I'm tasked with performing the following integral. Specifically, the problem asks the following:
Calculate the potential along the $z$-axis for the case where the upper hemisphere [of a sphere] is at potential $V_0$ and the lower at $-V_0$ by integrating: \begin{equation} \Phi(r,\theta,\phi)= r^{1/2}\partial_r r^{1/2} \int_{-1}^1 d\cos{\theta'} \int_0^{2\pi} \dfrac{d\phi'}{2\pi}\dfrac{af(\theta',\phi')}{\sqrt{r^2+a^2-2ra[\cos{\theta}\cos{\theta'}+\sin{\theta}\sin{\theta'}\cos{(\phi-\phi')}]}} \end{equation}
I can provide more context for the problem if necessary, but I really just need a hand getting started with this integral. I'm outright stuck from the beginning and could use a push in the right direction with integrating the $\phi'$ and $\theta'$ components.
Thank you!
Edit: Since SE broke up the integral on two lines, I'll write it this way to avoid confusion: \begin{equation} \Phi(r,\theta,\phi)= r^{1/2}\partial_r r^{1/2} \int_{-1}^1 d\cos{\theta'} \int_0^{2\pi} \dfrac{d\phi'}{2\pi}\dfrac{af(\theta',\phi')}{\sqrt{r^2+a^2-2ra\cos{\gamma}}} \end{equation} Where $\cos{\gamma} = \cos{\theta}\cos{\theta'}+\sin{\theta}\sin{\theta'}\cos{(\phi-\phi')}$.
Without explicit knowledge of $f$ you can not calculate the integral. However, observe that your integral still simplifies quite a bit since you are only tasked to find the potential along the z-axis which translates to $\theta=0$ or $\theta=\pi$.