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

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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.

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Notes

Acknowledgements

The author is grateful to D. Crowdy and T. DeLillo for helpful discussions, E.A. Krushevski for discussions concerning the results [41], and A.E. Malevich for the help in preparation of the code (see Appendix).

References

  1. 1.
    Baker, H.F.: Abelian Functions: Abel’s Theorem and the Allied Theory, Including the Theory of the Theta Functions. New York, Cambridge University Press (1995)MATHGoogle Scholar
  2. 2.
    Bell, S.: Finitely generated function fields and complexity in potential theory in the plane. Duke Math. J. 98, 187–207 (1999)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bell, S.: A Riemann surface attached to domains in the plane and complexity in potential theory. Houston J. Math. 26, 277–297 (2000)MATHMathSciNetGoogle Scholar
  4. 4.
    Bobenko, A.I., Klein Ch. (Eds.): Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics. Berlin, Springer (2011)MATHGoogle Scholar
  5. 5.
    Bogatyrev, A.: Extremal Polynomials and Riemann Surfaces. Berlin, Springer (2012)CrossRefMATHGoogle Scholar
  6. 6.
    Crowdy, D.: Conformal mappings between canonical multiply connected domains. Comput. Methods Funct. Theory 6(1), 59–76 (2006)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Crowdy, D.: Conformal mappings from annuli to canonical doubly connected Bell representations J. Math. Anal. Appl. 340, 669–674 (2007)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Crowdy, D.: Geometric function theory: a modern view of a classical subject. Nonlinearity 21, T205–T219 (2008)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Crowdy, D.: The Schwarz problem in multiply connected domains and the Schottky–Klein prime function. Complex Variables and Elliptic Equat. 53, 221–236 (2008)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Crowdy, D.: The Schottky–Klein prime function on the schottky double of planar domains. Comput. Methods Funct. Theory 10(2), 501–517 (2010)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Crowdy, D.G., Fokas, A.S., Green, C.C.: Conformal mappings to multiply connected polycircular arc domains. Comput. Methods Funct. Theory 11(2), 685–706 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Czapla, R., Mityushev, V., Rylko, N.: Conformal mapping of circular multiply connected domains onto slit domains. Electron. Trans. Numer. Anal. 39, 286–297 (2012)MathSciNetGoogle Scholar
  13. 13.
    DeLillo, T.K. Elcrat, A.R., Pfaltzgraff, J.A.: Schwarz–Christoffel mapping of multiply connected domains. J. d’Analyse Mathematique 94, 17–47(2004)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    DeLillo, T.K.: Schwarz–Christoffel mapping of bounded, multiply connected domains. Comput. Methods Funct. Theory 6(2), 275–300 (2006)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    DeLillo, T.K., Driscoll, T.A., Elcrat, A.R., Pfaltzgraff, J.A.: Computation of multiply connected Schwarz–Christoffel map for exterior domains. Comput. Methods Funct. Theory 6(2), 301–315 (2006)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    DeLillo, T.K., Driscoll, T.A., Elcrat, A.R., Pfaltzgraff, J.A.: Radial and Circular Slit Maps of Unbounded Multiply Connected Circle Domains. vol. A464, pp. 1719–1737. Proceedings of the Royal Society, London (2008)Google Scholar
  17. 17.
    Driscoll, T.A., Trefethen, L.N.: Schwarz-Christoffel Mapping. Cambridge University Press, Cambridge (2002)CrossRefMATHGoogle Scholar
  18. 18.
    Dubrovin, B.A.: Riemann surfaces and non-linear equations. RHD Publication, Izhevsk (2001) (in Russian)Google Scholar
  19. 19.
    Gakhov, F.D.: Boundary Value Problems. Nauka, Moscow, (1977) (3rd edn. in Russian); English translation of 1st edn.: Pergamon Press, Oxford (1966)MATHGoogle Scholar
  20. 20.
    Jarczyk, P., Mityushev, V.: Neutral coated inclusions of finite conductivity. Proc. Roy. Soc. London A, 468A, 954–970 (2012)CrossRefGoogle Scholar
  21. 21.
    Jeong, M., Mityushev, V.: The Bergman kernel for circular multiply connected domains. Pacifc J. Math. 233, 145–157 (2007)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Jeong, M., Oh, J.-W., Taniguchi, M.: Equivalence problem for annuli and Bell representations in the plane. J. Math. Anal. Appl. 325, 1295–1305 (2007)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Golusin, G.M.: Geometric Theory of Functions of Complex Variable. Nauka, Moscow 1966 (2nd edn. in Russian); English translation by AMS, Providence, RI (1969)Google Scholar
  24. 24.
    Kühnau, R.: Handbook of Complex Analysis: Geometric Function Theory. Elsevier, North Holland, Amsterdam (2005)Google Scholar
  25. 25.
    Mikhlin, S.G.: Integral Equations, Pergamon Press, New York (1964)MATHGoogle Scholar
  26. 26.
    Mityushev, V.V.: Solution of the Hilbert boundary value problem for a multiply connected domain. Slupskie Prace Mat.-Przyr. 9a, 37–69 (1994)Google Scholar
  27. 27.
    Mityushev, V.V.: Generalized method of Schwarz and addition theorems in mechanics of materials containing cavities. Arch. Mech. 47(6), 1169–1181 (1995)MATHMathSciNetGoogle Scholar
  28. 28.
    Mityushev, V.V.: Convergence of the Poincaré series for classical Schottky groups. Proc. Amer. Math. Soc. 126(8), 2399–2406 (1998)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Mityushev, V.V.: Hilbert boundary value problem for multiply connected domains. Complex Variables 35, 283–295 (1998)CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Mityushev, V.: Conductivity of a two–dimensional composite containing elliptical inclusions, Proc Royal Soc, London A465, 2991–3010 (2009)MathSciNetGoogle Scholar
  31. 31.
    Mityushev, V.: Riemann-Hilbert problems for multiply connected domains and circular slit maps. Comput. Methods Funct. Theory, 11(2), 575–590 (2011)CrossRefMATHMathSciNetGoogle Scholar
  32. 32.
    Mityushev, V.V.: New boundary value problems and their applications to invisible materials. In: Rogosin, S.V. (ed.) Analytic Methods of Analysis and Differential Equations: AMADE-2011: materials of the 6th International Conference dedicated to the memory of prof. A.A. Kilbas, pp. 141–146. Publishing House of BSU, Minsk (2012)Google Scholar
  33. 33.
    Mityushev, V.: \(\mathbb{R}\)–linear and Riemann–Hilbert Problems for Multiply Connected Domains. In: Rogosin, S.V., Koroleva, A.A. (eds.) Advances in Applied Analysis. pp. 147–176. Birkhäuser, Basel (2012)CrossRefGoogle Scholar
  34. 34.
    Mityushev, V.: Scalar Riemann-Hilbert Problem for Multiply Connected Domains, In: Rassias, Th.M., Brzdek, J. (eds.) Functional Equations in Mathematical Analysis, Springer Optimization and Its Applications, vol. 52, pp. 599–632. Springer Science+Business Media, LLC (2012)CrossRefGoogle Scholar
  35. 35.
    Mityushev, V.: Schwarz–Christoffel formula for multiply connected domains. Comput. Methods Funct. Theory 12(2), 449–463 (2012)CrossRefMATHMathSciNetGoogle Scholar
  36. 36.
    Mityushev, V.V., Rogosin, S.V.: Constructive methods to linear and non-linear boundary value problems of the analytic function. Theory and applications. Monographs and Surveys in Pure and Applied Mathematics, Chapman & Hall / CRC, Boca Raton etc. (2000)Google Scholar
  37. 37.
    Mityushev, V., Rylko, N.: A fast algorithm for computing the flux around non–overlapping disks on the plane. Mathematical and Computer Modelling, 57, 1350–1359 (2013)CrossRefMathSciNetGoogle Scholar
  38. 38.
    Mityushev, V., Pesetskaya, E., Rogosin, S.: Analytical Methods for Heat Conduction, in Composites and Porous Media in Cellular and Porous Materials Ochsner A., Murch G, de Lemos M. (eds.) Wiley-VCH, Weinheim (2008)Google Scholar
  39. 39.
    Schmies, M.: Computing Poincaré Theta series for Schottky groups. In Bobenko, A.I., Klein, Ch. (eds.): Computational Approach to Riemann Surfaces, Lecture Notes in Mathematics, pp. 165–182. Berlin, Springer (2011)Google Scholar
  40. 40.
    Vekua, I.N.: Generalized Analytic Functions. Nauka, Moscow (1988) (2nd edn. in Russian); English translation of 1st ed.: Pergamon Press, Oxford (1962)MATHGoogle Scholar
  41. 41.
    Zverovich, E.I.: Boundary value problems in the theory of analytic functions in Hölder classes on Riemann surfaces. Russ. Math. Surv. 26(1), 117–192 (1971)CrossRefGoogle Scholar
  42. 42.
    Zverovich, E.I.: The inversion Jacobi problem, its generalizations and some applications. Aktual’nye problemy sovremennogo analiza, pp. 69–83. Grodno University Publication, Grodno (2009) (in Russian)Google Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Computer Sciences and Computer MethodsPedagogical UniversityKrakowPoland

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