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2002 Nov

2002 Nov 20

Do as many problems as you can.

  1. Let \((X, d)\) be a metric topological space and \(C \subset X\) be a closed subspace. Define a function \(f\colon X \to \mathbb R\) by \(f(a) = \inf\{d(x, y) : y\in C\}\).\ (a) Show that \(|f(x)-f(y)| \leq d(x,y)\) for all \(x, y \in X\).\ (b) Suppose that \(K \subset X\) is compact and disjoint from \(C\). Prove that there is an \(\epsilon >0\) so that \(d(x,y) \geq \epsilon >0\) for all \(x \in K\), \(y\in C\).

  2. Define the sequence \(\{c_n\}\) as follows:

    \[c_n = \left\{ \begin{array}{ll}(-1)^n/4^{n+1}, & n = 0, 1, \dots,\\\\ (-1)^n n, & n = -1, -2, \dots. \end{array}\right.\]

    (a) Find the largest annulus \(A(r_1, r_2) = \{z : r_1 < |z| < r_2\}\) so that the Laurent series \(\sum_{n=-\infty}^\infty c_n z^n\) converges to a holomorphic function on \(A(r_1, r_2)\).\ (b) Express this holomorphic function as a rational function of \(z\).

  3. (a) State the Riemann Mapping Theorem and explain the meaning of the geometric statement.\ (b) Find a holomorphic bijection between the unit disc \(U\) and the semidisc \(G = \{z \in U : \mathrm{Im}\ z > 0\}\).

  4. Suppose that \(f(z)\) is an entire function and that \(\mathrm{Im}\ f(z)= 0\) for all \(0\leq x \leq 1\). Prove that \(\mathrm{Im}\ f(x)= 0\) for all real \(x\).

  5. Suppose that \(G\subset \mathbb C\) is a domain and that \(f_n\) is a holomorphie function on \(G\) for each \(n=1,2,\dots\). Suppose further that \(f(z) = \lim_{n\to \infty} f_n(z)\) exists for each \(z\in G\) and that the convergence in uniform on every compact subset of \(G\).\ (a) Prove that \(f\) is holomorphic on \(G\).\ (b) Prove that \(f_n' \longrightarrow f'\) uniformly on compact subsets of \(G\).\ (c) Suppose that instead of uniform convergence on every compact subset of \(G\) we only know that the sequence \(\{f_n\}\) is bounded on every compact subset of \(G\). Prove that \(f\) is holomorphic on \(G\).

  6. Let \(G\) be the domain consisting of the entire complex plane with the integers removed, i.e., \(G = \mathbb C \setminus \mathbb Z\). Prove that a bounded holomorphic function on \(G\) must be constant.