fourier transform

[thinwhiteduke]

hi there,

provided question: find the fourier transform of the function f(t) defined by f(t)= cos(t), for [-pi/2 < t < pi/2] and 0 for all other values. Are there any values of k where the formula that you get for f(k) is undefined? Investigate the value of f(k) at these places carefully using limits.

Ok so i've been working through my text book and i know the concepts etc and can do questions with f(t)= 1 etc but am stumped on where to start for this question? In the end will i be end up with a f(k) value containing t?

so, i first plug into f(k) eq and integrate to get

Uploaded with ImageShack.us. This seems very messy. I assume i'm doing it completely wrong so if someone can point me in the right direction it would be much appreciated.

cheers

 

[tkhunny]

No. It's not nearly messey enough!

\(\displaystyle \int \cos(t)\cdot e^{-ikt}\;dt = \int e^{-ikt}\;d(\sin(t)) = \sin(t)\cdot e^{-ikt} - \int \sin(t) d(e^{-ikt}) = \sin(t)\cdot e^{-ikt} + ik\cdot\int \sin(t)\cdot e^{-ikt} dt\)

Do it again and you'll be close.