Advertisement

Functions of Several Complex Variables

  • Raghavan Narasimhan

Abstract

In this chapter, we shall define holomorphic functions of several complex variables. The essentially local theory given in Chapter 1, §§3, 4 extends to these functions with little effort. We shall then prove two theorems which show that the behavior of functions of n complex variables, with n> 1, is, in some ways, radically different from that of functions of one variable.

Keywords

Compact Subset Complex Variable Analytic Continuation Real Hypersurface Bergman Kernel 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Cartan, H.: Séminaire, École Normale Supérieure, Paris, 1951/52. (Reprint: Benjamin, 1967.)Google Scholar
  2. [2]
    Cartan, H. and P. Thullen : Zur Theorie der Singularitäten der Funktionen mehrerer komplexer Veränderlichen : Regularitäts- und Konvergenzbereiche. Math. Annalen 106 (1932), 617–647.MathSciNetCrossRefGoogle Scholar
  3. [3]
    Chern, S. S. and J. Moser : Real hypersurfaces in complex manifolds. Acta Math. 133 (1974), 219–271.MathSciNetCrossRefGoogle Scholar
  4. [4]
    Diederich, K. and I. Lieb : Konvexität in der komplexen Analysis, Boston : Birkhäuser, 1981.zbMATHGoogle Scholar
  5. [5]
    Ehrenpreis, L.: A new proof and an extension of Hartogs’ theorem. Bull. Amer. Math. Soc. 67 (1961), 507–509.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    Fefferman, C. : The Bergman kernel and biholomorphic mappings of pseudoconvex domains. Inv. Math. 26 (1974), 1–65.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Grauert, H. and R. Remmert : Theorie der Steinschen Räume. Springer, 1977. (English translation: Theory of Stein spaces. Springer, 1979.)zbMATHCrossRefGoogle Scholar
  8. [8]
    Hartogs, F.: Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen Fortschreiten. Math. Ann. 62 (1906), 1–88.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Hartogs, F.: Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen. Sitzungsberichte Münchener Akademie, 36 (1906), 223–242.Google Scholar
  10. [10]
    Hörmander, L.: An introduction to complex analysis in several variables, 2nd ed. North-Holland, 1973.zbMATHGoogle Scholar
  11. [11]
    Krantz, S.: Function theory of several complex variables. New York: John Wiley, 1982.zbMATHGoogle Scholar
  12. [12]
    Narasimhan, R.: Several complex variables. Chicago: University of Chicago Press, 1971.zbMATHGoogle Scholar
  13. [13]
    Oka, K.: Sur les fonctions analytiques de plusieurs variables. Iwanami Shoten, Tokyo, 1961. (See note on this book in References: Chapter 6.)zbMATHGoogle Scholar
  14. [14]
    Remmert, R. and K. Stein : Eigentliche holomorphe Abbildungen. Math. Zeit. 73 (1960), 159–189.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    Tanaka, N.: On generalized graded Lie algebras and geometric structures. I. Jour. Math. Soc. Japan 19 (1967), 111–128.Google Scholar

Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • Raghavan Narasimhan
    • 1
  1. 1.Department of MathematicsThe University of ChicagoChicagoUSA

Personalised recommendations