Analytic Continuation beyond the Ideal Boundary

  • M. Shiba
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 9)


“Analytic continuation beyond the ideal boundary” is a generalization of the corresponding classical notion. The new notion will turn out to be natural and important if we consider not only plane domains but also Riemann surfaces (of finite genus). We survey the new concept in general with a number of examples and study certain simple cases in detail; we give conditions for a function on a noncompact Riemann surface of genus one to be meromorphically continuable beyond the ideal boundary. The classical theorem of Abel plays an important role for our discussion.


Riemann Surface Analytic Continuation Meromorphic Function Conformal Structure Compact Riemann Surface 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Ahlfors, L. V.: Complex Analysis, Third Ed., McGraw-Hill, New York, 1979.MATHGoogle Scholar
  2. [2]
    Ahlfors, L. V. and L. Sario: Riemann Surfaces, Princeton Univ. Press, Princeton, 1960.MATHGoogle Scholar
  3. [3]
    Behnke, H. und F. Sommer: Theorie der Analytischen Funktionen einer Komplexen Veränderlichen, Dritte Aufl., Springer, Berlin-GöttingenHeidelberg, 1965.CrossRefGoogle Scholar
  4. [4]
    Farkas, H. M. and I. Kra: Riemann Surfaces, Springer, Berlin-HeidelbergNew York, 1980. Second Ed.. Springer, Berlin-Heidelberg-New York, 1992.Google Scholar
  5. [5]
    Goluzin, G. M.: Geometric Theory of Functions of a Complex Variable (with a Supplement by V. I. Smirnov), Amer. Math. Soc., Providence, 1969. Original Russian Edition: Moscow, 1966.Google Scholar
  6. [6]
    Grunsky, H.: Lectures on Theory of Functions in Multiply Connected Domains, Vandenhoeck & Ruprecht, Göttingen-Zürich, 1978.MATHGoogle Scholar
  7. [7]
    Horiuchi, R. and M. Shiba: Deformation of a torus by attaching the Riemann sphere, J. reine angew. Math. 456 (1994), 135–149.MathSciNetMATHGoogle Scholar
  8. [8]
    Hurwitz, A. und R. Courant: Vorlesungen über Allgemeine Funktionentheorie und Elliptische Funktionen Vierte Aufl. mit einem Anhang von H. Röhrl, Springer, Berlin-Göttingen-Heidelberg-New York, 1964.CrossRefGoogle Scholar
  9. [9]
    Klein, F.: Über Riemann’s Theorie der Algebraischen Functionen und Ihrer Integrale, Teubner, Leipzig, 1882.Google Scholar
  10. [10]
    Krazer, A.: Lehrbuch der Thetafunktionen, Teubner, Leipzig, 1903. Chelsea Reprint: New York, 1970.Google Scholar
  11. [11]
    Kusunoki, Y.: Theory of Abelian integrals and its applications to conformal mappings, Mem. Coll. Sci. Univ. Kyoto, Ser. A. Math. 32 (1959), 235–258.MathSciNetMATHGoogle Scholar
  12. [12]
    Kusunoki, Y.: Theory of Functions — Riemann Surfaces and Conformal Mapping (in Japanese), Asakura, Tokyo, 1973.Google Scholar
  13. [13]
    Masumoto, M.: Conformal mappings of a once-holed torus, J. Anal. Math. 66 (1995), 117–136.MathSciNetMATHCrossRefGoogle Scholar
  14. [14]
    Masumoto, M.: Extremal lengths of homology classes on Riemann sutfaces, J. reine angew. Math. 508 (1999), 17–45.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Masumoto, M. and M. Shiba: Intrinsic disks on a Riemann surface, Bull. London. Math. Soc. 27 (1995), 371–379.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Rodin, B. and L. Sario: Principal Functions (with an Appendix by M. Nakai), Van Nostrand, Princeton, 1969.Google Scholar
  17. [17]
    Schmieder, G. und M. Shiba: Realisierungen des idealen Randes einer Riemannschen Fläche unter konformen Abschließungen, Archiv Math. 68 (1997), 36–44.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Schmieder, G. und M. Shiba: One-parameter variations of the ideal boundary and compact continuations of a Riemann surface, Analysis 18 (1998), 125–130.MathSciNetMATHGoogle Scholar
  19. [19]
    Schmieder, G. und M. Shiba: Realisierungen des idealen Randes einer Riemannschen Fläche unter konformen Abschließungen II, to appear in Annales UMCS, Sec. A.Google Scholar
  20. [20]
    Shiba, M.: On the Riemann-Roch theorem on open Riemann surfaces, J. Math. Kyoto Univ. 11 (1971), 495–525.MathSciNetMATHGoogle Scholar
  21. [21]
    Shiba, M.: Degeneration of Abelian integrals on an open Riemann surface to elliptic ones, J. Math. kyoto Univ. 21 (1981), 1–18.MathSciNetMATHGoogle Scholar
  22. [22]
    Shiba, M.: The Riemann-Hurwitz relation, parallel slit covering map, and con- tinuation of an open Riemann surface of finite genus.. Hiroshima Math. J. 14 (1984), 371–399.MathSciNetMATHGoogle Scholar
  23. [23]
    Shiba, M.: The moduli of compact continuations of an open Riemann surface of genus one, Trans. Amer. Math. Soc. 301 (1987), 299–311.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    Shiba, M.: The period matrices of compact continuations of an open Riemann surface of finite genus, “Proc. Holomorphic Functions and Moduli”, ed. by Drasin et al. Vol. I, Springer. 1989, pp. 237–246.Google Scholar
  25. [25]
    Shiba, M.: Conformal embeddings of an open Riemann surface into closed surfaces of the same genus, “Analytic Function Theory of One Complex Variable”, ed. by Yang et al. Pitman Res. Notes Math. 212(1989), pp. 287–298.Google Scholar
  26. [26]
    Shiba, M.: The euclidean, hyperbolic, and spherical spans of an open Rie- mann surface of low genus and the related area theorems, Kodai Math. J. 16 (1993), 118–137.MathSciNetMATHGoogle Scholar
  27. [27]
    Shiba, M.: Analytic continuations beyond the ideal boundary, Kokyuroku, Res. Inst. Math. Sci. Kyoto Univ. 1155 (2000), 120–127.MathSciNetMATHGoogle Scholar
  28. [28]
    Shiba, NI. and K. Shibata: Singular hydrodynamic continuations of finite Riemann surfaces, J. Math. Kyoto Univ. 25 (1985), 745–755.MathSciNetMATHGoogle Scholar
  29. [29]
    Shiba, NI. and K. Shibata: Hydrodynamic continuations of an open Riemann surface of finite genus, Complex Variables 8 (1987), 205–211.MathSciNetMATHCrossRefGoogle Scholar
  30. [30]
    Siegel, C. L.: Topics in Complex Function Theory, Vol. II, Automorphic Functions and Abelian Integrals (translated by A. Shenitzer and M. Tretkoff), Wiley-Interscience, 1971, 193pp. Original German Edition: Vorlesungen über ausgewählte Kapitel der Funktionentheorie, Teil II, Univ. Göttingen, mimeographed notes.Google Scholar
  31. [31]
    Springer, G.: Introduction to Riemann Surfaces, Addison Wesley, Reading, 1957. Second Ed.,: Chlsea, New York, 1981.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • M. Shiba
    • 1
  1. 1.Department of Applied MathematicsHiroshima UniversityHiroshimaJapan

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