MATH 396 PROBLEMS 11

IGOR KRIZ

Regular problems:

1. Prove that the function

f(x + iy) = \frac{x + 1}{(x + 1)^{2} + y^{2}} − i\frac{y}{(x + 1)^{2} + y^{2}}

is holomorphic where defined. What is its domain of definition?

2. Using the Caychy formula, calculate:

(a) ∫_{K} \frac{e^{x}}{(x−2)^{3}} dx where K is a small circle around 2 oriented counter-clockwise (b) ∫_{L} \frac{e^{z}+z}{z^{2}(z−1)}dz where L is the circle with center 0 and radius 6, oriented counter-

clockwise. [For this one, use the method of partial fractions (ignoring the numerator), to get denominators as in the Cauchy formula.]

3. Using the definitions of complex elementary functions, rewrite the following expressions of z = x + iy as f_{1} + if_{2} where f_{1}, f_{2} are real elementary functions of x and y (e.g., e^{z} = e^{x} cos(y) + ie^{x} sin(y)):

(a) e^{z^{2}}

(b) \frac{1}{z^{2}+e^{z}}.

4. Find the Taylor expansion of the function z ln(z) with center 1. What is its radius of convergence?

Challenge problem:

5. A point a ∈ ℂ is called an isolated singularity of a function f if f is holomorphic in some disk with center a, but with a removed (i.e. B_{ϵ}(a) \ {a}). a is called a removable singularity if f is bounded in a neighbourhood of a. a is called a pole if f(z)(z − a)^{n} has a removable singularity at a. Otherwise, a is called an essential singularity.

(a) Prove that if a is a removable singularity, then one can redefine f(a) in such a way that f becomes holomorhpic in B_{ϵ}(a). [Use the Cauchy formula.]

(b) Prove that if f is holomorphic at a, then 1/f is holomorphic or has a pole at a (i.e. does not have an essential singularity). [Use the Taylor formula.]

(c) Prove that the function e^{−1/z} has an essential singularity at 0.

(d) Prove that if a function has an essential singularity at a, then the image f[B_{ϵ}(a)] = {f(z)|z ∈ B_{ϵ}(a)} is dense in ℂ (i.e. every non-empty open set in ℂ intersects f[B_{ϵ}(a)]). [Hint: use contradiction and (b).]