To acquire a broad knowledge of holomorphic and meromorphic functions of one complex variable and of their main properties. To acquire good dexterity in complex integration and in the calculation of real definite integrals
teacher profile teaching materials
Series and Abel's theorem. Exponential and logarithms.
Elementary conformal mappings. Complex integration; Cauchy's theorem; Cauchy's formula.
Local properties of holomorphic functions (singularities, zeroes and poles; local mapping theorem and
maximum principle).
Residues.
Harmonic functions.
Series expansions (Weierstrass' theorem, Taylor's series).
Partial fractions and infinite products.
Supplementary arguments (depending on time): entire functions and Hadamard's theorem. Riemann zeta function.
Riemann mapping theorem.
Programme
The complex field. Holomorphic functions; Cauchy-Riemann equations.Series and Abel's theorem. Exponential and logarithms.
Elementary conformal mappings. Complex integration; Cauchy's theorem; Cauchy's formula.
Local properties of holomorphic functions (singularities, zeroes and poles; local mapping theorem and
maximum principle).
Residues.
Harmonic functions.
Series expansions (Weierstrass' theorem, Taylor's series).
Partial fractions and infinite products.
Supplementary arguments (depending on time): entire functions and Hadamard's theorem. Riemann zeta function.
Riemann mapping theorem.
Core Documentation
Ahlfors, Lars V, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. International Series in Pure and Applied Mathematics. McGraw-Hill Book Co., New York, 1978. xi+331 pp. ISBN 0-07-000657-1Reference Bibliography
[L] Lang, Serge Complex analysis. (English summary) Fourth edition. Graduate Texts inMathematics, 103. Springer-Verlag, New York, 1999. xiv+485 pp. ISBN 0-387-98592-1 [P] Pap, Endre Complex Analysis Through Examples and Exercises Kluwer Texts in the Mathematical Sciences, V. 21 (Hardcover, 1999) [E] M. Evgrafov, Coll, Recueil de problèmes sur la théorie des fonctions analytiques, Traduction francaise, Editions Mir, 1974.Type of delivery of the course
60 hours of lectures including parts of exercises. A fundamental part of the lessons is the interaction between students and teacher and in particular the feedback on the degree of learning.Type of evaluation
The assessment is based on a written paper on fundamental topics discussed in class and on an oral test which consists of a discussion of the written and of a possible discussion on other aspects of the course content.