Cauchy Integral Formula and Electrodynamics

sinyud

Is it possible to solve for an E field from a charge density function using the Cauchy Integral Formulas from complex variables?

Cauchy Integral Formula about a closed loop in the complex plane
(Integral[f[z]/ (z-z0)^(n+1)dz = 2 pi i /n! d^n f(z0)/dz ])

that is the n derivative of f with respect to z evaluated at z0

CharlesP

If I remember that was a curl type formula and you need a divergence type formula. It may not require math that complicated. It is just that doing the integral might be awkward. In the formula E=Integral (rho/r^2) dr.

sinyud

I was thinking that just like the Gauss's theorem (the surface integral version of the Div[E] = rho/ epsilon) picks out charges which are in effect mathematical singularities, so to the cauchy residue theorem picks out every 1/z of a function.

Can this similarity be used to solve electrodynamics problems in two dimensions?