# How to Fourier transform this expression?

1. Aug 10, 2014

### Steve Drake

I have this expression:
$$f(\tau) = 4 \pi \int \omega ^2 P_2[\cos (\omega \tau)] P(\omega) \, \mathrm{d}w \quad [1]$$ where $P_2$ is a second order Legendre polynomial, and $P(\omega)$ is some distribution function.

Now I am told that, given a data set of $f(\tau)$, I can solve for $P(\omega)$ by either assuming a model for it or Fourier transforming Eq. [1]. I can do this by assuming a distribution, eg Gaussian, then putting it in the integral, but I do not understand how I can obtain $P(\omega)$ directly via Fourier transforming. How could I do this in say MATLAB or Mathematica?

Thanks

2. Aug 16, 2014