Solving electrostatic, rotationally symmetric 3D problem with conformal mapping?

[Gerenuk]

I heard that one can solve 2D problem with conformal mapping of complex numbers.
Is it possible to use this method for 3D axial-rotationally symmetric problems (which are effectively 2D with a new term in the differential equation)?

 

[marcusl]

Azimuthally symmetric 3D problems have different solutions than 2D ones, so solution approaches are not interchangeable. Take a cut through a cylinder including the axis--it looks like a rectangle, but radial solutions are Bessel functions. The sine/cosine solutions you'd get from solving it as a 2D rectangle are flat out wrong.

 

[Gerenuk]

I know that.
I was wondering if one can still use some kind of complex variable method such as conformal mapping to treat this 2D problem.

 

[marcusl]

To be clear on what you mean by "this" 2D problem:
Solution of 2D problem in a planar boundary like rectangle--yes
Solution of 3D azimuthally symmetric problems--no.

Actually, Weber recounts that Maxwell evaluated the capacitance of parallel plates with guard rings by approximating as a 2D conjugate-function solution valid far from the axis. The general answer is no.

Weber, Electromagnetic Fields, Vol. 1: Mapping and Fields, Wiley, 1950.