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2004 Apr

2004 April 19

Instructions. Use a separate sheet of paper for each new problem. Do as many problems as you can. Complete solutions to five problems will be considered as an excellent performance. Be advised that a few complete and well written solutions will count more than several partial solutions.

Notation. \(D(z_0, R) = \{z \in \mathbb C : |z-z_0| < R\}\), \(R>0\). For an open set \(G\subseteq \mathbb C\), \(H(G)\) will denote the set of functions that are analytic in \(G\).

  1. Let \(\gamma\) be a rectifiable curve and let \(\phi \in C(\gamma^\ast)\). (That is, \(\phi\) is a continuous complex-valued function defined on the trace, \(\gamma^\ast\), of \(\gamma\).)

    \[\text{Let } \ F(z) = \int_\gamma \frac{\phi(w)}{w-z}\ dw,\ z \in \mathbb C - \gamma^\ast. \quad \text{ Prove that } \ F'(z) = \int_\gamma \frac{\phi(w)}{(w-z)^2}\ dw,\ z \in \mathbb C - \gamma^\ast,\]

    without using Leibniz's Rule.

  2. (a) State the Casorati-Weierstrass Theorem.\ (b) Evaluate the integral

    \[I = \frac{1}{2\pi i}\int_{|z|=R} (z-3)\sin \left( \frac{1}{z+2}\right)\ dz,\quad \text{ where } R\geq 4.\]

  3. Let \(f(z)\) be an entire function such that \(f(0) = 1\), \(f'(0)=0\) and \(0< |f(z)| \leq e^{|z|},\) for all \(z\in \mathbb C\). Prove that \(f(z) = 1\) for all \(z\in \mathbb C\).

  4. Let \(C\) be an arbitrary circle through \(-1\) and \(1\). Suppose that \(z_1\) and \(z_2\) are two points that satisfy \(z_1z_2=1\) and do not lie on the circle \(C\). Show that one of these points lies inside \(C\) and the other lies outside \(C\).

  5. Show that there is no one-to-one analytic function that maps \(G = \{z : 0 < |z| < 1\}\) onto the annulus \(\Omega = \{z : r < |z| < R \}\), where \(r > 0\).

  6. (a) State a theorem that gives a sufficient condition for a family \(\mathcal F\) of analytic functions to be normal in a domain \(G\).\ (b) Let \(\mathcal F \subseteq H(D)\) be a family of analytic functions in the open unit disk \(D=D(0,1)\). Let \(\{M_n\}\) be a sequence of positive real numbers such that \(\limsup_{n\to \infty} (M_n)^{1/2} < 1\). If for each \(f(z) = \sum_{n=0}^\infty a_n z^n \in \mathcal F\), \(|a_n| \leq M_n\) for all \(n\), prove that \(\mathcal F\) is a normal family.

  7. Is there a harmonic function \(u(z)\) defined on the open unit disk, \(D(0, 1)\), such that \(u(z_n)\to \infty\) whenever \(|z_n| \to 1^-\)? Prove your answer.

  8. Let \(G\) be a simply connected domain with at least \(2\) boundary points. Let \(S = \{\psi \in H(G)\ |\ \psi \colon G \to D(0, 1),\ \psi \text{ is one-to-one}\}\). Prove, without using the Riemann Mapping Theorem, that the set \(S\) is nonempty.