I get the standard result for the following integration:
$$ \int_{-\infty}^{\infty}\frac{\exp ik[\sqrt{s^2+x^2}]}{\sqrt{s^2+x^2}}\,dx=i\pi H_0^1 (ks)$$.
Where, $H_0^1(x)$ stands for the Hankel function of first kind with order $0$.
I can not derive this standard result. Would you kindly help me?
Further, I want to find out the same integral but within different limit (finite range) such as $$\int_{-L}^L\frac{\exp ik[\sqrt{s^2+x^2}]}{\sqrt{s^2+x^2}}\,dx$$ where $L$ is some finite length.
Would you kindly suggest me how I can get the desired value from the given result. Thanking you