Homework 3

Complex analysis, lecture 2

(Due February 23, 2026 by 11:59 PM)

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. Catching substantive errors may earn small numbers of challenge points.

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. For each of the following harmonic functions, determine if it has a harmonic conjugate, and find it if so.

(a)

$u(x,y)=\frac{x}{x^{2}+y^{2}}$ on $\mathbb{R}^{2}\setminus\{(0,0)\}$ ;
(b)

$u(x,y)=3x^{2}y-y^{3}$ on $\mathbb{R}^{2}$ .

The above problem is directed towards Objective 3 (harmonic functions).

Problem 2. Let $p(z)$ be a nonconstant polynomial with complex coefficients. Use the harmonic maximum principle for $h(z)=\frac{1}{p(z)}$ to show that the equation $p(z)=0$ has a solution in $\mathbb{C}$ .

To simplify the algebra, you can assume the following simple bound: if $p(z)=a_{d}z^{d}+\cdots+a_{0}$ is of degree $d$ , then there exists a positive real constant $C$ such that for all $|z|$ sufficiently large we have $|p(z)|\geq C|z|^{d}$ . (You are free to prove this result if you’d like to, but do not need to.)

(Recall that this is what we reduced the fundamental theorem of algebra to, so you have now proven the fundamental theorem of algebra!)

The above problem is directed towards Objective 3 (harmonic functions).

Problem 3. Let $D\subseteq\mathbb{C}$ be a domain, and $g(t,z)$ be a continuous function $[a,b]\times D\to\mathbb{C}$ for real numbers $a<b$ . For each fixed $t$ , suppose that $g_{t}(z)=g(t,z)$ is a harmonic function on $D$ . Show that

G(z)=\int_{a}^{b}g(t,z)\,dt

is a harmonic function on $D$ .

Note that although it is tempting to directly differentiate under the integral sign, it is not immediately obvious that the partial derivatives of $g_{t}$ vary continuously with $t$ , which is required in order to differentiate under the integral sign. Instead, use the characterization of harmonic functions by the mean value property.

The above problem is directed towards Objective 3 (harmonic functions).

Problem 4. Let $f:\mathbb{C}\to\mathbb{C}$ be the function given by $f(x+iy)=\cos x+i\sin y$ . Let $\gamma$ be a path in $\mathbb{C}$ from $\gamma(0)=z_{0}$ to $\gamma(1)=z_{1}$ . Is the integral $\int_{\gamma}f(z)\,dz$ independent of the choice of path $\gamma$ from $z_{0}$ to $z_{1}$ ?

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