Trying to solve the following integral, $$\int_{-\infty}^\infty \frac{\alpha y\cdot e^{-\alpha\sqrt{x^2 + y^2}}}{\sqrt{x^2 + y^2}}dx $$ where $\alpha$ is a constant.
this is an integral over $dx$ of the partial derivative of the original function $ 1-e^{-\alpha\sqrt{x^2+y^2}}$ by $dy$:
$$ \frac{d(1-e^{-\alpha\sqrt{x^2+y^2}})}{dy} $$
Thanks!
$x=y\sinh t$ leads to the modified Bessel's (a.k.a. Macdonald's) $K_0$: assuming $\alpha,y>0$, $$\int_{-\infty}^\infty\frac{e^{-\alpha\sqrt{x^2+y^2}}}{\sqrt{x^2+y^2}}\,dx=\int_{-\infty}^\infty e^{-\alpha y\cosh t}\,dt=2K_0(\alpha y).$$