Needs to calculate the result of an integrand, as
$$ I=\int^{2\pi}_{0} \frac{e^{ik\sqrt{r^2_1+r^2_2-2r_1r_2\cos\theta+(z_1-z_2)^2}}}{\sqrt{r^2_1+r^2_2-2r_1r_2\cos\theta+(z_1-z_2)^2}} \, d\theta $$
which becomes the problem as,
$$ I=\int^{2\pi}_{0}\frac{e^{ik\sqrt{a-b\cos\theta}}}{\sqrt{a-b\cos\theta}} \, d\theta $$
I do appreciate it if you know,
1) is there a close form or an approximate close form of this integrand?
2) If no, how can I efficiently integral it by numerical integration?
Thanks a lot.