I was trying to evaluate this function:
$$\int^{\infty}_{-\infty} \frac{e^{-ik\sqrt{p^2+y^2}}}{\sqrt{p^2+y^2}}dy$$
Using contour integration, this integral should be possible. But using the section 6.677 of I. S.Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (9th integral) I have found the answer to be
$$- i \pi \text{H}^{(2)}_0(kp)$$
Mathematica evaluates this integral to be
$$-2 \text{Hypergeometric0F1Regularized}^{(1,0)}\left(1,-\frac{1}{4} k^2 p^2\right)+(-2 \log (k p)-i \pi +\log (4)) J_0(k p)$$
Superscript $(1,0)$ indicates that the function is differentiated wrt the first parameter of the function.
Now, mathematica evaluates these expressions to be the same (it is disappointing that mathematica failed to find the simpler notation using the Hankel function)
I am curious if anyone has any explanation as to how (perhaps even why) this identity holds.