In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function) along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued.
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| - Monodromiesatz (de)
- Théorème de monodromie (fr)
- Monodromy theorem (en)
- Теорема о монодромии (ru)
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| - Der Monodromiesatz ist ein wichtiger mathematischer Satz aus dem Gebiet der Funktionentheorie und beschreibt die Homotopie-Invarianz der analytischen Fortsetzung einer holomorphen Funktion. (de)
- Le théorème de monodromie est un outil puissant d'analyse complexe pour étendre une propriété locale (de germes) à une propriété globale (de fonction). On l'utilise par exemple dans certaines preuves des théorèmes de Picard pour inverser globalement la fonction j (invariant modulaire) aux points où sa dérivée est non nulle, alors que l'inversion n'est a priori que locale. (fr)
- Теорема о монодромии дает достаточное условие существования прямого аналитического продолжения аналитической функции, то есть существования иной аналитической на большем множестве функции, совпадающей с изначальной на первоначальной области определения. (ru)
- In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function) along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued. (en)
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| - Der Monodromiesatz ist ein wichtiger mathematischer Satz aus dem Gebiet der Funktionentheorie und beschreibt die Homotopie-Invarianz der analytischen Fortsetzung einer holomorphen Funktion. (de)
- In complex analysis, the monodromy theorem is an important result about analytic continuation of a complex-analytic function to a larger set. The idea is that one can extend a complex-analytic function (from here on called simply analytic function) along curves starting in the original domain of the function and ending in the larger set. A potential problem of this analytic continuation along a curve strategy is there are usually many curves which end up at the same point in the larger set. The monodromy theorem gives sufficient conditions for analytic continuation to give the same value at a given point regardless of the curve used to get there, so that the resulting extended analytic function is well-defined and single-valued. Before stating this theorem it is necessary to define analytic continuation along a curve and study its properties. (en)
- Le théorème de monodromie est un outil puissant d'analyse complexe pour étendre une propriété locale (de germes) à une propriété globale (de fonction). On l'utilise par exemple dans certaines preuves des théorèmes de Picard pour inverser globalement la fonction j (invariant modulaire) aux points où sa dérivée est non nulle, alors que l'inversion n'est a priori que locale. (fr)
- Теорема о монодромии дает достаточное условие существования прямого аналитического продолжения аналитической функции, то есть существования иной аналитической на большем множестве функции, совпадающей с изначальной на первоначальной области определения. (ru)
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