Second Level Degree in Mathematics
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COMPLEX ANALYSIS - 6 CFU
Teacher
Pietro Polesello
Scheduled Period
I Year - 1 Semester | 28/09/2020 - 16/01/2021
Hours: 48 (24 esercitazione, 24 lezione)
Prerequisites
- Undergraduate courses in Calculus and Geometry
- Elementary notions on complex functions of one complex variable. In particular:
Cauchy-Riemann identities and complex differentiation; holomorphic functions. Line integrals of complex functions and their homotopy invariance.
Logarithm of a path and winding number. Cauchy formula for a circle. Analiticity of holomorphic functions.
Zero-set of a holomorphic function; the identity theorem.
Laurent series and isolated singularities. Residue theorem and its use for the computation of integrals.
(All these notions will be recalled in the first lectures.)
Target skills and knowledge
Advanced notions on complex functions of one complex variable (in particular: the main properties of holomorphic/meromorphic functions on the plane and on the extended plane and the different ways to represent/construct them - by means of series, integrals or infinite products; the study of the conformal maps between regions in the plane, and of the Gamma and Zeta functions), with applications (in particular: the Prime Number theorem).
Examination methods
Written exam (exercises, theoretical exercises, statements and proofs; duration: 2h30) with possible additional oral exam to improve the mark.
Assessment criteria
The student must be able to solve exercises and has to know the most important statements of the theory, with proofs.
Course contents
- The Argument principle and applications
- Conformal maps and the Riemann Mapping theorem
- The Schwarz reflection principle
- Runge's theory and applications
- Infinite products and the Weierstrass factorization theorem
- Partial Fraction Decompositions and Mittag-Leffler's theorem
- Principal ideals of holomorphic functions
- Some special functions (Gamma, Zeta)
- The Prime Number theorem
Planned learning activities and teaching methods
Lectures with the tablet (possibly only online), with assignments of exercises (solved during the lectures).
Additional notes about suggested reading
Additional references:
- slides of the lectures given in class
- selected exercises with solutions
- Giuseppe De Marco, Selected Topics of Complex Analysis, self - published (2012)
- Giuseppe De Marco, Basic Complex Analysis, self published (2011)
- Reinhold Remmert, Classical Topics in Complex Function Theory. Graduate Texts in Mathematics, Springer-Verlag, Berlin (1991)
- Reinhold Remmert, Theory of Complex Functions. Graduate Texts in Mathematics, Springer-Verlag, Berlin (1991)
Textbooks (and optional supplementary readings)
- Jean-Pierre Schneiders, Fonctions de Variables Complexes, Université de Liège, self published, 2010 The pdf will be available from the course's home page
- Rudin, Walter, Real and complex analysis, New York, McGraw-Hill, 1974.
- Gamelin, Theodore W., Complex analysisTheodore W. Gamelin, New York [etc.], Springer, 2001.
- Ash, Robert B.; Novinger, W. Phil, Complex variablesRobert B. Ash, W. Phil Novinger, Mineola, Dover publications, 2007.