2006 November 13

Notation. $\mathbb C$ is the set of complex numbers, $D = \{z\in \mathbb C : |z|<1\}$ is the open unit disk, $\Pi^+$ and $\Pi^-$ are the upper and lower half-planes, respectively, and, given an open set $G\subset \mathbb C$, $H(G)$ is the set of holomorphic functions on $G$.

1. (a) Suppose that $f \in H(D \setminus \{0\})$ and that $|f(z)| < 1$ for all $0<|z|<1$. Prove that there is $F\in H(D)$ with $F(z) = f(z)$ for all $z\in D\setminus \{0\}$.
(b)} State a general theorem about isolated singularities for holomorphic functions.
1. (a) Explicitly construct, through a sequence of mappings, a one-to-one holomorphic function mapping the disk $D$ onto the half disk $D \cap \Pi^+$.
(b) State a general theorem concerning one-to-one mappings of D onto domains $\Omega\subset \mathbb C$.
1. (a) State the Schwarz lemma.
(b) Suppose that $f\in H(\Pi^+)$ and that $|f(z)|<1$ for all $z\in \Pi^+$. If $f(i)=0$ how large can $|f'(i)|$ be? Find the extremal functions.
(cf. '95 Apr #6)

2. (a) State Cauchy's theorem and its converse. (b) Suppose that $f$ is a continuous function defined on the entire complex plane. Assume that
(i) $f\in H(\Pi^+ \cup \Pi^-)$
(ii) $f(\bar{z}) = \overline{f(z)}$ all $z\in \mathbb C$.
Prove that $f$ is an entire function.

1. (a) Define what it means for a family $\mathcal F \subset H(\Omega)$ to be a normal family. State the fundamental theorem for normal families.
(b) Suppose $f\in H(\Pi^+)$ and $|f(z)|<1$ all $z\in \Pi^+$. Suppose further that $\lim_{t\to 0+} f(it) = 0$. Prove that $f(z_n) \rightarrow 0$ whenever the sequence $z_n \rightarrow 0$ and $z_n \in \Gamma$ where $\Gamma = \{ z\in \Pi^+ : |\mathrm{Real}\ z| \leq \mathrm{Imag}\ z\}$. (Hint. Consider the functions $f_t(z) = f(tz)$ where $t>0$.)