Solution of the Cauchy Problem for Laplace's Equation in Axially Symmetric Systems

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Abstract
The determination of the electrodes for an electron gun reduces to solving the Cauchy problem for Laplace's equation. Using Green's Theorem, an analytic solution for the case of axial symmetry is obtained in terms of a line integral involving the product of the analytically continued values of the voltage and normal electric fields at the beam edge by an appropriate weighting function. Since an explicit solution is given in terms of line integrals, numerical results should be obtained far more accurately and quickly than those obtained by integrating Laplace's equation directly by finite-difference methods.

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