I have two questions about integral transform:
1. I would like to know how to derive the Fourier sine transform of $x^{-1}e^{-ax^2}$, $|\arg a|<\frac{\pi}{2}$. i.e. $\int_{0}^{\infty}x^{-1}e^{-ax^2} \sin(xy)dx$,$y>0$
2. I would like to know how to write $\int_{-\infty}^{\infty}\frac{\exp\left(-\pi^2 t^2+2i\pi x t\right)}{t} dt=\int_{-\infty}^{\infty}i\pi\; \text{sgn}(x-t) \frac{1}{\sqrt{\pi}}e^{-t^2} dt$
Also, are there any books about the derivation of different integral transform? What I can find is a table of different integral transform, but there is no book which explain derivations of different integral transform.