1. The problem statement, all variables and given/known data

I am trying to directly calculate the electric field (using Coulomb) at some arbitrary point P(0,0,z). The charge is evenly distributed over the surface of a sphere (radius R, charge density σ). Here I use θ for the polar angle and p for the azimuthal angle.

I will leave out the messy details, but I know by symmetry only projection onto z-axis is relevant. I also determined the angle ψ (that between separation vector π and the z-axis) in terms of z,R,θ, and π.