Fourier Intergrals and transforms

[Brewer]

How do I do a Fourier integral (and what's the point of them??)

I've been asked to evaluate

$$F(\omega) = \frac{1}{\sqrt{2\pi}}\int dte^{-\alpha t}cos\omega t$$

and I've not the foggiest idea what to do. I thought I could just go about doing in the integral by parts (limits are 0 and infinity by the way), but on further research I don't think I can do that can I?

 

[d_leet]

Try integration by parts.

 

[mathman]

cos(u)=(exp(iu)+exp(-iu))/2

Substitute into your integral and you will have the sum of two exp integrals. I presume you can do that.

 

[Brewer]

Yes that's familliar. I have done this integral before - just never dawned on me to use the substitution.

I was also a little confused when it called it a Fourier Integral. I thought it was going to be a lot more complex than it appears to be now.