Zeros of the isomonodromic tau functions in constructive conformal mapping of polycircular arc domains: the n-vertex case

The prevertices of the conformal map between a generic, n -vertex, simply connected, polycircular arc domain and the upper half plane are determined by finding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarith...

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Bibliographic Details
Published in:Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2022-01, Vol.55 (2), p.25201
Main Authors: Carneiro da Cunha, Bruno, Abarghouei Nejad, Salman, Anselmo, Tiago, Nelson, Rhodri, Crowdy, Darren G
Format: Article
Language:English
Subjects:
conformal mapping
Fredholm determinants
isomonodromic tau functions
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Summary:The prevertices of the conformal map between a generic, n -vertex, simply connected, polycircular arc domain and the upper half plane are determined by finding the zeros of an isomonodromic tau function. The accessory parameters of the associated Fuchsian equation are then found in terms of logarithmic derivatives of this tau function. Using these theoretical results a constructive approach to the determination of the conformal map is given and the particular case of five vertices is considered in detail. A computer implementation of a construction of the isomonodromic tau function described by Gavrylenko and Lisovyy (2018 Commun. Math. Phys. 363 1–58) is used to calculate some illustrative examples. A procedural guide to constructing the conformal map to a given polycircular arc domain using the method presented here is also set out.
ISSN:1751-8113
1751-8121
DOI:10.1088/1751-8121/ac3f88