# Homework Help: Fourier Transform of Distribution

1. Jan 17, 2014

### VVS

Hi,

I hope somebody can help me with this one.

1. The problem statement, all variables and given/known data
Compute the Fourier Transform of the distribution x-a

2. Relevant equations
The Fourier Transform of a distribution is just the distribution evaluated with the Fourier Transform of a test function.

3. The attempt at a solution
See this pdf View attachment Übung 27.pdf
I used integration by parts but now I am stuck, because I have to evaluate the integral of the Foureir Transform of the test function.

Thanks
VVS

2. Jan 17, 2014

### vanhees71

I'd rather regularize the distribution, e.g., defining
$$\tilde{f}_{\epsilon}(k)=\int_{\mathbb{R}} \mathrm{d} x \exp(-\mathrm{i} k x) (x-a) \exp(-\epsilon x^2).$$
Then you only need to know that
$$\delta_{\epsilon}(k)=\frac{1}{2 \sqrt{\pi \epsilon}} \exp \left (-\frac{k^2}{4 \epsilon} \right )$$
is a smoothed $\delta$ distribution, i.e.,
$$\lim_{\epsilon \rightarrow 0^+} \delta_{\epsilon}(k)=\delta(k).$$