# Homework Help: Fourier Transform

1. Sep 18, 2010

### kreil

1. The problem statement, all variables and given/known data
I have to find the fourier transform of

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

2. Relevant equations
Fourier Transform is given by

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

3. 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?

2. Sep 19, 2010

### Thaakisfox

Use the residue theorem.

3. Sep 19, 2010