Why Can't I Calculate This Inverse Fourier Transform Correctly?

[cathode-ray]

Homework Statement

Hi!

I tried to get the inverse Fourier transform of the function:

X(j\omega)=1/(jw+a)​

for a>0, using the integral:

x(t)=(1/2\pi)\int_{-\infty}^{+\infty} X(j\omega)e^{j\omega t}d\omega​

I know that the inverse Fourier transform of X(j\omega) is:

x(t)=e^{-at}u(t), a>0​

but when i tried to calculate the integral i got:

x(t)=(1/2\pi)\int_{-\infty}^{+\infty} e^{j\omega t}/(jw+a)​

,and i wasnt able to get that integral using any of the techniques i know. What am i doing wrong or isn't possible to get the inverse Fourier transform of that function this way?

 

[susskind_leon]

I guess you have to look into the residue theorem here. Let me know if you need more instructions.

 

[cathode-ray]

Thanks a lot :D. I always forgot that theorem to calculate integrals. It should work. I am going to try it and if i have some problem i will say something.