Laurent series and residue theorem 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. Find the Laurent series expansion of the function $f(z)=\frac{2z}{z^2+4z+3}$ on the annulus .

Problem 2. Classify the singularities of $z^2{e}^{\frac{1}{z^2}}$, including at the point at infinity. For any poles, find their order.

Problem 3. Find the first few terms of the Laurent series of $\frac{1}{\sin z}$ near $z_0=0$: if

$$\frac{1}{\sin z}=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{a}_n{z}^n,$$

compute $a_n$ for all $n\le 1$.

Problem 4. Compute ${\mathrm{Res}}_0(z^2\cos (1/z))$.

Problem 5. Compute the residues of $f(z)=\frac{\pi \cot (\pi z)}{z^2}$ at each integer $z=n$. (Watch out for the case $n=0$!)

Problem 6. Compute ${\int }_{|z|=1}\frac{1}{z^2\sin z}𝑑z$.

Problem 7. Compute ${\int }_{|z|=2}\frac{e^{iz}}{\cos z}𝑑z$.

Problem 8. Compute ${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{x^2}{x^4+4}𝑑x$.

Problem 9. Compute ${\int }_{-\pi }^{\pi }\frac{1}{5-4\cos \theta }𝑑\theta$.