Fourier Transform of Distribution

[VVS]

Hi,

I hope somebody can help me with this one.

Homework Statement

Compute the Fourier Transform of the distribution x-a

Homework Equations

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

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

 

[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.,
$$\lim_{\epsilon \rightarrow 0^+} \delta_{\epsilon}(k)=\delta(k).$$