I have faced some difficulties to do the following integral
$$\int_0^l\int_0 ^{2\pi} \frac {\exp (i k \sqrt{(r^2 +x_0^2+2 r x_0 \sin \phi+d^2)})}{\sqrt{r^2 +x_0^2+ 2r x_0 \sin \phi+d^2} } \, r \,dr \, d\phi$$
Where, $x_0$, $r$ and $d$ have the same dimension of length.Further, $k$ and $x_0$ is considered as constants.
while trying to understand some cases related to diffraction and scattering problem.
1) Would you kindly tell me how I can get a closed form result or give me some hint or text such that on going through which I can do it by myself.
2) Further, we can write,
$$ \frac {\exp (i k \sqrt{(r^2 +x_0^2+2 r x_0 \sin \phi+d^2)})}{\sqrt{r^2 +x_0^2+ 2r x_0 \sin \phi+d^2} }=\frac{1}{2d} \frac{d}{dd}\exp i k\sqrt {(r^2 +x_0^2+2 r x_0 \sin \phi+d^2}$$
Would you kindly tell me is there any way, that $\exp i k\sqrt {(r^2 +x_0^2+2 r x_0 \sin \phi+d^2}$
can be expressed in terms of special function such as Bessel function or Hankel function or etc.
Thanking you.