Analytical Error Estimation of Conformal Mappings Using Complex Bessel Functions under Perturbed Boundaries

TL;DR

This paper develops a theoretical framework for conformal mappings using complex Bessel functions in regions with randomly perturbed boundaries, providing error estimates and stability analysis.

Contribution

It introduces a new analytical error estimation method for complex Bessel function-based conformal mappings under stochastic boundary perturbations.

Findings

Established existence and uniqueness of the conformal mapping.

Derived an error formula showing stability under boundary perturbations.

Proved asymptotic convergence of the mapping error for small perturbations.

Abstract

This paper studies the theoretical construction and analytic error estimation of complex Bessel function-based conformal mappings in regions with randomly perturbed boundaries. First, we construct a conformal mapping applicable to such boundary conditions and prove the existence and uniqueness of the mapping. On this basis, an analytical error estimation method is proposed to quantify the effect of the magnitude of the boundary perturbation on the accuracy of the mapping. By deriving the error formula, we show the stability of the complex Bessel function under perturbed boundary conditions and prove the asymptotic convergence of the mapping error under small perturbation conditions. This study provides new theoretical support for conformal mapping under complex boundary conditions and reveals the potential of complex Bessel functions in dealing with stochastic boundary problems.