TITLE: Fourier transform of a real function is real
QUESTION [11 upvotes]: I was trying to find the Fourier transform of the function
$$x \mapsto \frac1{x^2 - 2x  +2}$$
and I keep getting something with non-zero imaginary part. But the Fourier transform of a real function should be real, right? So I must be making a mistake?
What is a proof that the Fourier transform of a real function is real?

REPLY [15 votes]: The following are equivalent: 

$f(-x)=\overline{f(x)}$ for a.e. $x\in\mathbb R$
$\hat f(\xi)\in\mathbb R$ for a.e. $x\in\mathbb R$

The proof is immediate from 
$$
\overline{\hat f(\xi)} = \int_{\mathbb R} \overline{f(x)}e^{2\pi i x \xi}\,dx 
 = \int_{\mathbb R} \overline{f(-x)}e^{-2\pi i x \xi}\,dx
$$