First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
MATHEMATICS
Course unit
COMPLEX ANALYSIS
SCN1037789, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2018/19

Information on the course unit
Degree course Second cycle degree in
MATHEMATICS
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2018/19
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination COMPLEX ANALYSIS
Website of the academic structure http://matematica.scienze.unipd.it/2018/laurea_magistrale
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Branch PADOVA
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Lecturers
Teacher in charge PIETRO POLESELLO MAT/05

Mutuated
Course unit code Course unit name Teacher in charge Degree course code
SCN1037789 COMPLEX ANALYSIS PIETRO POLESELLO SC1172

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/05 Mathematical Analysis 6.0

Course unit organization
Period First semester
Year 1st Year
Teaching method frontal

Type of hours Credits Teaching
hours
Hours of
Individual study
Shifts
Practice 3.0 24 51.0 No turn
Lecture 3.0 24 51.0 No turn

Calendar
Start of activities 01/10/2018
End of activities 18/01/2019

Examination board
Board From To Members of the board
6 Analisi Complessa - a.a. 2018/2019 01/10/2018 30/09/2019 POLESELLO PIETRO (Presidente)
D'AGNOLO ANDREA (Membro Effettivo)
BARACCO LUCA (Supplente)
MARASTONI CORRADO (Supplente)
ROSSI FRANCESCO (Supplente)

Syllabus
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 unit 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 (on the blackboard and with the tablet) and exercises.
Additional notes about suggested reading: Additional references:
- slides of the lectures given in class
- 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 [etc.]: McGraw-Hill, 1986. Cerca nel catalogo
  • Gamelin, Theodore W., Complex analysis. New York [etc.]: Springer, 2001. Cerca nel catalogo
  • Ash, Robert B.; Novinger, W. Phil, Complex variables. Mineola: Dover publications, 2007. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • One Note (digital ink)
  • Webpage