Inverse Fourier Transform of $\frac{b}{\pi(x^2+b^2)}$: Solving the Problem

[buttersrocks]

Homework Statement

I need to take the inverse Fourier transform of

$$\frac{b}{\pi(x^2+b^2)}$$

Homework Equations

f(t)=$$\int_{-\infty}^{\infty}e^{itx}\frac{b}{\pi(x^2+b^2)}dx$$

It might be useful that $$\frac{2b}{\pi(x^2+b^2)}=\frac{1}{b+ix}+\frac{1}{b-ix}$$

The Attempt at a Solution

I know the result is $$e^{(-b|t|)}$$, and I can get from $$e^{(-b|t|)}$$ to
$$\frac{b}{\pi(x^2+b^2)}$$, but how do I do it in reverse if I didn't already know the pair existed? This doesn't require complex integration does it?

 

[genneth]

I have to admit my first thought was a contour integral...

In my experience, things involving |x| tend to require them.

 

[pam]

It is a standard contour integral. Close the contour with a semicircle above the real axis.

 

[buttersrocks]

Okay guys, thanks, that is what I was thinking, but the book I'm in doesn't have anything else involving complex integration, so I assumed that I was just missing a trick.