I am trying to calculate the following Fourier sine transform:
\begin{equation} f(x,y)=\int_0^{\infty}\, \frac{e^{-y \sqrt{b^2+\xi^2}}}{\xi} \,\sin(x\,\xi)\,d\xi \end{equation} with $b>0$ and $y>0$.
Bateman in his book "Tables of integral transforms" Vol. 1 (pp. 16, Eq. (26)) gives that:
\begin{equation} g(x,y)=\int_0^{\infty}\, e^{-y \sqrt{b^2+\xi^2}}\,\cos(x\,\xi)\,d\xi=\frac{y\,b}{\sqrt{x^2+y^2}}\,K_1[b \sqrt{x^2+y^2}] \end{equation} where $K_1[]$ is 1st order modified Bessel function of the second kind.
It appears that $\partial_x{f(x,y)}=g(x,y)$. However, I cannot find the primitive of $g(x,y)$.
Has anyone seen this integral before? Any ideas would be appreciated.