Credits: 9

Type: advanced

Objectives:

The study of complex dynamics and the basic principles of the delta bar operator and approximation theorems, infinite products and their applications; the domains of holomorphy of the functions of several complex variables and the differences in the case of one variable with the consideration of integral representations.

Knowledge:

• Complex dynamics : iteration of holomorphic and meromorphic functions; analytical conjugation; Julia and Fatou sets.
• Approximation theorems : Pompeiu's integral representation formula; Runge's theorem; the delta bar equation; the Mittag-Leffler theorem.
• Infinite products: Weierstrass formula of the gamma function, Stirling's formula; Zeros of holomorphic functions; Blaschke products.
• Functions of several complex variables: basic properties;domains of holomorphy; formulas of integral representation, power series, Hartogs' theorem.

Subject-specific competences:

To show understanding in the following areas: the use of Montel's theorem: iteration theory and Julia sets; approximation with rational functions and construction of meromorphic functions with prescribed singular parts; the Mittag-Leffler theorem; construction of holomorphic functions with prescribed zeros, with the Weierstrass theorem and Blaschke products; the holomorphic functions of several variables; power series of several variables.