Practice final exam

Complex analysis, lecture 2

(April 30, 2026)

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

Problem 1. Write $(1+i\sqrt{3})^{3}$ in Cartesian and polar coordinates.

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

Problem 2. Find the branch points of the function $f(z)=\sqrt{z^{2}+2z}$ . What are the corresponding phase factors at each branch point?

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

Problem 3. Use the Cauchy–Riemann equations to determine the set of all points where $f(x+iy)=x^{2}+iy^{2}$ is complex differentiable. On what domain if any is $f$ analytic?

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

Problem 4. Show that if $f(z)$ is analytic on a domain $D$ , and its modulus $|f(z)|$ is constant on $D$ , then $f(z)$ must be constant.

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

Problem 5. Verify that $u(x,y)=e^{x}\cos y$ is harmonic on $\mathbb{R}^{2}$ , and find a harmonic conjugate $v(x,y)$ for it.

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

Problem 6. Let $u(x,y)$ be a harmonic function on a domain $D$ . Prove that the function $g(z)=u_{x}(x,y)-iu_{y}(x,y)$ is an analytic function on $D$ .

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

Problem 7. Let $\gamma$ be the upper half of the unit circle $|z|=1$ , oriented from $1$ to $-1$ . Compute

\int_{\gamma}\overline{z}\,dz.

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

Problem 8. Let $\gamma$ be the straight line segment from $0$ to $i\sqrt{\pi}$ . Compute

\int_{\gamma}z\cos(z^{2})\,dz.

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

Problem 9. Let $C$ be the circle $|z|=1$ . Compute

\int_{C}\frac{e^{z}}{z(z-2)}\,dz.

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

Problem 10. Let $f$ be analytic inside and on a simple closed contour $\gamma$ . Show that for any $z_{0}$ strictly inside $\gamma$ ,

\int_{\gamma}\frac{f^{\prime}(z)}{z-z_{0}}\,dz=\int_{\gamma}\frac{f(z)}{(z-z_{% 0})^{2}}\,dz.

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

Problem 11. Suppose $f$ is analytic in the open disk $D=\{z\in\mathbb{C}:|z|<2\}$ and satisfies $|f(z)|\leq 5$ for all $z\in D$ . Find an upper bound for $|f^{(3)}(0)|$ . (For full credit, give the best possible bound, though you don’t need to prove it is the best one.)

The above problem is directed towards Objective 6 (consequences of Cauchy’s theorem and formula).

Problem 12. Show, without using Taylor series methods such as the identity theorem, that there does not exist an entire function $f:\mathbb{C}\to\mathbb{C}$ such that for all $|z|>1$ , $f(z)=\frac{1}{z}$ .

The above problem is directed towards Objective 6 (consequences of Cauchy’s theorem and formula).

Problem 13. Find the Taylor series of $f(z)=\frac{1}{(1-z)^{2}}$ centered at $z_{0}=0$ . What is its radius of convergence?

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

Problem 14. Find the Taylor expansion of $f(z)=\frac{1}{1-z}$ at infinity.

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

Problem 15. Find the Laurent expansion of $f(z)=\frac{1}{z^{2}+1}$ valid in the punctured disk $0<|z-i|<2$ .

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

Problem 16. Find and classify the isolated singularities of $f(z)=\frac{e^{z}-1}{z^{2}}$ . For any poles, state their order.

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

Problem 17. Using the residue theorem, compute

\int_{|z|=2}\frac{1}{z^{3}(z+4)}\,dz.

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

Problem 18. Using the residue theorem, compute

\int_{0}^{2\pi}\frac{1}{3+2\cos\theta}\,d\theta.

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

Problem 19. Use Rouché’s theorem to find the number of zeros (with multiplicity) of the polynomial $p(z)=z^{5}+4z^{2}+1$ inside the unit disk $|z|<1$ .

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

Problem 20. Let $p(z)$ be a polynomial of degree $n\geq 2$ . Prove that $p$ cannot be an analytic bijection from $\mathbb{C}$ to $\mathbb{C}$ .

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