Fourier transform of a real function is real

Question

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?

Answer 1

The following are equivalent:

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

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