A Transform Pair for Doubly Connected Domains

Abstract

A new transform-based technique that generalizes the unified transform method is developed for bounded doubly connected domains as a novel way to numerically solve boundary value problems for holomorphic functions and solutions to the Laplacian. This work builds on the transform methods for multiply connected circular domains developed by Crowdy (2015, IMA J., 80, 1902-1931) and the methods for simply connected bounded domains developed by Hulse et al. (2025, Proc. A, 481 (2319)). The Szegő kernel of the annulus and a corresponding transformation law is pivotal in the derivation of this new technique. The modified Schwarz problem and a boundary value problem for a point vortex inside of a doubly connected domain are implemented in detail to demonstrate the effectiveness of this new method.