Practice midterm 1

Complex analysis, lecture 2

(February 10, 2026)

As usual, be sure to include your method, and remember to write your name.

Problem 1. Find a square root of $1-i$ .

The above problem is directed towards Objective 1 (complex numbers, algebra, and functions).

Problem 2. Where are the branch points of $\frac{\sqrt{z^{2}-4}}{z}$ ? What are the phase factors at each?

The above problem is directed towards Objective 1 (complex numbers, algebra, and functions).

Problem 3. Show that if $f:\mathbb{C}\to\mathbb{C}$ is an analytic function such that $\operatorname{Re}(f(z))=\operatorname{Im}(f(z))$ for all $z$ , then $f$ is constant.

The above problem is directed towards Objective 2 (analytic functions and the Cauchy–Riemann equations).

Problem 4. Verify using the Cauchy–Riemann equations that $f(z)=z^{2}$ is analytic on $\mathbb{C}$ .

The above problem is directed towards Objective 2 (analytic functions and the Cauchy–Riemann equations).

Problem 5. Let $h(z)=|z|^{2}$ , as a function $\mathbb{C}\to\mathbb{C}$ . Show that $h$ does not satisfy the mean value property.

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

Problem 6. Let $\mathbb{D}=\{(x,y)\in\mathbb{R}^{2}:x^{2}+y^{2}<1\}$ , and let $u:\mathbb{D}\to\mathbb{R}$ be given by $u(x,y)=x^{2}-y^{2}$ . Show that $u$ is harmonic, and determine whether or not it has a harmonic conjugate; if so, find one.

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