# Proving inverse Fourier transform of 1/(1+x^2) = 1/(1+x^2)

1. Apr 11, 2017

### Vitani11

1. The problem statement, all variables and given/known data
F(t) = sqrt(π/2)e-t for t>0
F(t) = sqrt(π/2)et for t<0
In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)eitxdt and show that it equals 1/(1+x2)
2. Relevant equations
F(t) = sqrt(π/2)e-t for t>0
F(t) = sqrt(π/2)et for t<0
1/sqrt(2π) ∫F(t)eitxdt
3. The attempt at a solution
I want to use complex numbers but these functions are analytic and so I can not do this using residues. A naive approach to integrating with respect to t is straightforward but the limits cause the result to be infinity. How should I approach this properly?

2. Apr 11, 2017

### Ray Vickson

Break up the real line into $-(\infty, 0)$ and $(0,\infty)$, then do two separate integrals. Each integral is really easy.