Teaching plan for the course unit



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General information


Course unit name: Complex Analysis of One or More Variables

Course unit code: 568181

Academic year: 2021-2022

Coordinator: Jorge Marzo Sanchez

Department: Department of Mathematics and Computer Science

Credits: 6

Single program: S



Estimated learning time

Total number of hours 150


Face-to-face and/or online activities



-  Lecture





(in-person and online classes)


-  Lecture with practical component





(in-person and online classes)

Supervised project


(Final assignment.)

Independent learning


(Problem study and resolution.)





Students should have completed a subject in complex variables.



Competences to be gained during study


— Capacity for rigorous analysis of theoretical or practical problems under conditions of uncertainty, for the purpose of expanding knowledge and later developing research and working in multidisciplinary contexts. 
— Ability to access the bibliographic items available that are required to achieve the above.
— Understanding of rigorous mathematical arguments and the capacity for rigorous expression in mathematical language.
— Capacity for the exchange of ideas when working on a group project.





Learning objectives


Referring to knowledge

— To understand elements of potential theory, harmonic and subharmonic functions, and the relationship with holomorphic functions.


— To understand the role of the Cauchy-Riemann equation in the study of holomorphic functions.


— To identify the basic differences between the theory of functions of a complex variable and that of several variables.


Referring to abilities, skills

— To perform introductory mathematical research in this field, starting with the appropriate bibliographical sources.



Teaching blocks


1. The inhomogeneous Cauchy-Riemann equation and applications

2. Harmonic functions and the Dirichlet problem

3. Subharmonic functions and zeros of holomorphic functions

4. Functions of several variables; Local theory



Teaching methods and general organization


The different subject areas are introduced in lecture classes. A collection of problems relating to each teaching block is handed out, and the procedures for solving them are explained and discussed in class; Students are required to submit completed answers to some of the problems; Once the answers have been marked, some of them may be discussed in class.



Official assessment of learning outcomes


Assessment is based on the submitted answers to problems. There may be also a final project that would have to be presented. 
In cases where a pass grade is not achieved the students can go to a final examination with theoretical and practical questions.


It consists of a final examination with theoretical and practical questions.



Examination-based assessment

The single assessment consists of a final examination with theoretical and practical questions.


It consists of a final examination with theoretical and practical questions.



Reading and study resources

Consulteu la disponibilitat a CERCABIB


M. Andersson: Topics in Complex Analysis, Universitext: Tracts in Mathematics, Springer 1997

J. Bruna, J. Cufí: Complex Analysis, EMS Textbooks in Mathematics 2013

S. Krantz: Function theory of several complex variables , AMS-chelsea 1992

L. Hörmander: Complex analysis in several variables, North Holland 1973

T. Ransford: Potential Theory in the Complex Plane, London Math Society.

W. Rudin: Real and Complex Analysis McGraw-Hill, New. York, 1966

Web page

J. Korevaar-J. Wiegerinck: Lecture notes several complex variables