
Course unit
COMPLEX ANALYSIS
SCN1037789, A.A. 2018/19
Information concerning the students who enrolled in A.Y. 2018/19
Mutuated
Course unit code 
Course unit name 
Teacher in charge 
Degree course code 
SCN1037789 
COMPLEX ANALYSIS 
PIETRO POLESELLO 
SC1172 
ECTS: details
Type 
ScientificDisciplinary 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 
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)

Prerequisites:

 Undergraduate courses in Calculus and Geometry
 Elementary notions on complex functions of one complex variable. In particular:
CauchyRiemann 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.
Zeroset 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 MittagLeffler'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, SpringerVerlag, Berlin (1991)
 Reinhold Remmert, Theory of Complex Functions. Graduate Texts in Mathematics, SpringerVerlag, Berlin (1991) 
Textbooks (and optional supplementary readings) 

JeanPierre Schneiders, Fonctions de Variables Complexes. UniversitÃ© de LiÃ¨ge: self published, 2010. The pdf will be available from the course's home page

Rudin, Walter, Real and complex analysis. New York [etc.]: McGrawHill, 1986.

Gamelin, Theodore W., Complex analysis. New York [etc.]: Springer, 2001.

Ash, Robert B.; Novinger, W. Phil, Complex variables. Mineola: Dover publications, 2007.

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

