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Problem 1 Let $f$ be entire with $f(x)$ real-valued for $x\in\mathbb R$. Define $g:\mathbb R\to\mathbb R$ by $g(y) = \Re f(iy)$. Then $g$ is even. Problem 2 Let $f$ be entire with $f(x)$ real-valued for $x\in\mathbb R$. Define $h:\mathbb R\to\mathbb R$ by $h(y) = \Im f(iy)$. Then $h$ is an odd function. Problem 3 Verify the previous problems with the functions $\sin z, \cos z, 1 + z^2 + z^3$. Problem 4 Let $f$ be entire with $f(x)$ real-valued for $x\in\mathbb R$ and $f(iy)$ purely imaginary for $y\in\mathbb R$. Then $f$ is odd. Problem 5 What can you say if, instead, $f(iy)$ is also real valued for $y\in\mathbb R$?
Due April 24th Problem 1 Let $U,V\subset\mathbb C$ open, $f:U\to V$ holomorphic and $u:V\to\mathbb C$ harmonic. The $u\circ f$ is harmonic in $U$. Problem 2 Let $f\in C(\partial \mathbb D)$ and $\psi\in\text{Aut}(\mathbb D)$. Then $\mathscr P(f\circ\psi) = \mathscr Pf\circ\psi$. (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 $\bar{\mathbb D}$, holomorphic in $\mathbb D$ and equal to $f(e^{it}) = e^{-it}$ on the boundary, is wrong: Let $u\in C(\bar{\mathbb D})$ be holomorphic in $\mathbb D$ with $u|_{\partial\mathbb D} = f$. Then $v(z) = u(z) - 1/z$ is holomorphic in $\mathbb D\setminus\{0\}$, vanishing on the boundary of $\mathbb D$. Since the set $\partial\mathbb D$ has an accumulation point and \$latex \mathbb D\setminu