Il corso fa parte del programma Mundus Master ALGANT, per cui le lezioni si terranno in inglese.
Course description:
Advanced notions on complex functions of one complex variable, with applications (48h, 6 credits).
Syllabus:
The Argument principle and applications.
Conformal maps and the Riemann Mapping theorem.
Runge's theory and applications.
Infinite products and the Weierstrass factorisation theorem.
Partial Fraction Decompositions and Mittag-Leffler's theorem.
Principal ideals of holomorphic functions.
Some special functions (Gamma, Zeta).
The Prime Number theorem.
The Schwarz reflection principle.
NEW:
Detailed syllabus 2020-2021
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; analyticity 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.
Office hours: by e-mail appointment.
Schedule: first semester 2020.
10h30-12h15, Thursday and Friday (from Thursday October 1 to Friday December 18), on Zoom.
The 24th (and last) lecture of the course will be on January 8 2021 (no lecture on January 7 2021).
To subscribe the Zoom lectures click here. The links to the Zoom lectures will be also available on the Moodle page.
Exams: written exam (exercises, theoretical exercises, statements and proofs; duration: 2h30) with possible additional oral examination.
Written examination online modality. Instruction for students
- January 28, 2021; 9h30, online via Zoom (deadline for registration: 27/1/2021). Registration/oral exam: February 1, 2021; 14h30, online via Zoom.
- February 12, 2021; 9h30, online via Zoom (deadline for registration: 11/2/2021). Registration/oral exam: February 22, 2021; 9h30, online via Zoom.
- July 6, 2021; 9h30, online via Zoom (deadline for registration: 5/7/2021). CHANGE OF DATE: Registration/oral exam: July 9, 2021; 9h30.
- September 1, 2021; 14h30, online via Zoom (deadline for registration: 31/8/2021). Registration/oral exam: September 7, 2021; 14h30, online via Zoom.
- February, 2022; on request.
Exams informations and registration: UNIWEB
You can have a double-sided printed Formulae (last version) during the written examination. You are allowed to handwrite your own formulas in the blank part of the second page. Only one double-sided printed Formulae per person.
Check Student's page for detailed examination texts (with solutions).
Textbooks:
Other useful references:
- 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)
- Giuseppe De Marco - Selected Topics of Complex Analysis (2012)
- Giuseppe De Marco - Basic Complex Analysis (2011)
Student's page
Links:
Last modified: June 10, 2021