Ricardo A. Sáenz

Problem 1 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex g:\mathbb R\to\mathbb R$ by $latex g(y) = \Re f(iy)$. Then $latex g$ is even. Problem 2 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$. Define $latex h:\mathbb R\to\mathbb R$ by $latex h(y) = \Im f(iy)$. Then $latex h$ is an odd function. Problem 3 Verify the previous problems with the functions $latex \sin z, \cos z, 1 + z^2 + z^3$. Problem 4 Let $latex f$ be entire with $latex f(x)$ real-valued for $latex x\in\mathbb R$ and $latex f(iy)$ purely imaginary for $latex y\in\mathbb R$. Then $latex f$ is odd. Problem 5 What can you say if, instead, $latex f(iy)$ is also real valued for $latex y\in\mathbb R$?

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\setminu...