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 $latex y\in\mathbb R$?
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