Algebraic Curves by Maxim E. Kazaryan & Sergei K. Lando & Victor V. Prasolov

Author:Maxim E. Kazaryan & Sergei K. Lando & Victor V. Prasolov
Language: eng
Format: epub
ISBN: 9783030029432
Publisher: Springer International Publishing

9.1 Mittag-Leffler’s Problem

Mittag-Leffler’s problem was the starting point that gave rise to the theory of sheaves. It is formulated as follows.

Given a collection of principal parts of local functions at poles on a curve, determine whether they are the principal parts of a meromorphic function that is defined on the whole curve and has no other poles.

The statement of the problem should be made more precise, and we proceed to necessary definitions. For simplicity, we begin by considering the case of simple poles. So, let C be a complex curve and x ∈ C; we consider meromorphic functions defined in a neighborhood of x and having a simple pole or no pole at x. Two functions f 1, f 2 are said to have the same principal part at x if their difference has no pole at x. Clearly, “having the same principal part” is an equivalence relation on the set of such functions. The principal part of a function f at x is the equivalence class containing f.

If we introduce in a neighborhood of x a local coordinate z centered at x, then every function f with a simple pole or no pole at x can be written as a Laurent series

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