Help with Fourier Transform integration

[Moomax]

Homework Statement

Find the Fourier transform of f(t) = 1 / (t^2 +1)

Homework Equations

F(w) = Integral f(t) * e^-jwt dt

The Attempt at a Solution

Hi guys, so I've been having problems trying to solve Fourier transforms. It seems that slapping the e^-jwt makes it hard to integrate for me. Like for instance in the above problem my first instinct was to try and do it by Integration by parts with u = 1/(t^2 +1) and v = INT(e^-jwt) however, if you do it that way, you get stuck in a never ending loop of integration by parts since the u term doesn't "die off".

In addition to the above problem, I've been having some trouble solving for Fourier transforms in general. The integration has been leaving me with scenarios like the one above and I just don't know what to do. Can someone offer me some guidance on how to tackle these sort of problems? Thanks, it's really appreciated!

 

[Dick]

That looks like a job for a contour integral. Have you used the residue theorem before? You need it for a lot of Fourier problems. Might be a good time to review.

 

[Moomax]

hhmmm so I did some reading on the residue theorem but am not really sure how to apply it if for instance the integral was t^2+4 instead of t^2+1 on the bottom.

So in other words, another integral of interest would be: INT(-inf to inf) of exp(-jwt)/(t^2+4) dt.

I'm guessing to solve this new integral i'd have to use residue as well but I'm not really sure how to apply it here. On the residue theorem page on wikipedia, they give an example solving the original integral I asked, but I can't figure it out what to do if the bottom number was changed from 1 to 4 mainly because I can't figure out what the res() function is really doing.

 

[Dick]

If you change 1 to 4 then all that changes is the factorization becomes (t^2+4)=(t+2i)(t-2i) so the poles move to +2i and -2i. Nothing else changes. Using the residue theorem is not that easy if you are new to it. All I can suggest is searching for more stuff on the web, or consulting a mathematical methods type book in the library for a more leisurely introduction. Try working out a few problems in some detail. Post again here with the details of your attempt if you have problems.