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Accessory parameters in conformal mapping: exploiting the isomonodromic tau function for Painlevé VI
Tiago Anselmo1,2, Rhodri Nelson2, Bruno Carneiro da Cunha1
1Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife, Brazil.
Proceedings. Mathematical, Physical, and Engineering Sciences
|September 18, 2018
Summary
We developed a new numerical method to solve the accessory parameter problem in conformal mapping using isomonodromic tau functions. This approach aids in creating accurate maps to circular arc quadrilaterals and related shapes.
Area of Science:
- Complex Analysis
- Geometric Function Theory
- Mathematical Physics
Background:
- Conformal mapping is crucial for transforming shapes while preserving angles.
- The accessory parameter problem arises in constructing these maps, particularly for complex regions.
- Existing methods may be limited, especially for regions with curved boundaries.
Purpose of the Study:
- To introduce a novel numerical method for solving the accessory parameter problem.
- To address conformal mapping to circular arc quadrilaterals and related geometries.
- To leverage isomonodromic tau functions and their known asymptotic expansions.
Main Methods:
- Exploiting the isomonodromic tau function associated with the Painlevé VI equation.
- Extracting monodromy data for the target domain.
- Employing a numerical approach based on established asymptotic expansions.
Main Results:
- A viable numerical method for solving the conformal mapping accessory parameter problem.
- The approach successfully captures the Schwarz-Christoffel accessory parameter problem.
- Demonstrated effectiveness through explicit examples.
Conclusions:
- The presented method offers a powerful new tool for constructing conformal maps.
- The technique shows promise for extension to circular arc polygons with more sides.
- Connects conformal mapping problems with advanced concepts in mathematical physics.
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