Measure, Integration & Real Analysis · Problem 11C.6

Probability Measures — Sheldon Axler

Mathematics — Measure Theory · fourier-transform-computation

Problem

Suppose

f(x) = \begin{cases} xe^{-2\pi x} & \text{if } x > 0, \\ 0 & \text{if } x \leq 0. \end{cases}

Show that

\hat{f}(t) = \dfrac{1}{4\pi^2(1+it)^2}

for all

t \in \mathbf{R}