# Cauchy theorem and fourier transform

1. Nov 12, 2007

1. The problem statement, all variables and given/known data
Hi,
I have this problem and I dont know how to finish it:

Using the Cauchy Theorem, prove that the fourier tranform of $$\frac{1}{(1+t^2)}$$ is
$$\pi.e^{-2.\pi.|f|}$$ .( you must show the intergration contour) Stetch the power spectrum.

I applied the fourier transform formula but then tried to break down the
1/(1+t^2) but I get stuck to apply the Cauchy theorem.
Please can I have some help?

Thank you
B

2. Nov 12, 2007

### Kummer

You want to find, $$\int_{-\infty}^{\infty} e^{-2\pi i ft}\frac{1}{1+t^2}dt$$. Use the popular semi-circular contour and proceede by Cauchy-Goursat theorem. I can post all the detail but it is too long. It is better if you start doing the problem and we help when you need it.

3. Nov 12, 2007

But first, just say I have this function $$f(z)=e^{-\omega*i.z}\frac{1}{1+z^2}$$
I tried to break it down $$f(z)= \frac{e^{-i\omega z}}{z^2+1} = \frac{\frac{1}{2i}e^{-i\omega z}}{z-i} - \frac{\frac{1}{2i}e^{-i\omega z}}{z+i}$$