Solving Fourier Transform of f(x)=β2/(β2+x2)

[kreil]

Homework Statement

I have to find the Fourier transform of

$$f(x)=\frac{\beta^2}{\beta^2+x^2}$$

Homework Equations

Fourier Transform is given by

$$F(k) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^{\infty} e^{-ikx}f(x) dx$$

The Attempt at a Solution

I'm having trouble with the integration after I separate into two integrals using partial fractions:

$$F(k)=\frac{\beta^2}{\sqrt{2\pi}}\int_{-\infty}^{\infty} \frac{e^{-ikx}}{\beta^2+x^2}dx$$

Note
$$\frac{1}{\beta^2+x^2}=\frac{1}{2i\beta} \left( \frac{1}{x-i\beta} - \frac{1}{x+i\beta} \right)$$

$$F(k)=\frac{1}{\sqrt{2\pi}} \frac{\beta}{2i} \left[ \int_{-\infty}^{\infty} \frac{e^{-ikx}}{(x-i\beta)} dx - \int_{-\infty}^{\infty} \frac{e^{-ikx}}{(x+i\beta)} dx \right]$$


Are there any suggestions on how to proceed?

 

[Thaakisfox]

Use the residue theorem.

 

[kreil]

excellent advice, thank you