A Mathematical Analysis of the Complex Dipole Simulation Method

Abstract

We propose the complex dipole simulation method (CDSM) which approximates a holomorphic function by linear combination of $1 / (z - \zeta)$ with the use of its boundary values. In this paper, we treat a function $f$ which is holomorphic in $\Omega$ and continuous on $\overline{\Omega}$ in the case where $\Omega$ is a disk or the exterior domain of a disk. Then we establish the following fact: if $f$ is holomorphic in some neighborhood of $\overline{\Omega}$, the error of an approximate function $f^{(N)}$ decays exponentially with respect to $N$, where $N$ is the number of the charge points.