Poincaré $$\boldsymbol{\alpha }$$ -Series for Classical Schottky Groups

Vladimir V. Mityushev ()
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Vladimir V. Mityushev: Pedagogical University, Department of Computer Sciences and Computer Methods

A chapter in Analytic Number Theory, Approximation Theory, and Special Functions, 2014, pp 827-852 from Springer

Abstract: Abstract The Poincaré α-series ( $$\alpha \in {\mathbb{R}}^{n}$$ ) for classical Schottky groups are introduced and used to solve Riemann–Hilbert problems for n-connected circular domains. The classical Poincaré θ 2-series is a partial case of the α-series when α vanishes. The real Jacobi inversion problem and its generalizations are investigated via the Poincaré α-series. In particular, it is shown that the Riemann theta function coincides with the Poincaré α-series. Relations to conformal mappings of the multiply connected circular domains onto slit domains and the Schottky–Klein prime function are established. A fast algorithm to compute Poincaré series for disks close to each other is outlined.

Keywords: Classical Schottky Group; Schottky Klein Prime Function; Jacobi Inversion Problem; Riemann-Hilbert Problem; Theta Functions (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4939-0258-3_33

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DOI: 10.1007/978-1-4939-0258-3_33

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