I am trying to solve the following integral in spherical 3D coordinates
$$\mathcal{I}(\theta,\phi)=
\int_0^{2\pi}d\phi_{\Delta}\int_{\delta}^{\Delta/2}\sin\theta_{\Delta}d\theta_{\Delta}
\frac{1}{[1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]}$$
where $(\theta_{\Delta},\phi_{\Delta})$ are the spherical angular coordinates that I wish to integrate with respect to, whereas $(\theta,\phi)$ are simply some free angular spherical coordinates (of some other vector).
I would like to learn some hints on computing it. I have tried switching the integration order, writing integrand as another integral, namely
$$\frac{1}{[1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]}=
\int_{-\infty}^0d\alpha \exp\Big(\alpha
[1-\sin\theta_{\Delta}\sin\theta\cos(\phi_{\Delta}-\phi)-\cos\theta_{\Delta}\cos\theta]\Big)$$
But nothing seems to work. Can someone provide me with a hint? Also, what happens in the $\delta\rightarrow0$ limit?
Thanks
Best Answer
Note that the denominator is $1$ minus the scalar product of the two unit vectors corresponding to $(\theta,\phi)$ and $(\theta_\Delta,\phi_\Delta)$. If you were to integrate over the entire solid angle, the result wouldn’t depend on $(\theta,\phi)$ (since the scalar product is invariant under rotations). So in that case you could choose a convenient value for $(\theta,\phi)$ to evaluate the integral, for instance $\theta=\phi=0$. That would lead to
$$ \int_0^{2\pi}\mathrm d\phi_\Delta\int_0^\pi\mathrm d\theta_\Delta\frac{\sin\theta_\Delta}{1-\cos\theta_\Delta}\;. $$
The denominator is quadratic in $\theta_\Delta$, while the numerator is only linear, so this integral diverges at $\theta_\Delta=0$.
Whether this singularity occurs in your truncated version of the integral depends on whether $(\theta,\phi)$ lies within the region of integration. The integral is only defined for values of $(\theta,\phi)$ outside this region. The limit $\delta\to0$ doesn’t have any special significance, since the singularity doesn’t occur at $\theta_\Delta=0$ but at $(\theta_\Delta,\phi_\Delta)=(\theta,\phi)$.
As regards evaluating this integral in closed form, I doubt that that’s possible, but you never know.