Complex Analysis: Theory and Applications · Problem 10.5.1

Applications of Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · steady-state-temperatures

Problem

Show that

H(x,y,z) = \frac{1}{\sqrt{x^2+y^2+z^2}}

satisfies Laplace's equation

H_{xx}+H_{yy}+H_{zz}=0

in three-dimensional Cartesian space, but that

h(x,y) = \frac{1}{\sqrt{x^2+y^2}}

does not satisfy equation

h_{xx}+h_{yy}=0

in two-dimensional Cartesian space.