Homework 7

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. Suppose $f:D\to ℂ$ is an analytic function on a domain $D$, and for some point $z_0\in D$ we have $f^{(n)}(z_0)=0$ for all $n\ge 1$. Show that $f$ is constant on $D$.

The above problem is directed towards Objective 7 (power series).

Problem 2. Find Laurent series expansions centered at $0$ for:

1. $f(z)=\frac{1}{z^2-z}$ for ;

2. $g(z)=\frac{z-1}{z+1}$ for $|z|>1$.

The above problem is directed towards Objective 8 (Laurent series).

Problem 3. For each of the following functions, find all the isolated singularities, and classify them as removable, poles, or essential singularities. If they are poles, determine their order.

1. $f(z)=\frac{z}{{(z^2-1)}^2}$

2. $g(z)=z^2\sin (1/z)$

The above problem is directed towards Objective 8 (Laurent series).