## Questions

1.  True or False: If $f$ is analytic and bounded in $\mathbb{H}$, then
    $f$ is constant on $\mathbb{H}$.

    

    False: Take $f(z) = e^{-z}$, where $|f(z)| \leq 1$ in $\mathbb{H}$.

    

2.  Compute $\int_{-\infty}^{\infty} \frac{\sin x}{x(x^2+a^2)}dx$

    

    Two semicircles needed to avoid singularity at zero. Limit equals
    the residue at zero, solution is
    $\pi (\frac{1}{a^2} - \frac{e^{-a}}{a^2})$.

    

3.  Compute $\int_0^{2\pi} \frac{1}{2+\cos\theta}d\theta$

    

    Cosine sub, solution is $\frac{2\pi}{\sqrt{3}}$

    

4.  Find the first three terms of the Laurent expansion of
    $\frac{e^z+1}{e^z-1}$.

    

    Equals $2z^{-1} + 0 + 6^{-1}z + \ldots$

    

5.  Compute $\int_{S_1} \frac{1}{z^2+z-1}dz$

    

    Equals $i\frac{2\pi}{5}$

    

6.  True or false: If f is analytic on the unit disk
    $E = \{z : |z| < 1\}$, then there exists an $a \in E$ such that
    $|f (a)| \geq |f (0)|$.

    

    True, by the maximum modulus principal. Suppose otherwise. Then
    $f(0)$ is a maximum of $f$ inside $S_1$. But by the MMP, $f$ must
    attain its maximum on $\partial S_1$.

    

7.  Prove that if $f(z)$ and $f (\bar{z})$ are both analytic on a domain
    D, then f is constant on D

    

    Analytic $\implies$ Cauchy-Riemann equations are satisfied. Also
    have the identity $f' = u_x + iv_x$, and $f' = 0$ $\implies$ $f$ is
    constant.