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2006 Nov

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.

  2. (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 \(\Omega\subset \mathbb C\).

  3. (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)

  4. (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.

  5. (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 \(t>0\).)