PROBLEMS FOR ASSIGNMENT 4
1. Problem 1
For each of the following functions, determine all its zeroes and all its isolated singular points
in the extended complex plane C (that is, the point ∞ must be investigated as well!). As long as
a point is a zero or a pole, determine the order. Explain your answer!
a) f(z) = 1−cos(nzn+5)
(sin z)n+4 · e
1
π−z
b) f(z) =
√
z sin 1√
z2n+5
(explain also why the function is holomorphic outside its isolated
singular points; the choice of Arg z for the two roots is the same).
Remark. I remind that a removable singularity can be considered at the same time as an
isolated zero! In this case, also determine the order.
2. Problem 2
Find Taylor/Laurent expensions of the function f(z) in all (!) possible discs and annuli with
the center z = 0:
f(z) =
zn+3
z(z − n)(z − 2n)
After doing so, determine the types of isolated singularity at the points z = 0 and z = ∞
(including the order, if applicable).
Hint. Use partial fractions.
3. Problem 3
Using the theory of Laurent series, calculate
lim
→0 |z|=
zn
e1/z
1