The Location of Singularities of Two-Dimensional Harmonic Functions. I: Theory
ROBERT F. MILLAR · SIAM Journal on Mathematical Analysis · 1970
A procedure is developed for locating singularities of two-dimensional, exterior harmonic functions $\Phi $. The method is capable of extension to more general differential equations and, in principle at least, to higher dimensions. It utilizes the fact that $\Phi $ may be expanded in Fourier series about a point $P_0 $ external to the boundary C. On the circle of convergence lies at least one singularity of $\Phi $. The envelope formed by the circles of convergence as $P_0 $ describes a closed curve about C will bound the singularities. The radius of the circle of convergence is obtained by determining the asymptotic behavior of the Fourier coefficients. For the Dirichlet problem, this is made possible by expressing as the potential of a double layer, of density $\mu $, on C. The asymptotic behavior of the coefficients is governed by the singularities of $\mu $ in the plane of the complex arclength parameter s. Properties of $\mu $ are found by using the Fredholm integral equation of the second kind which it satisfies; the explicit solution is not required.