Classical Electrodynamics · Problem 2.19

Boundary-Value Problems in Electrostatics: II — John David Jackson

Physics — E&M · green-function-annular-region

Problem

Show that the two-dimensional Green function for Dirichlet boundary conditions for the annular region, b≤ρ≤cb \leq \rho \leq c (concentric cylinders) has the expansion G=ln⁡(ρ<2/b2) ln⁡(c2/ρ>2)ln⁡(c2/b2)+2∑m=1∞cos⁡[m(ϕ−ϕ′)]m[1−(b/c)2m](ρ<m−b2m/ρ<m)(1/ρ>m−ρ>m/c2m)G = \frac{\ln(\rho_<^2/b^2)\,\ln(c^2/\rho_>^2)}{\ln(c^2/b^2)} + 2\sum_{m=1}^{\infty}\frac{\cos[m(\phi - \phi')]}{m[1 - (b/c)^{2m}]}(\rho_<^m - b^{2m}/\rho_<^m)(1/\rho_>^m - \rho_>^m/c^{2m})

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