Cauchy theorem and fourier transform

[brad sue]

Homework Statement

Hi,
I have this problem and I don't 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

 

[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.

 

[brad sue]

Thank you Kummer

this iswhere my problem is. I don't get the part of the contour.
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 $$f(z)= \frac{e^{-i\omega z}}{z^2+1} = \frac{\frac{1}{2i}e^{-i\omega z}}{z-i} - \frac{\frac{1}{2i}e^{-i\omega z}}{z+i}$$
Please, what Iam am doing from here?