I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$.
For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I would like to know, whether there exists some property that relates the inverse Fourier transform to the inverse Laplace transform. I have the hope, that in this case there might exist something, since my $\tilde f$ only depends on the absolute value of $s$.
Let me be a bit more precise: Consider \begin{align} \tilde f(s) = \frac{1}{|s|^\alpha+1}. \end{align} Then the inverse Laplace transform is given in terms of the Mittag-Leffler function, i.e. \begin{align} f(t) = t^{\alpha-1}E_{\alpha,\alpha}(-t^\alpha) \end{align} However, I need the inverse Fourier transform. The whole question is placed in the context of fractional calculus and Green's functions for fractional differential equations. Any help is appreciated.
Hint:
$\dfrac{1}{|s|^\alpha+1}=\dfrac{1}{|s|^\alpha\left(1+\dfrac{1}{|s|^\alpha}\right)}=\dfrac{1}{|s|^\alpha}\sum\limits_{n=0}^\infty(-1)^n|s|^{-\alpha n}=\sum\limits_{n=0}^\infty(-1)^n|s|^{-\alpha(n+1)}$