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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 $y\in\mathbb R$?