Functions with Prescribed Principal Parts

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

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 
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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

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

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