Ir al contenido principal

Homework 11: Complex Analysis

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 $latex y\in\mathbb R$?

Comentarios