Complex Analysis
Keywords |
Classification |
Keyword |
OFICIAL |
Mathematics |
Instance: 2024/2025 - 2S
Cycles of Study/Courses
Teaching Staff - Responsibilities
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