Fourier transform of complicated function

Question

I am looking for a way to find the Fourier transform of the function $f(x) = x^{ax}$. The issue is I don't know where to begin the problem, is there a way to simply the above function and find Fourier tranform? Any help is appreciable.

Answer 1

It should be mentioned that there exists a stable probability $\mu(dy)$ of parameter 1 (with the same Levy measure as the Cauchy distribution but restricted to $(-\infty,0)$ such that $$x^{ax}=\int_{-\infty}^{\infty}e^{xy}\mu(dy).$$ This $\mu$ is called a Landau distribution and is described in Wikipedia.

I do not understand well the question:'find the Fourier transform of $x^{ax}$' since clearly this density has no Fourier transform in the ordinary sense. If it is in the sense of a more general theory, like Schwartz distributions, I am not competent but guess that the result could have something to do with the density of the Landau distribution.