Homework 8

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 than 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 disk of radius $2$ centered at $z_0=0$. Compute the following integrals using the residue theorem:

1. ${\int }_{\partial D}\frac{1}{z{e}^z-z}𝑑z$

2. ${\int }_{\partial D}\frac{z}{z^4-1}𝑑z$

The above problem is directed towards Objective 9 (the residue theorem).

Problem 2. Use the residue theorem to evaluate the following integrals.

1. ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\cos (x)}{{(x^2+1)}^2}𝑑x$

2. ${\int }_0^{2\pi }\frac{\cos \theta }{2+\cos \theta }𝑑\theta$

The above problem is directed towards Objective 9 (the residue theorem).

Problem 3. Fix a real number $a>1$, and choose the principal branch of $z^a$, i.e. the branch whose values on positive real numbers $z$ are positive real numbers. By integrating this branch around a wedge-shaped region of angle $2\pi /a$, show that

$${\int }_0^{\mathrm{\infty }}\frac{1}{1+x^a}𝑑x=\frac{\pi }{a\sin (\pi /a)}.$$

The above problem is directed towards Objective 9 (the residue theorem).

Note: students doing the final project do not need to do the remaining problems, which target Objective 10.

Problem 4. Let $f(z)=\frac{z^9+2z^5-2z^4+z+3}{2z-1}$. The goal of this problem is to count the number of zeros of $f$ with real part greater than $0$.

1. Let for some large real number $R$, so that its boundary is given by the arc of radius $R$ from angle $-\pi /2$ to $\pi /2$ together with the line segment connecting $Ri$ to $-Ri$. Show that (for $R$ sufficiently large) $f$ has no zeros or poles on $\partial D$.

2. Write $g(z)=z^9+2z^5-2z^4+z+3$ and $h(z)=2z-1$, so that $f(z)=\frac{g(z)}{h(z)}$. Compute the increase in argument of $h(z)$ as $z$ moves along the arc. Recall that you can make an approximation for $R$ sufficiently large.

3. Compute the increase of argument of $h(z)$ as $z$ moves along the line segment, from $Ri$ to $-Ri$.

4. Compute the increase in argument of $g(z)$ as $z$ moves along the arc.

5. Compute the increase of argument of $g(z)$ as $z$ moves along the line segment.

6. Putting everything together and using $\arg f=\arg g-\arg h$, compute the increase in argument of $f$ along $\partial D$. Using the argument principle and by studying the poles of $f$ directly, find the number of zeros of $f$ (with multiplicity) inside $D$, and by taking the limit as $R\to \mathrm{\infty }$ the number of zeros in the right half-plane.

The above problem is directed towards Objective 10 (special topics).

Problem 5. Using Rouché’s theorem, find the number of solutions to $z^5+3z^2+1=0$, counting multiplicity, with .

The above problem is directed towards Objective 10 (special topics).

Problem 6. Let $f:D\to ℂ$ be a non-constant analytic function, and let $A\subset D$ be a subdomain. Show that $f(A)$ is a domain.

If $\partial A$ is also contained in $D$, then we can evaluate $f$ on it as well. Give an example where $f(\partial A)$ is not equal to $\partial f(A)$; that is, find $D$, $f$, $A$, and a point $z\in \partial A$ such that either $f(z)\notin \overline{f(A)}$ or $f(z)$ is in $f(A)$ itself, rather than on its boundary.

The above problem is directed towards Objective 10 (special topics).