I am completely struck with this painful complex integral. Can somebody please look at and see if you can do this? The integral is following. Please note that the integration is with respect to k.
$$\int_{0}^{\infty}\frac{ke^{ik[z\cos{\theta}+r\sin{\theta}\sin{\phi}]}}{ik^3\cos{\theta}+\sin^2{\theta}} dk$$
Note that the integrand has poles located both in upper and lower half planes and also involves a complex exponential. And thus we can not introduce a branch cut to solve it. I cannot extend over real line as well since the integrand is not even.
Something that I know from a different perspective (physical problem) that this integral must depend on the sign of z.(which also gets reflected in the location of poles, I guess). Any help or tricks to solve this painful integral are welcome.
Edit: I am adding more information regarding the integral as that may help in solving this. In particular, I am writing down the actual 3-D integral where the above occurred. It is
$$\int_{0}^{\pi}e^{-i\phi}d\phi\int_{0}^{\pi}\sin^2\theta\cos\theta d\theta\int_{0}^{\infty}\frac{ke^{ik[z\cos{\theta}+r\sin{\theta}\sin{\phi}]}}{ik^3\cos{\theta}+\sin^2{\theta}} dk$$
My idea was to solve the k-integral and thus reduce one dimension so as to do numerical evaluation of the remaining 2-D integral as a function of r and z. Note that $0<r<\infty$ and $-\infty<z<\infty$. Essentially I do like to get rid of limit of $\infty$ due to slow convergence issues during numerical evaluation.
Any help regarding this is appreciated.
Note that if $\theta=0$, then the integral diverges because it behaves like $1/k^2$ near $k=0$.
If I have not made a mistake, Maple evaluates the integral (in terms of exponential integrals $\mathrm{Ei}_1$) as $$ \sum_Q\frac{(z\cos\theta+r\sin\theta\sin\phi)e^Q\mathrm{Ei}_1(Q)}{3 Q \cos\theta} $$ where the sum is over the three cube roots $Q$ of $$ -{\frac { \left( \cos \left( \theta \right) \right) ^{4}\sin \left( \theta \right) \sin \left( \phi \right) \left( \cos \left( \phi \right) \right) ^{2}{r}^{3}+3\, \left( \cos \left( \theta \right) \right) ^{5} \left( \cos \left( \phi \right) \right) ^{2}{r}^{2}z- \left( \cos \left( \theta \right) \right) ^{4}\sin \left( \theta \right) \sin \left( \phi \right) {r}^{3}+3\, \left( \cos \left( \theta \right) \right) ^{4}\sin \left( \theta \right) \sin \left( \phi \right) r{z}^{2}-2\, \left( \cos \left( \theta \right) \right) ^ {2}\sin \left( \theta \right) \sin \left( \phi \right) \left( \cos \left( \phi \right) \right) ^{2}{r}^{3}-3\, \left( \cos \left( \theta \right) \right) ^{5}{r}^{2}z+ \left( \cos \left( \theta \right) \right) ^{5}{z}^{3}-6\, \left( \cos \left( \theta \right) \right) ^{3} \left( \cos \left( \phi \right) \right) ^{2}{r}^{2}z+2 \, \left( \cos \left( \theta \right) \right) ^{2}\sin \left( \theta \right) \sin \left( \phi \right) {r}^{3}-3\, \left( \cos \left( \theta \right) \right) ^{2}\sin \left( \theta \right) \sin \left( \phi \right) r{z}^{2}+\sin \left( \theta \right) \sin \left( \phi \right) \left( \cos \left( \phi \right) \right) ^{2}{r}^{3}+6\, \left( \cos \left( \theta \right) \right) ^{3}{r}^{2}z- \left( \cos \left( \theta \right) \right) ^{3}{z}^{3}+3\,\cos \left( \theta \right) \left( \cos \left( \phi \right) \right) ^{2}{r}^{2}z-\sin \left( \theta \right) \sin \left( \phi \right) {r}^{3}-3\,\cos \left( \theta \right) {r}^{2}z}{\cos \left( \theta \right) }} $$ When $\theta=0$ we get $Q=0$ and the answer involves dividing by $0$.
Here, the exponential integral $\mathrm{Ei}_1$ is defined as $$ \mathrm{Ei}_1(x) = \int_1^\infty \frac{e^{-tx}}{t}\;dt . $$