Inverse Fourier Transform

fannushofi

I want to ask you how to calculate the Inverse Fourier Transform (IFT) of

$image=http://latex.codecogs.com/gif.latex?F_%7Bd%7D%5Cleft%20%28%20%5Ctheta%20%5Cright%20%29%3D%5Cleft%20%28%20cos%5Cleft%20%28%20%5Ctheta%20%5Cright%20%29%20%5Cright%20%29%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D&hash=c568fdd2f173a5f438943e4a58e36171$

where,

$image=http://latex.codecogs.com/gif.latex?%5Cfrac%7B-%5Cpi%20%7D%7B2%7D%3C%5Ctheta%3C%5Cfrac%7B%5Cpi%20%7D%7B2%7D&hash=6029bb6963d94641d4e3837848dbbd29$

so the IFT is,

$image=http://latex.codecogs.com/gif.latex?i_%7Bm%7D%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%7D%5Cint_%7B%5Cfrac%7B-%5Cpi%20%7D%7B2%7D%7D%5E%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7DF_%7Bd%7D%5Cleft%20%28%20%5Ctheta%20%5Cright%20%29e%5E%7B-jm%5Cpi%20cos%5Cleft%20%28%20%5Ctheta%20%5Cright%20%29%7Dd%5Ctheta&hash=55ce2e004e1fd26b554c6643d9464f2d$

assume that
$image=http://latex.codecogs.com/gif.latex?%5Comega%20%3Dcos%5Cleft%20%28%20%5Ctheta%20%5Cright%20%29&hash=dfaa5f54e5205b8c701545b7b1b537dd$
. So,

$image=http://latex.codecogs.com/gif.latex?i_%7Bm%7D%3D%5Cfrac%7B1%7D%7B2%5Cpi%20%7D%5Cint_%7B%5Cfrac%7B-%5Cpi%20%7D%7B2%7D%7D%5E%7B%5Cfrac%7B%5Cpi%20%7D%7B2%7D%7D%5Comega%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%20e%5E%7B-jm%5Cpi%20%5Comega%7Dd%5Comega&hash=ecfa974e00fe0950703d3f25e4b12445$

from here I can't go further, because I have trouble how to integrate that function with the limitation.
Anyone can help me??

Thx

chiro

MHF Helper
Hey fannushofi.

Hint: Look at the Gamma integral but adjust it for a complex argument.