I am having trouble with some Fast Fourier Transforms.

I have been using both matlab and FFTW in c.

The problem is that I want to calculate the potential of a system from the overdensity. The equation in Fourier space is simple

phi(k)= -del(k)/(k^2) ,

where del is the overdensity, phi is the potential and k is the wavenumber.

To test this I use a simple del(x) = x, and feed this in to the fft. I then run through the k values and divide through by k squared. When k =0, I am not sure what to do as obviously you can't divide by zero.

I have tried a variety of things such as setting phi(k) = 0 when k=0, or just leaving phi(k) = del(k)..both of which I am sure are wrong.

When I reverse FFT this phi(k), my result is always a sine wave with a wavelength the size of the x range, where as I am expecting (1/6)*x^3.

I am having trouble finding the answer online. Maybe I am searching for the wrong thing. Can anyone help?

Best wishes,