Practice problems

The following are practice problems for your own use. They will not be collected or marked; the idea is to provide more practice at the more computational aspects of the class, if desired, since the homeworks often focus on more theoretical aspects. I suggest focusing on problems targeted at aspects you feel less confident about.

Problem 1. Write $z=\frac{1+i}{1-i}$ in both Cartesian and polar form.

Problem 2. Express the complex conjugate $\overline{e^{iz}}$ in terms of $\overline{z}$.

Problem 3. Find the (finite) branch points and phase factors of ${(z^2-1)}^{1/3}$.

Problem 4. Where is $f(z)=z\cdot \mathrm{Re}(z)$ complex differentiable? Where is it analytic?

Problem 5. Show that $u(x,y)=x^2+y^2$ cannot be the real part of an analytic function.

Problem 6. Let $u$ be harmonic on the unit disk with boundary values $u(e^{i\theta })=\sin \theta$.

1. Find $u(0)$.

2. Show that in fact $u(z)=\mathrm{Im}(z)$.

Problem 7. Evaluate ${\int }_{\gamma }{z}^2𝑑z$ where $\gamma$ is the path $\gamma (t)=t^2+it$ from $0$ to $1+i$ (for $0\le t\le 1$).

Problem 8. Evaluate ${\int }_{\gamma }\overline{z}𝑑z$ where $\gamma$ is the line segment from $0$ to $1+i$.

Problem 9. Evaluate ${\int }_{|z|=2}\frac{e^z}{z-1}𝑑z$.

Problem 10. Evaluate ${\int }_{|z|=1}\frac{\sin z}{z^3}𝑑z$.

Problem 11. Evaluate ${\int }_{|z-i|=1}\frac{1}{z^2+1}𝑑z$ using Cauchy’s integral formula.

Problem 12. Suppose $f$ is an entire function such that $|f(z)|\le 7{|z|}^3$ for all $z\in ℂ$. Show that $f(z)=a{z}^3$ where $|a|\le 7$.

Problem 13. Let $g(z)={\int }_0^1\frac{\sin (zt)}{t}𝑑t$. Show that $g(z)$ is an entire function.

Problem 14. Find the Laurent series of $f(z)=\frac{1}{z^2-4}$ on $\{z:|z|>2\}$.

Problem 15. Find and classify all isolated singularities of $f(z)=\frac{1}{e^z-1}-\frac{1}{z}$.

Problem 16. Compute the residue of $f(z)=\frac{e^z}{z^2+1}$ at $z=i$.

Problem 17. Compute the residue of $f(z)=\frac{\cos z}{z^3}$ at $z=0$.

Problem 18. Evaluate ${\int }_{|z-i|=1}\frac{1}{z^2+1}𝑑z$ using the residue theorem. (If you also did the problem using Cauchy’s formula, hopefully your answers agree!)

Problem 19. Evaluate ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{x\sin x}{x^2+4}𝑑x$.

Problem 20. Evaluate ${\int }_0^{2\pi }\frac{1}{5-4\cos \theta }𝑑\theta$ via the residue theorem.

Problem 21. Let $f(z)=z^4-6z+3$. Use Rouché’s Theorem to find the number of roots with .

Problem 22. Calculate $\frac{1}{2\pi i}{\int }_{|z|=2}\frac{f^{\prime }(z)}{f(z)}𝑑z$ where $f(z)=z^3-1$.

Problem 23. Suppose $f$ is an analytic function on a connected domain $D$. Prove that the image $f(D)$ cannot be a line segment in the complex plane.