Classical Topics in Complex Function Theory pp 125-143 | Cite as
Functions with Prescribed Principal Parts
Chapter
Abstract
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 d v ∈ T 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 q v at each point d v.
Keywords
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.
Preview
Unable to display preview. Download preview PDF.
Bibliography
- [A]Alling, N. L.: Global ideal theory of meromorphic function fields, Trans. Amer. Math. Soc. 256, 241–266 (1979).MathSciNetMATHCrossRefGoogle Scholar
- [BStl]Behnke, H. and K. STEIN: Analytische Funktionen mehrerer Veränderlichen zu vorgegebenen Null-und Polstellenflächen, Jber. DMV 47, 177–192 (1937).Google Scholar
- [BSt2]Behnke, H. and K. Stein: Elementarfunktionen auf Riemannschen Flä- chen…, Can. Journ. Math. 2, 152–165 (1950).MathSciNetMATHCrossRefGoogle Scholar
- [BT]Behnke, H. and P. Thullen: Theorie der Funktionen mehrerer komplexer Veränderlichen, 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
- [Bo]Bourbaki, N.: Commutative Algebra, Chapter 7, Divisors, Addison-Wesley, Reading, 1972.Google Scholar
- [Cal]Cartan, H.: OEuvres 1, Springer, 1979.Google Scholar
- [Ca2]Cartan, H.: OEuvres 2, Springer, 1979.Google Scholar
- [Co]Cousin, P.: Sur les fonctions de n variables complexes, Acta Math. 19, 1–62 (1895).MathSciNetMATHCrossRefGoogle Scholar
- [D]Domar, Y.: Mittag-Leffler’s theorem, Dept. Math. Uppsala University, Re-port No. 1, 1981.Google Scholar
- [FL]Fischer, W. and I. LIEB: Funktionentheorie, Vieweg u. Sohn, Braun-schweig, 1980.Google Scholar
- [GRi]Grauert, H. and R. Remmert: Theory of Stein Spaces, trans. A. Huck-Leberry, Grundl. math. Wiss. 236, Springer, 1979.Google Scholar
- [GR2]Grauert, H. and R. Remmert: Coherent Analytic Sheaves, Grundl. math. Wiss. 265, Springer, 1984.Google Scholar
- [Hel]Helmer, O.: Divisibility properties of integral functions, Duke Math. Journ. 6, 345–356 (1940).MathSciNetCrossRefGoogle Scholar
- [Hen]Henriksen, M.: On the ideal structure of the ring of entire functions, Pac. Journ. Math. 2, 179–184 (1952).MathSciNetMATHCrossRefGoogle Scholar
- [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
- [Hu]Hurwitz, A.: Sur l’intégrale finie d’une fonction entière, Acta Math. 20, 285–312 (1897); Math. Werke 1, 436–459.Google Scholar
- [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
- [N]Narasimhan, R.: Complex Analysis in One Variable, Birkhäuser, 1985.Google Scholar
- [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
- [02]OKA, K.: Sur quelques notions arithmétiques, Bull. Soc. Math. France 78, 1–27 (1950); Coll. Pap. 80–108, Springer 1984.Google Scholar
- [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
- [Rub]Rubel, L. A.: Linear compositions of two entire functions, Amer. Math. Monthly 85, 505–506 (1978).MathSciNetCrossRefGoogle Scholar
- [Rü]RÜCkert, W.: Zum Eliminationsproblem der Potenzreihenideale, Math. Ann. 107, 259–281 (1933).MathSciNetCrossRefGoogle Scholar
- [Sch]Schilling, O. F. G.: Ideal theory on open Riemann surfaces, Bull. Amer. Math. Soc. 52, 945–963 (1946).MathSciNetMATHCrossRefGoogle Scholar
- [Wed]Wedderburn, J. H. M.: On matrices whose coefficients are functions of a single variable, Trans. Amer. Math. Soc. 16, 328–332 (1915).MathSciNetMATHCrossRefGoogle Scholar
- [Wei]weierstrass, K.: Math. Werke 2.Google Scholar
Copyright information
© Springer Science+Business Media New York 1998