The following integral is part of a calculation I have been working on:
$$ \int_{-\infty}^{+\infty} e^{ibx}x^{-a}\Gamma(a,ix)dx $$
where a is a strictly natural number and $b=1/4$, $i$ is the imaginary unit and $\Gamma$ is the incomplete gamma function. Integration by parts is not helpful, since the argument will appear again in the process, no matter which combination between the three terms we choose. I have been stuck on this for quite a while, any advice is welcome!
For $a<1$, Mathematica gives for $$I=\int_{-\infty}^{+\infty} e^{ibx}\,x^{-a}\,\Gamma(a,ix)\,dx$$ $$I=e^{-\frac{1}{2} i \pi a}\, b^{a-1} \,\sin (\pi a) \left(B_{b^2}\left(\frac{1}{2}-\frac{a}{2},0\right)+B_{b^2}\left(1-\frac{a}{2 },0\right)\right)$$