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Cours scientifiques - FMA_3F013_EP : Complex Analysis

Domaine > Mathématiques.

Descriptif

The central object of the course are complex functions of the complex variable. We will study the notion of complex differentiability, also known as holomorphicity, and the main properties of holomorphic functions. In particular, we will see that

  • the integral of a holomorphic function along closed curves is always 0
  • holomorphic functions are infinitely many times differentiable
  • holomorphic functions that coincide on any arbitrarily small disc in an open connected set coincide on the full set.

It will be clear pretty soon that complex-differentiability is a much stronger requirement than the usual differentiability with respect to real variables.

We will also study the different singularities a complex function of the complex variable might have, the famous residue formula and its applications.

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Vous devez avoir validé l'équation suivante : UE FMA_2F002_EP

Format des notes

Numérique sur 20

Littérale/grade américain

Pour les étudiants du diplôme Programmes d'échange internationaux

Pour les étudiants du diplôme Bachelor of Science de l'Ecole polytechnique

Le rattrapage est autorisé (Note de rattrapage conservée écrêtée à une note seuil de 10)
    L'UE est acquise si Note finale >= 9
    • Crédits ECTS acquis : 5 ECTS

    La note obtenue rentre dans le calcul de votre GPA.

    Programme détaillé

    This course will cover the following topics:

    1. Holomorphic functions: definitions and first properties
    2. Cauchy-Riemann equations and complex differentiability
    3. Integration over paths
    4. Cauchy integral formula
    5. Liouville's theorem
    6. Zeros of holomorphic functions
    7. Analytic continuation
    8. The maximum principle
    9. Meromorphic functions
    10. The complex logarithm
    11. Sequences, series and products of holomorphic functions

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