arXiv subjects

Jesse J. Hulse

Publications and source records attributed to Jesse J. Hulse.

3 recordsLinked to original sources

arXiv2026

A Transform Pair for Doubly Connected Domains

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) and the methods for simply connected bounded domains developed by H., Lanzani, Llewellyn Smith, and Luca (2025, Proc. A, 481). The Szeg\"{o} kernel of the annulus and a corresponding transformation law is pivotal in the derivation of this new technique. The modified Schwarz problem for two domains is implemented to demonstrate the effectiveness of this new method.

math.CV↗

arXiv2025

A Formula for the Pluricomplex Green Function of the Bidisk

In this paper, we derive a formula for the pluricomplex Green function of the bidisk with two poles of equal weights. In 2017, Kosiński, Thomas, and Zwonek proved the Lempert function and the pluricomplex Green function are equal on the bidisk, and their description of Lempert function was pivotal in computing the formula for the pluricomplex Green function. We divide the bidisk into two open regions, where the formula is found explicitly on the first region, and the other region is the union of a family of hypersurfaces. On each hypersurface, the formula is explicit up to a unimodular constant that is the root of a sixth degree polynomial. This derived formula for the bidisk leads to an explicit formula for the Carathéodory metric on the symmetrized bidisk up to a fourth degree polynomial. In 2004, Agler and Young found a formula for Carathéodory metric for the symmetrized bidisk that involves a supremum over the unimodular constants. The formula derived in this paper matches Agler and Young's formula, but the unimodular constant is determined by a 4th degree polynomial instead of the before mentioned supremum.

math.CV↗

arXiv2025

The Unified Transform Method: beyond circular or convex domains

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace's equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655--687), where new transform-based techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegő kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

math.CV↗