Help with calculation of a integral $ \int_c (x^2 + y^2)^{-1/2} \mathrm{e}^{-i k\sqrt{x^2 + y^2}}\,\mathrm{d}x $ (WolframAlpha fails)

Question

I am trying to solve two definite integrals. WolframAlpha also fails. Could you please help me:

$$ \int_c (x^2 + y^2)^{-1/2} \mathrm{e}^{-i k\sqrt{x^2 + y^2}}\,\mathrm{d}x $$

and $$ \int_c ( \frac{\partial }{\partial y} ( (x^2 + y^2)^{-1/2} \mathrm{e}^{-i k\sqrt{x^2 + y^2}}))\,\mathrm{d}x $$

So what I need is are 2 integrals along a not closed curve $c$, where $ x,y,c,k\in \mathbb{R} $. As I wrote in my comment, it might happen, that $k \in \mathbb{C}$. But this case may be ignored for the beginning.

Here Links to WolframAlpha: link1, link2

Answer 1

By setting $x=y\sinh\theta$, the question boils down to finding

$$ \int e^{-iky\sinh\theta} d\theta $$ that is not an elementary integral, but is trivially related with the Bessel/Hankel functions.