Numerical Solution of Laplace's Equation, Given Cauchy Conditions

Usually Laplace's equation ▽
²
ψ=0 (or Poisson's equation ▽
²
ψ=F) must be solved with conditions given all around the boundary of the region in question. Yet in such specific engineering problems as the design of electron guns,
^{1, 2}
solutions are sought in an open region with the Cauchy boundary condition.** With Cauchy conditions, Laplace's equation is “unstable” in that an exponential growth of errors occurs during numerical analysis by methods of finite differences. An expression that gives the order of magnitude of the propagated errors could therefore be of considerable value as a “rule of thumb” where these methods are used, particularly for digital computer programmers. This communication explains how this “rule of thumb” has been obtained.