Solving Fourier Transform of $\frac{1}{x^2+a^2}$

[matematikuvol]

Homework Statement

Find Fourier transform of function

$$f(x)=\frac{1}{x^2+a^2}$$, $$a>0$$

Homework Equations

$$\mathcal{F}[\frac{1}{x^2+a^2}]=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ikx}dx}{x^2+a^2}$$

The Attempt at a Solution

Two different case

$$k>0$$

and

$$k<0$$

How to solve integral

$$\mathcal{F}[\frac{1}{x^2+a^2}]=\frac{1}{\sqrt{2\pi}}\int^{\infty}_{-\infty}\frac{e^{-ikx}dx}{x^2+a^2}$$

Probably using complex analysis?! I forget this. I have two poles $$ia$$ and $$-ia$$. How to integrate this? Is there some other method without using complex analysis?

 

[obafgkmrns]

"Is there some other method without using complex analysis?"

Nope. But complex analysis isn't that bad -- it just seems like black magic until you get used to it. You might want to look at the example at http://en.wikipedia.org/wiki/Residue_theorem .