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Homework 10: Complex Analysis

Due April 24th

Problem 1

Let $latex U,V\subset\mathbb C$ open, $latex f:U\to V$ holomorphic and $latex u:V\to\mathbb C$ harmonic. The $latex u\circ f$ is harmonic in $latex U$.

Problem 2

Let $latex f\in C(\partial \mathbb D)$ and $latex \psi\in\text{Aut}(\mathbb D)$. Then $latex \mathscr P(f\circ\psi) = \mathscr Pf\circ\psi$. (A sketch of the proof of this result is given in the text --Theorem 10.2.0--; give the details.)

 Problem 3

Explain why the following "proof" that there is no continuous function on $latex \bar{\mathbb D}$, holomorphic in $latex \mathbb D$ and equal to $latex f(e^{it}) = e^{-it}$ on the boundary, is wrong:
Let $latex u\in C(\bar{\mathbb D})$ be holomorphic in $latex \mathbb D$ with $latex u|_{\partial\mathbb D} = f$. Then $latex v(z) = u(z) - 1/z$ is holomorphic in $latex \mathbb D\setminus\{0\}$, vanishing on the boundary of $latex \mathbb D$. Since the set $latex \partial\mathbb D$ has an accumulation point and $latex \mathbb D\setminus\{0\}$ is connected, then $latex v$ vanishes everywhere. Thus $latex u(z) = 1/z$, a contradiction.

Problem 4

Explain why the following "proof" of $latex \mathscr C f = \mathscr P f$, for $latex f\in C(\partial \mathbb D)$, is incorrect:
Since $latex \mathscr C f$ is holomorphic, then it is harmonic. Since $latex \mathscr P f$ is the only harmonic function in the disk with boundary values given by $latex f$, then $latex \mathscr C f$ must be equal to $latex \mathscr P f$.


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