Complex Analysis

Code:

M2008

Acronym:

M2008

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2023/2024 - 2S

Active?	Yes
Responsible unit:	Department of Mathematics
Course/CS Responsible:	Bachelor in Mathematics

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:B	0	Official Study Plan	3	-	6	48	162
L:CC	7	study plan from 2021/22	2	-	6	48	162
			3
L:F	16	Official Study Plan	2	-	6	48	162
			3
L:G	0	study plan from 2017/18	2	-	6	48	162
			3
L:M	94	Official Study Plan	2	-	6	48	162
L:MA	0	Official Study Plan	3	-	6	48	162
L:Q	0	study plan from 2016/17	3	-	6	48	162

Teaching language

Portuguese

Objectives

The student should know:  the basic results about sequences and series of complex numbers, continuity and derivability of complex functions of a complex variable, as well as its application to the computation of integrals of complex functions of a real variable.

Learning outcomes and competences

The student should know:  the basic results about sequences and series of complex numbers, continuity and derivability of complex functions of a complex variable, as well as its application to the computation of integrals of complex functions of a real variable.

Working method

Presencial

Pre-requirements (prior knowledge) and co-requirements (common knowledge)

Multivariable Calculus.

Program

1. Complex numbers
The complex number fiels. Topology of the complex plane.

2. Holomorphic functions and analytic functions
Differentiation. Power series. Analytic functions

3. Cauchy theory
Paths and loops. Cauchy's theorem and its applications. Analiticity of the holomorphic functions.

4. Singularities of analytic functions
Laurent series of analytic functions. Regular points and isolated singularities. The residue theorem and its applicatiosn.

Mandatory literature

Aníbal Coimbra A. de Matos; Curso de análise complexa. ISBN: 9789725921159

Complementary Bibliography

Serge Lang; Complex analysis. ISBN: 0-387-98592-1
Reinhold Remmert; Theory of complex functions. ISBN: 0-387-97195-5
David C. Ullrich; Complex made simple. ISBN: 9780821844793

Teaching methods and learning activities

Lectures and classes: The contents of the syllabus are presented in the lectures, where examples are given to illustrate the concepts. There are also practical lessons, where exercises and related problems are solved. All resources are available for students at the unit’s web page.

keywords

Physical sciences > Mathematics > Mathematical analysis

Evaluation Type

Distributed evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	50,00
Teste	50,00
Total:	100,00

Amount of time allocated to each course unit

designation	Time (hours)
Estudo autónomo	114,00
Frequência das aulas	48,00
Total:	162,00

Eligibility for exams

Attending classes is not compulsory: students are not required to attend the classes.

Calculation formula of final grade

There will be two tests, each corresponding approximately to half the syllabus. The second test may take place on the day for which the normal exam is scheduled. Students who do not pass through both tests can, in the second exam (which will be divided into two parts), use the classification obtained in one of the tests and only take the part of the exam corresponding to the other test. This possibility does not apply to anyone who improves their grade.

Grades above 17 will only be awarded after making an extra test.