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

[Vitani11]

Homework Statement

F(t) = sqrt(π/2)e^-t for t>0
F(t) = sqrt(π/2)e^t for t<0
In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)e^itxdt and show that it equals 1/(1+x²)

Homework Equations

F(t) = sqrt(π/2)e^-t for t>0
F(t) = sqrt(π/2)e^t for t<0
1/sqrt(2π) ∫F(t)e^itxdt

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?

 

[Ray Vickson]

Homework Statement

F(t) = sqrt(π/2)e^-t for t>0
F(t) = sqrt(π/2)e^t for t<0
In other words the question asks to solve this integral: 1/sqrt(2π) ∫F(t)e^itxdt and show that it equals 1/(1+x²)

Homework Equations

F(t) = sqrt(π/2)e^-t for t>0
F(t) = sqrt(π/2)e^t for t<0
1/sqrt(2π) ∫F(t)e^itxdt

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?

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