Runge Theory for Compact Sets

  • Reinhold Remmert
Part of the Graduate Texts in Mathematics book series (GTM, volume 172)


In discs B, all holomorphic functions are approximated compactly by their Taylor polynomials. In particular, for every fO(B) and every compact set K in B, there exists a sequence of polynomials pnsuch that lim |f - Pn|K = 0. In arbitrary domains, polynomial approximation is not always possible; in C×, for example, there is no sequence of polynomials pn that approximates the holomorphic function l/z uniformly on a circle γ, for it would then follow that
$$$$2\pi i = \int_\gamma{\frac{{d\varsigma }}{\varsigma }}= \lim \int_\gamma{{p_n}\left( \varsigma\right)d\varsigma }= 0$$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B]
    Burckel, R. B.: An Introduction to Classical Complex Analysis, vol. 1, Birkhäuser, 1979.Google Scholar
  2. [C]
    Cartan, H.: Œuvres 1, Springer, 1979.Google Scholar
  3. [Gail]
    Gazer, D.: Vorlesungen über Approximation im Komplexen, Birkhäuser, 1980.Google Scholar
  4. [Gaie]
    Gazer, D.: Approximation im Komplexen, Jber. DMV86, 151–159 (1984).MathSciNetGoogle Scholar
  5. [Gau]
    Gauthier, P.: Un plongement du disque unité, Sém. F. Norguet, Lect. Notes 482, 333–336 (1975).Google Scholar
  6. [HR]
    Hartogs, F. and A. ROSENTHAL: Über Folgen analytischer Funktionen, Math. Ann. 100, 212–263 (1928).MathSciNetCrossRefGoogle Scholar
  7. [ML]
    Mittag-Leffler, G.: Sur une classe de fonctions entières, Proc. 3rd Int. Congr. Math., Heidelberg 1904, 258–264, Teubner, 1905.Google Scholar
  8. [Mo]
    Montel, P.: Sur les suites infinies de fonctions, Ann. Sci. Ec. Norm. Sup. (3)24, 233–334 (1907).zbMATHGoogle Scholar
  9. [N]
    Newman, D. J.: An entire function bounded in every direction, Amer. Math. Monthly 83, 192–193 (1976).MathSciNetCrossRefGoogle Scholar
  10. O] Osgood, W. F.: Note on the functions defined by infinite series whose terms are analytic functions of a complex variable; with corresponding theorems for definite integrals, Ann. Math. (2)3, 25–34 (1901–1902).Google Scholar
  11. [Pr]
    Pringsheim, A.: Über eine charakteristische Eigenschaft sogenannter Trep- penpolygone und deren Anwendung auf einen Fundamentalsatz der Funktionentheorie, Sitz. Ber. Math.-Phys. Kl. Königl. Bayrische Akad. Wiss. 1915, 27–57.Google Scholar
  12. [PS]
    Pôlya, G. and G. Szegö: Problems and Theorems in Analysis, 2 volumes, trans. C. E. BILLIGHEIMER, Springer, New York, 1976.CrossRefGoogle Scholar
  13. [Rub]
    Rubel, L. A.: How to use Runge’s theorem, L’Enseign. Math. (2)22, 185–190 (1976) and Errata ibid. (2)23, 149 (1977).Google Scholar
  14. [Run]
    Runge, C.: Zur Theorie der analytischen Functionen, Acta Math. 6, 245–248 (1885).Google Scholar
  15. [Sz]
    Saks, S. and A. Zygmund: Analytic Functions, trans. E. J. SCOTT, 3rd ed., Elsevier, 1971.Google Scholar

Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Reinhold Remmert
    • 1
  1. 1.Mathematisches InstitutWestfälische Wilhelms—Universität MünsterMünsterGermany

Personalised recommendations