Classical Electrodynamics · Problem 2.21

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

Physics — E&M · cauchy-theorem-poisson-integral

Problem

Use Cauchy's theorem to derive the Poisson integral solution. Cauchy's theorem states that if

F(z)

is analytic in a region

R

bounded by a closed curve

C

, then

\frac{1}{2\pi i}\oint_C \frac{F(z')\,dz'}{z' - z} = \begin{cases} F(z) & \text{if } z \text{ is inside} \\ 0 & \end{cases} R

Hint: You may wish to add an integral that vanishes (associated with the image point) to the integral for the point inside the circle.