How to Solve this Complex Integral

Question

The following integral

$$\frac{2\pi}{ir}\int_{0}^{\infty}ke^{-i\omega_kt}(e^{ikr}-e^{-ikr})dk$$

arises in finding the probability amplitude: $\langle x|e^{-i\hat{H}t}|x=0\rangle$. Here $\vec{x} = r\hat{z}$; $k$ is the angular wave number; $\omega_k = \sqrt{m^2+k^2}$, and $m$ is the mass of the particle

Kindly point me to some web resources or some textbook where I could learn to solve/decipher this integral; or why it can or cannot be solved and its consequences.

Answer 1

Hints:

1. Expand the parenthesis, so you get two integrals
2. The first integral is $$\int_0^\infty ke^{-i\omega_k t}e^{ikr}dk$$ See how it is related to $$\frac{d}{dr}\int_0^\infty e^{-i\omega_k t}e^{ikr}dk$$
3. Apply a similar procedure for the second integral