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

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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?
 
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"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 .
 
Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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