Homework 3: Complex Analysis

Due February 21st

Problem 1

1. Verify explicitly that $\text{Ind}(\partial D(z_0,r),z_0) = 1$ for all $z_0\in\mathbb C$ and $r>0$.
2. For $z_0\in\mathbb C$ and $r>0$,
   
   $\text{Ind}(\partial D(z_0,r),z) = \begin{cases} 1 & |z-z_0| < r\\ 0 & |z-z_0| > r. \end{cases}$

Problem 2

1. If $f,g$ are holomorphic near $z_0$ and $f$ has a simple zero at $z_0$, find an expression for the residue of $g/f$ at $z_0$.
2. If $f$ has a simple pole at $z_0$ and $g$ is holomorphic near $z_0$, then
   
   $\text{Res}(fg,z_0) = g(z_0)\text{Res}(f,z_0)$.
3. If $f$ is holomorphic near $z_0$ and $g(z) = f(z)/(z-z_0)^n$, then
   
   $\text{Res}(g,z_0) = \dfrac{f^{(n-1)}(z_0)}{(n-1)!}$.

Problem 3

1. Use the residue theorem to show that $\displaystyle \int_{-\infty}^\infty \frac{dx}{1+x^2} = \pi$. (Hint: For $R>0$, consider the upper semicircle $\gamma$ with radius $R$ and diameter over the real line, calculate $\int_\gamma f(z)dz$ and let $R\to\infty$.)
2. Let $P,Q$ polynomials over $\mathbb R$, $\text{degree}(Q)\ge\text{degree}(P)+2$, and suppose $Q$ does not have real roots. Then $\displaystyle \int_{-\infty}^\infty \frac{P(x)}{Q(x)} dx$ is $2\pi i$ times the sum of the residues of $P(z)/Q(z)$ in the upper half-plane.

Problem 4

1. Find $\displaystyle \lim_{\epsilon\to0^+,R\to\infty} \Big( \int_{-R}^{-\epsilon} + \int_\epsilon^R\Big) \frac{e^{it}}{t} dt$. (Hint: Consider the same curve $\gamma$ as in the previous exercise, with a small dent of radius $\epsilon$ around $0$.)
   
2. Use the previous result to calculate the integral $\displaystyle \lim_{R\to\infty} \int_{-R}^R \frac{\sin t}{t} dt$.

Problem 5

Suppose $P(z)$ is polynomial over $\mathbb C$ of degree $n\ge2$ with distinct roots $z_1, z_2, \ldots, z_n$. Then

$\displaystyle \sum_{j=1}^n \frac{1}{P'(z_j)} = 0$.

(Explain first why each $P'(z_j)\not=0$.)