Fourier Transformation Problem

[jk89]

Hello there! This is my first post, and I've got to say what a great forum you run here. Its helped me out many times before and for that I'm really grateful.

I have a problem with one of the Maths homeworks I've been set. The question has two parts the first part being:

If we have a function f(x) which is defined to be:
-e^(ax) for x<0
e^(-ax) for x>=0

Find its Fourier transformation with
f~(p) = (1/root(2*pi))*integral(f(x)*e^(-ipx)dx) in the limits of infinity and -infinity.

This part i think I've cracked, my answer is:
f~(p)=-(1/root(2*pi))*(2ip/(p^2+a^2)

Now then next step is a bit tricker:

2) Use the theory of Fourier transformations and your answer to the first part to determine the function g(x):

where g(x) is defined to be:

g(x) = integral((p*e^(ipx))/(p^2+a^2)dp) in the limits of infinity and -infinity.

By inspecting the integral i know that the Fourier transform of g(x) is:
g~(x) = p/(p^2+a^2)

However i can't seem to work it back from there any tips would be greatly appreciated but i don't want the full answer(spoils the fun!)

Cheers!

 

[Amok]

g(x) is the inverse Fourier transform of the function:

$$\frac{p}{p^2 + a^2}$$

Which is the Fourier transform of f, which means that g equals...