## 2007 April 16

Notation. $\mathbb C$ is the set of complex numbers, $D = \{z\in \mathbb C: |z|<1\}$, and if $G\subset \mathbb C$ is an open set, then $H(G)$ is the set of holomorphic functions on $G$.

1. Give the Laurent series expansion of $\frac{1}{z(z-1)}$ in the region $A \equiv \{z\in \mathbb C: 2< |z+2| < 3\}$.

2. Suppose for all $n\in \mathbb N$ that the function $f_n$ is holomorphic in $D$ and satisfies $|f_n(z)|<1$ for all $z \in D$. Also suppose that $\lim_{n\to \infty} \mathrm{Im}\ f_n(x) = 0$ for all $x\in (-1,0)$.
(a) Prove: $\lim_{n\to \infty} \mathrm{Im}\ f_n(1/2) = 0$.
(b) Give a complete statement of the convergence theorem that you use in part (a).

3. Use the residue theorem to evaluate $\int_{-\infty}^{\infty} \frac{1}{1+x^4}\ dx$.

4. Present a function $f$ that has all of the following properties:
(i) $f$ is one-to-one and holomorphic on $D$.
(ii) $\{f(z): z\in D\} = \{w \in \mathbb C: \mathrm{Re}\ w > 0, \ \mathrm{Im}\ w > 0\}$.
(iii) $f(0) = 1+i$.

5. (a) Prove: If $f: D \rightarrow D$ is holomorphic and $f(1/2) = 0$, then $|f(0)| \leq 1/2$.
(b) Give a complete statement of the maximum modulus theorem that you use in part (a).

6. Prove: If $G$ is a connected open subset of $\mathbb C$, any two points of $G$ can be connected by a parametric curve in $G$.