue theorem 715 Resistance 91 Resonance 86 Response 28, 84 Restoring force 62 Resultant of forces 367 Riccati equation 34 Riemann 618
sphere 710
surface 746 Right-hand
derivative 484
limit 484 Right-handed 377 Risk 1095 /?L-circuit 97, 240 flLC-circuit 95, 241, 244, 499 Robin problem 558. 587 Rodrigues's formula 181 Romberg integration 829 Root 610 Root test 671
Rotation 381, 385, 414, 734, 764
Rounding 782
Row
echelon form 294
-equivalent 292, 298
operations 292
scaling 838
space 300
sum norm 849
vector 274 Runge-Kutta-Fehlberg 893 Runge-Kutta methods 892, 904 Runge-Kutta-Nystrom 906
s
Saddle point 143 Sample 997, 1045 covariance 1085
118
Index
Sample (Cont.)
distribution function 1076
mean 996, 1045
moments 1046
point 997
range 994
size 997, 1045
space 997
standard deviation 996, 1046
variance 996, 1045 Sampling 1004, 1023, 1073 SAS 991
Sawtooth wave 248, 493, 505 Scalar 276, 364
field 384
function 384
matrix 284
multiplication 276, 368
triple product 381 Scaling 838 Scheduling 987 Schoenberg 810 Schrodinger 242 Schur's inequality 869 Schwartz 242 Schwarz inequality 326 Secant 627, A62
method 794 Second
Green's formula 466
shifting theorem 235 Sectionally continuous
(see Piecewise continuous) Seidel 846 Self-starting 898 Self-sustained oscillations 157 Separable differential equation 12 Separation of variables 12, 540 Sequence 664, A69 Series 666, A69
addition of 680
of Bcsscl functions 213, 583
binomial 689
convergence of 171, 666
differentiation of 174, 680
double Fourier 576
of eigenfunctions 210
Fourier 211, 480, 487
geometric 167, 668, 673, 687, 692
harmonic 670
Series (Cont.)
hypergeometric 188 infinite 666, A70 integration of 680 Laurent 701, 712 Maclaurin 683 multiplication of 174. 680 of orthogonal functions 210 partial sums of 171, 666 power 167, 673 real A69
remainder of 171, 666, 684
sum of 171, 666
Taylor 683
trigonometric 479
value of 171, 666 Serret-Frenet formulas
(see Frenet formulas) Set of points 613 Shifted data problem 232 Shifting theorems 224, 235, 528 Shortest
path 959
spanning tree 967 Shrine of the Book 814 Sifting 242
Significance level 1059 Significant
digit 781
in statistics 1059 Sign test 1081 Similar matrices 350 Simple
curve 391
event 997
graph 955
pole 708
zero 709 Simplex
method 944
table 945 Simply connected 640, 646 Simpson's rule 821 Simultaneous
corrections 850
differential equations 124
linear equations (see Linear systems)
Sine
of a complex variable 627, 688, 742 hyperbolic 688
Index
Sine (Cont.)
integral 509, 690, A65, A95
of a real variable A60 Single precision 782 Single-valued relation 615 Singular
at infinity 711
matrix 315
point 183, 686, 707
solution 8, 50
Sturm-Iiouville problem 206 Singularity 686, 707 Sink 464, 765, 973 SI system: Front cover Size of a sample 997, 1045 Skew-Hermitian 357, 361 Skewness 1020
Skew-symmetric matrix 283, 345
Sky diver 12
Slack variable 941
Slope field 9
Smooth
curve 421, 638
piecewise 421, 448, 639
surface 448 Sobolev 242 Soft spring 159 Software 778, 991 Solution
of a differential equation 4, 46, 105, 536
general 6, 48, 106, 138
particular 6, 48, 106
singular 8, 50
space 304
steady-state 88
of a system of differential equations 136
of a system of equations 288
vector 288 SOR 851 Sorting 969, 993 Source 464, 765, 973 Span 300 Spanning tree 967 Sparse
graph 957
matrix 812, 912
system of equations 846 Spectral
density 520
mapping theorem 344, 865
Spectral (Cont.)
radius 848
representation 520
shift 344, 865, 874 Spectrum 324, 542, 864 Speed 394 Sphere 446
Spherical coordinates 588, A71 Spiral 399
point 144 Spline 811 S-PLUS, SPSS 992 Spring 62 Square
error 503
matrix 274
membrane 575
root 792
wave 211, 480, 488, 492 Stability 31, 148, 783, 822, 922
chart 148 Stagnation point 763 Staircase function 248 Standard
basis 328, 369
deviation 1016
form of a linear ODE 26, 45, 105 Standardized random variable 1018 Stationary point 937 Statistical
inference 1044
tables A96-A106 Steady 413, 463
state 88 Steepest descent 938 Steiner 399, 457 Stem-and-leaf plot 994 Stencil 912
Step-by-step method 886 Step function 234 Step size control 889 Stercographic projection 711 Stiff
ODE 896
system of ODEs 907 Stirling formula 1008, A64 Stochastic
matrix 285
variable 1011 Stokes's theorem 469
120
Index
Straight line 375, 391
Stream function 762
Streamline 761
Strength of a source 767
Strictly diagonally dominant 868
String 204, 538, 594
Student's r-distribution 1053, A100
Sturm-Liouville problem 203
Subgraph 956
Submarine cable equations 594 Submatrix 302
Subsidiary equation 220, 230 Subspacc 300
invariant 865 Success 1021 Successive
corrections 850
overrelaxation 851 Sum (see Addition) Sum of a series 171, 666 Superlinear convergence 795 Superposition principle 106, 138 Surface 445
area 435, 442, 454
integral 449
normal 406 Surjective mapping 729 Symmetric matrix 283, 345 System of
differential equations 124-165, 258-263, 902 linear equations (see Linear system) units: Front cover
T
Tables
on differentiation: Front cover
of Fourier transforms 529-531
of functions A94-A106
of integrals: Front cover
of Laplace transforms 265-267
statistical A96-A106 Tangent 627, A62
to a curve 392, 397
hyperbolic 629, A62
plane 406, 447
vector 392 Tangential acceleration 395 Target 973 Tarjan 971
Taylor 683
formula 684
series 683 Tchebichef (see Chebyshev) f-distribution 1053, A100 Telegraph equations 594 Termination criterion 791 Termwise
differentiation 696
integration 695
multiplication 680
Test
chi-square 1077
for convergence 667-672
of hypothesis 1058-1068
nonparametric 1080 Tetrahedron 382 Thermal
conductivity 465, 552
diffusivity 465, 552 Three
-eights rule 830
-sigma limits 1028 Timetabling 987 Torricelli's law 15 Torsion of a curve 397 Torsional vibrations 68 Torus 454 Total differential 19 Trace of a matrix 344, 355, 864 Trail 959
Trajectories 35, 133, 141
Transfer function 230
Transformation
of Cartesian coordinates A84 by a complex function 729 of integrals 437, 439, 459, 469 linear 281
linear fractional 734 orthogonal 346 similarity 350 unitary 359
of vector components A83 Transient state 88 Translation 365. 734 Transmission line equations 593 Transpose of a matrix 282 Transpositions 1081, A106 Trapezoidal rule 817 Traveling salesman problem 960
Index
Tree 966 Trend 1081 Trial 997
Triangle inequality 326, 372, 608 Triangular matrix 283 Tricomi equation 551, 552 Tridiagonalization 875 Tridiagonal matrix 812, 875, 914 Trigonometric
approximation 502
form of complex numbers 607, 624
functions, complex 626, 688
functions, real A60
polynomial 502
series 479
system 479, 482 Trihedron 398 Triple
integral 458
product 381 Trivial solution 27, 304 Truncation error 783 Tuning 543 Twisted curve 391 Two-dimensional
distribution 1032
normal distribution 1090
random variable 1032
wave equation 571 Two-sided test 1060 Type of a differential equation 551 Type I and II errors 1060
u
UCL 1068
Unconstrained optimization 937 Uncorrelated 1090 Undamped system 62 Underdamping 66 Underdelcrmined system 292 Underflow 782
Undetermined coefficients 79, 117, 160 Uniform
convergence 691
distribution 1015, 1017, 1034 Union of events 998 Uniqueness
differential equations 37, 73. 107, 137
Dirichlet problem 774
Uniqueness (Cant.) Laurent series 705 linear equations 303 power series 678
Unit
binormal vector 398 circle 611 impulse 242 matrix 284 normal vector 447 principal normal vector 398 step function 234 tangent vector 392, 398 vector 326 Unitary-matrix 357
system of vectors 359
transformation 359 Unstable (see Stability) Upper control limit 1068
V
Value of a series 171, 666 Vandermonde determinant 112 Van dcr Pol equation 157 Variable
complex 614
random 1010, 1032
standardized random 1018
stochastic 1011 Variance of a
distribution 1016
sample 996, 1045 Variation of parameters 98, 118, 160 Vector 274, 364
addition 276, 327, 367
field 384
function 384
moment 380
norm 853
product 377
space 300, 323, 369
subspace 300 Velocity 394
field 385
potential 762
vector 394 Venn diagram 998 Verhulst 31
122
Index
Vertex 955
coloring 987
exposed 983
incidence list 957 Vibrations [see Oscillations) Violin string 538 Vizing's theorem 987 Volta 92 Voltage drop 92 Volterra 154, 201, 253 Volume 435 Vortex 767 Vorticity 764
W
Waiting time problem 1013 Walk 959
Wave equation 536, 539, 569, 929 Weber 217
equation 217
Weber (Cont.)
functions 201 Website see Preface Weierstrass 618, 696
approximation theorem 797
M-test 696 Weighted graph 959 Weight function 205 Well-conditioned 852 Wessel 605
Wheatstone bridge 296 Work 373, 423
integral 422 Wronskian 75, 108
Z
Zero
of analytic function 709 matrix 276 vector 367
Some Constants
e = 2.71828 18284 59045 23536 Ve = 1.64872 12707 00128 14685 e2 = 7.38905 60989 30650 22723
77 = 3.14159 26535 89793 23846 tt2 = 9.86960 44010 89358 61883 Vir" = 1.77245 38509 05516 02730
logic t = 0.49714 98726 94133 85435 In 77= 1.14472 98858 49400 17414 log10<? = 0.43429 44819 03251 82765 In 10 = 2.30258 50929 94045 68402
V'2 = 1.41421 35623 73095 04880
V 2
1.25992 10498 94873 16477
V3 = 1.73205 08075 68877 29353 X3/^ = 1.44224 95703 07408 38232 ln 2 = 0.69314 71805 59945 30942 ln 3 = 1.09861 22886 68109 69140
y =   0.57721 56649 01532 86061 ln y = -0.54953 93129 81644 82234 (see Sec. 5.6) 1° =   0.01745 32925 19943 29577 rad 1 rad = 57.29577 95130 82320 87680° = 57° 17'44.806"
Polar Coordinates
x = r cos
y = r sin ß
r = Vx2 + -2
y~ tan dxdx = rdrdO
Series
so
= S xm    (|*| < 1)
m=0
E ^7
x
rŕ/, = O
(2m +1)!
(-D'V
(2m)!
m—0
=c    /_i \mv2m
ln (1 - jc) = - 2 - (W < D
, m
m—l
" (-l)mx2mTl
arctan x = X - (x < 1)
2m + 1
Greek Alphabet
u	Alpha	v	Nu
ß	Beta	t	Xi
%r	Gamma	o	Omicron
S, A	Delta	77	Pi
e, e	Epsilon	P	Rho
	Zeta	(r, S	Sigma
"'?	Eta	7	Tau
, #, O	Theta	v, Y	Upsilon
t	Iota	4>, y,	Phi
K	Kappa	y	Chi
A, A	Lambda	i/y, Ý	Psi
M	Mu	w, fl	Omega
i	j	k
"i		
bi	b-2	b3
Vectors
a*b = axbx + a2b2 + a3b3
a x b =
ŕ) f      d f       d f grad/= Vf = ^-[ + ~f} + ~k
ó X riV d z
dt>i      dv2 dv3 div v = V • v =--1---1--
dx      dy dz
curi v = V x v =
i	j	k
d	d	d
dx	dy	dz
	v2	V3