Homework 4

As usual, you may use any resources to solve these problems except where stated otherwise, with the exception of computational software/generative AI and posting these problems anywhere to be answered by others. Collaboration is encouraged, but everyone should write their own solutions. Write the names of any collaborators or sources used at the top of your homework. If you did not use any sources, write “sources used: none.”

If you find any errors in either the homework or the lecture notes, please let me know, even if you are unsure whether it is an error or not.

As on most math problems, the mathematics is the issue, not the answer: whether you have a correct method is more important then whether you get to the correct number at the end, so include your method!

You do not have to simplify your answers completely (so for example $\frac{2}{2}$ is fine), but you do need to do all the computations (so for example if the problem is “find the largest value of $f(x)$,” the answer “$f(3)$” is incomplete; you would also need to evaluate $f$ at $3$).

Problem 1. Let $D$ be the triangle , and let $\gamma$ be its boundary (in the counterclockwise direction as usual).

1. Compute ${\int }_{\gamma }\text{Re(z)}𝑑z$.

2. Find the $ML$ bound for the above integral. Is it sharp in this case?

The above problem is directed towards Objective 4 (complex integration).

Problem 2. Suppose $P(z)=a_n{z}^n+\mathrm{\cdots }+a_{1}z+a_0$ is a non-constant polynomial (so $n\ge 1$, and $|P(z)|\to \mathrm{\infty }$ as $|z|\to \mathrm{\infty }$) with complex coefficients $a_i$. Let $Q(z)=\frac{1}{z}(P(z)-P(0))=a_n{z}^{n-1}+\mathrm{\cdots }+a_{2}z+a_1$. Then $P(z)=a_0+zQ(z)$, so

$$\frac{a_0+zQ(z)}{zP(z)}=\frac{P(z)}{zP(z)}=\frac{1}{z}.$$

Integrating both sides around a circle of sufficiently large radius $R$, use Cauchy’s theorem and the ML bound to show that $P(z)=0$ must have some solution with for $R$ sufficiently large.

This gives another proof of the fundamental theorem of algebra!

The above problem is directed towards Objectives 4 and 5 (complex integration and Cauchy’s integral theorem and formula).

Problem 3. Evaluate the integral

$${\oint }_{|z|=r}\frac{\sin (z)}{z}𝑑z.$$

The above problem is directed towards Objective 5 (Cauchy’s integral theorem and formula).

Problem 4. Let $u:D\to ℝ$ be a harmonic function on a domain $D\subset {ℝ}^2$, and fix a point $z_0\in D$ and a disk of radius $r$ centered at $z_0$ and contained in $D$. Give a new proof of the mean value property for $u$,

$$\frac{1}{2\pi }{\int }_0^{2\pi }u(z_0+r{e}^{i\theta })𝑑\theta =u(z_0),$$

using Cauchy’s integral formula.

The above problem is directed towards Objective 5 (Cauchy’s integral theorem and formula).