Help finding ths Fourier transform

[gony rosenman]

Homework Statement

find the Fourier transform of the following function in two ways , once using direct computation , and second using the convolution therom .

Homework Equations

Acos(w₀t)/(d²+t²)

The Attempt at a Solution

I tried first to solve directly . used Euler's identity and got
∫e^-it(w₀+w)/(d²+t²) + ∫e^it(w₀-w)/(d²+t²)

but I think it was a deadend , unsolvable integral .

I got a hint from an internet source saying to use the Lorentzian transform but I am not familiar with it

then I tried going first from the convolution approach , which means I find the separate F transform of cos(w₀t) and of 1/(d²+t²) and then calculate the convolution of them in order to get the FT of the multiplication oh them .

I couldn't do that as well because I didn't manage to integrate e^-iwt/(d²+t²) and so can't find that Fourier transform as well

any direction ,or explained hint would be greatly appreciated!

 

[phyzguy]

This integral can be done using partial fractions, expanding the denominator as (d+t) * (d-t). It then gives a sum of terms of the exponential integral (Ei) function. Or you could just look it up.