Complex Analysis

Code:

M2008

Acronym:

M2008

Level:

200

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2024/2025 - 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	0	study plan from 2021/22	2	-	6	48	162
			3
L:F	11	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

Students should come to grasp the basic concepts and techniques of the theory of functions of one complex variable, such as power series developments and Cauchy's theory. This study should equip the students with better skills to deal with the main objects and techniques of mathematical analysis.

Learning outcomes and competences

See the above paragraph.

Working method

Presencial

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

Real Analysis I, Real Analysis II

Program

Complex numbers and complex functions. Topology of the complex plane. Limits and continuity. Holomorphic functions and Cauchy-Riemann equations. Power series: radius of convergence, differentiability of functions defined by series. Analytic functions. Exponential, logarithm and trigonometric functions. Path integrals. Cauchy's integral formula. Theorems of Liouville, Goursat and Morera. Holomorphic functions are analytic. Laurent series. Riemann´s extension theorem for analytic functions. Casorati-Weierstrass's theorem. Residue theorem. The argument principle. Rouché's theorem. Using residues to compute certain real integrals.

Mandatory literature

Soares, Marcio G.; Cálculo em uma variável complexa, IMPA, 1999. ISBN: 85-244-0144-3
Matos Aníbal Coimbra A. de; Curso de análise complexa. ISBN: 9789725921159
Smirnov Gueorgui V.; Análise complexa e aplicações. ISBN: 972-592-152-6
Neto Alcides Lins; Funcoes de uma variavel complexa. 2ªed. ISBN: 85-244-0087-0

Teaching methods and learning activities

Theoretical lectures and problem-solving classes.

Evaluation Type

Evaluation with final exam

Assessment Components

designation	Weight (%)
Exame	100,00
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

Class attendance is not compulsory.

Calculation formula of final grade

Final examination only.

Special assessment (TE, DA, ...)

Any type of special evaluation can be either an oral or written examination, or a combination of both.

Observations

In case there is reasonable doubt as to the fairness of the result obtained in a written examination, the students may be subjected to an oral examination that may override or invalidate altogehter the result of the written examination