# Help with Difficult Integral - Green's Function and Finding the Potential

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 $$\cos{\gamma} = \cos{\theta}\cos{\theta'}+\sin{\theta}\sin{\theta'}\cos{(\phi-\phi')}$$.

• What's $f(\theta',\phi')$? – maxmilgram Feb 10 at 21:55
• @maxmilgram Some arbitrary funciton of $f$. – Kosta Feb 10 at 21:58

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 $$\theta=\pi$$.