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

In: Analytic Number Theory, Approximation Theory, and Special Functions

Author

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  • Vladimir V. Mityushev

    (Pedagogical University, Department of Computer Sciences and Computer Methods)

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.

Suggested Citation

  • Vladimir V. Mityushev, 2014. "Poincaré $$\boldsymbol{\alpha }$$ -Series for Classical Schottky Groups," Springer Books, in: Gradimir V. Milovanović & Michael Th. Rassias (ed.), Analytic Number Theory, Approximation Theory, and Special Functions, edition 127, pages 827-852, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4939-0258-3_33
    DOI: 10.1007/978-1-4939-0258-3_33
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