Functions with Prescribed Principal Parts

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


If h is meromorphic in the region D, its pole setP(h) is locally finite in D. By the existence theorem 4.1.5, every set that is locally finite in D is the pole set of some function h ∈M(D) (see also 3.1.5(1)). We now pose the following problem:

Let T = {d1, d2, ...} be a set that is locally finite in D, and let every point dvT be somehow assigned a “ finite principal part”\({q_v}\left( z \right) = \sum\nolimits_{u = 1}^{mv} {avu} {\left( {z - {d_v}} \right)^{ - u}} \ne 0\). Construct a function meromorphic in D that has T as its pole set and moreover has principal part qv at each point dv.


Holomorphic Function Meromorphic Function Ideal Theory Principal Part Principal Ideal 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A]
    Alling, N. L.: Global ideal theory of meromorphic function fields, Trans. Amer. Math. Soc. 256, 241–266 (1979).MathSciNetCrossRefGoogle Scholar
  2. [BStl]
    Behnke, H. and K. STEIN: Analytische Funktionen mehrerer Veränderlichen zu vorgegebenen Null-und Polstellenflächen, Jber. DMV47, 177–192 (1937).zbMATHGoogle Scholar
  3. [BSt2]
    Behnke, H. and K. Stein: Elementarfunktionen auf Riemannschen Flä- chen…, Can. Journ. Math. 2, 152–165 (1950).CrossRefGoogle Scholar
  4. [BT]
    Behnke, H. and P. Thullen: Theorie der Funktionen mehrerer komplexerVeränderlichen, 2nd ed., with appendices by W. BARTH, O. Forster, H. Holmann, W. KAUP, H. Kerner, H.-J. Reiffen, G. Scheja, and K. Spallek, Erg. Math. Grenzgeb. 51, Springer, 1970.Google Scholar
  5. [Bo]
    Bourbaki, N.: Commutative Algebra, Chapter 7, Divisors, Addison-Wesley, Reading, 1972.zbMATHGoogle Scholar
  6. [Cal]
    Cartan, H.: OEuvres 1, Springer, 1979.Google Scholar
  7. [Ca2]
    Cartan, H.: OEuvres 2, Springer, 1979.Google Scholar
  8. [Co]
    Cousin, P.: Sur les fonctions de n variables complexes, Acta Math. 19, 1–62 (1895).MathSciNetCrossRefGoogle Scholar
  9. [D]
    Domar, Y.: Mittag-Leffler’s theorem, Dept. Math. Uppsala University, Re-port No. 1, 1981.Google Scholar
  10. [FL]
    Fischer, W. and I. LIEB: Funktionentheorie, Vieweg u. Sohn, Braun-schweig, 1980.CrossRefGoogle Scholar
  11. [GRi]
    Grauert, H. and R. Remmert: Theory of Stein Spaces, trans. A. Huck-Leberry, Grundl. math. Wiss. 236, Springer, 1979.Google Scholar
  12. [GR2]
    Grauert, H. and R. Remmert: Coherent Analytic Sheaves, Grundl. math. Wiss. 265, Springer, 1984.Google Scholar
  13. [Hel]
    Helmer, O.: Divisibility properties of integral functions, Duke Math. Journ. 6, 345–356 (1940).MathSciNetCrossRefGoogle Scholar
  14. [Hen]
    Henriksen, M.: On the ideal structure of the ring of entire functions, Pac. Journ. Math. 2, 179–184 (1952).MathSciNetCrossRefGoogle Scholar
  15. [Her]
    Hermite, C.: Sur quelques points de la théorie des fonctions (Extrait d’une lettre de M. Hermite à M. Mittag-Leffler), Journ. reine angew. Math. 91, 54–78 (1881); OEuvres 4, 48–75.Google Scholar
  16. [Hu]
    Hurwitz, A.: Sur l’intégrale finie d’une fonction entière, Acta Math. 20, 285–312 (1897); Math. Werke 1, 436–459.MathSciNetCrossRefGoogle Scholar
  17. [ML]
    Mittag-Leffler, G.: Sur la représentation analytique des fonctions mono-gènes uniformes d’une variable indépendante, Acta Math. 4, 1–79 (1884).MathSciNetCrossRefGoogle Scholar
  18. [N]
    Narasimhan, R.: Complex Analysis in One Variable, Birkhäuser, 1985.Google Scholar
  19. [O1]
    Oka, K.: II. Domaines d’holomorphie, Journ. Sci. Hiroshima Univ., Ser. A, 7, 115–130 (1937); Coll. Pap., trans. R. NARASIMHAN, 11–23, Springer, 1984.Google Scholar
  20. [02]
    OKA, K.: Sur quelques notions arithmétiques, Bull. Soc. Math. France 78, 1–27 (1950); Coll. Pap. 80–108, Springer 1984.Google Scholar
  21. [P]
    Pringsheim, A.: Über die Weierstrass’sche Produktdarstellung ganzer tran-szendenter Funktionen und über bedingt convergente unendliche Produkte, Sitz. Ber. math.-phys. Kl. Königl. Bayer. Akad. Wiss. 1915, 387–400.Google Scholar
  22. [Rub]
    Rubel, L. A.: Linear compositions of two entire functions, Amer. Math. Monthly 85, 505–506 (1978).MathSciNetCrossRefGoogle Scholar
  23. [Rü]
    RÜCkert, W.: Zum Eliminationsproblem der Potenzreihenideale, Math. Ann. 107, 259–281 (1933).MathSciNetCrossRefGoogle Scholar
  24. [Sch]
    Schilling, O. F. G.: Ideal theory on open Riemann surfaces, Bull. Amer. Math. Soc. 52, 945–963 (1946).MathSciNetCrossRefGoogle Scholar
  25. [Wed]
    Wedderburn, J. H. M.: On matrices whose coefficients are functions of a single variable, Trans. Amer. Math. Soc. 16, 328–332 (1915).MathSciNetCrossRefGoogle Scholar
  26. [Wei]
    weierstrass, K.: Math. Werke 2.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