PROBLEMS FOR THE COMPLEX ANALYSIS COURSE
Problems marked by *** count for 6 points, those marked by ** count for 4 points, those marked
by * count for 3 points, the others count for 2 points.
1. Is there a holomorphic function f(z) in {|z| < 1} such that f(1/n) = (−1)n/n for n ∈ N?
2. Is there a sequence of complex polynomials Pn(z) which converges uniformly on the circle
{|z| = 1} to the function f(z) = 1/z?
3. Describe all harmonic functions u in a domain D such that u3 is also harmonic.
4*. Let f(z) be a holomorphic function in {0 < |z| < 1} and the function |z||f(z)| is bounded.
Prove that f extends holomorphically to {|z| < 1}.
5**. Let f(z) be a holomorphic function in {0 < |z| < +∞} and for all z we have
|f(z)| ≤ |z| + 1/ |z|.
Prove that f is constant.
6. Let f(z) = u(z) + iv(z) be a holomorphic function in a domain D, and u2 + v3 = 1 holds.
Prove that f is constant.
7*. Let D = {0 < |z| < 1} and f(z) be a holomorphic function in D which is continuous up to
the boundary of D and vanishes at the boundary. Prove that f is indentically 0.
8. Describe all C∞ functions f(z) in C, such that ∂f
∂¯z = z.
9. For which values of R > 0 is the set {|z2 − 1| < R} connected?
10. Find the residue at 0 for the function f(z) = exp (ctg z).
11*. Let f(z) be a holomorphic function in C and F(R) ⊂ R. Prove that f(¯z) = f(z) for all z.
12**. By considering integrals over circles {|z| = πn + π/2} from the function f(ξ) =
ctgξ
ξ(ξ−z) ,
prove the expansion
ctg z =
1
z
+
∞
n=1
1
z − πn
+
1
z + πn
for all z = πn, n ∈ Z.
13. Can the function f(x) = x ln(1 + x) be extended holomorphically from the positive ray R+
to a domain in complex plane? To the entire complex plane?
14*. Does there exist a function f(z) holomorphic in {|z| > 0} such that for all z we have
|f(z)| > exp(1/|z|)?
15*. Prove that the function f(z) :=
n≥1
zn! is holomorphic in the disc {|z| < 1} but cannot be
extended holomorphically to a neighborhood of any point a in the boundary of the disc.
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2 PROBLEMS FOR THE COMPLEX ANALYSIS COURSE
16*. Let y(x) be the solution of the differential equation y = y2 + e−1/x2
with the initial value
y(0) = 0. Can y(x) be extended holomorphically from the real line to the complex plane? (The
function e−1/x2
here is extended smoothly to 0 with the valued 0).
17*. Prove that if a sequence of holomorphic polynomials converges uniformly in the boundary
of the disc D = {|z| < 1}, then it converges uniformly in the whole closed disc.
18**. Let f(z) be a continuous function in C, which is holomorphic in C \ R. Prove that f is
holomorphic in C.
19. Prove the Minimum Modulus Principle: if f(z) is a nonvanishing function holomorphic in a
domain D such that |f(z)| has a local minimum at a point a ∈ D, then f is constant.
20*. Let u(z) be a harmonic function f(z) in the annulus {1 < |z| < 2} which equals 0 on the
internal and 1 on the external circles. Prove that u(z) = a ln |z| for an appropriate a.
21***. Let f ∈ O(B1(0)), f(0) = 0, f (0) = 1. Denote the image domain of B1(0) under f by
Ω. Prove that λ(Ω) ≥ π, where λ denotes the area (so, the area of the initial domain can only
increase!).
Hint. Use the Mean Value Theorem.
22. Prove that there exists a holomorphic function f(z) in {|z| > 1} such that for every z, f(z)
equals to one of the values of
√
1 + z2.
23*. Prove that if f(z) is a holomorphic function in {|z| < 1} and Im f(z) > 0 for all z, then
there exists a holomorphic function g(z) in the same domain such that f(z) = eg(z).
24*. Let u(x, y) be a harmonic polynomial. Prove that it is the real part of a complex polynomial.
25*. Can the function f(z) = 1/z2 be approximated by a normally converging sequence of
complex polynomials in the domain {1 < |z| < 2}?
Hint. Argue as in Problem 2.
26. A function f(z) is holomorphic in {|z| > 1} and is bounded there from below: |f(z)| ≥ M > 0
for all z. Prove that there exists a (finite or infinite) limz→∞ f(z).
27. Construct a biholomorphic map from B1(0) onto the domain {Im z > Re z}.
28*. Prove that the group of linear-fractional automorphisms of the upper half-plane {Im z > 0}
consists of the maps
z →
az + b
cz + d
, a, b, c, d ∈ R, ad − bc = 0.
29*. Let u be a harmonic function in the right half-plane {Re z > 0}, continuous up to the
boundary, which is zero on the boundary and has the zero limit at ∞. Prove that u ≡ 0.
Hint. Use linear-fractional transformations.
30*. Let u be a bounded harmonic function in the annulus D = {0 < |z| < 1}. Assume, in
addition, that the harmonically conjugated function v is single-valued in D. Prove that u extends
harmonically to the unit disc B1(0).
Hint. Argue as in the proof of the maximum principle.