Complex and Fourier Analysis

Code:

M212

Acronym:

M212

Keywords
Classification	Keyword
OFICIAL	Mathematics

Instance: 2015/2016 - 2S

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

Cycles of Study/Courses

Acronym	No. of Students	Study Plan	Curricular Years	Credits UCN	Credits ECTS	Contact hours	Total Time
L:AST	0	Plano de Estudos a partir de 2008	3	-	7,5	-
L:B	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:F	0	Plano de estudos a partir de 2008	3	-	7,5	-
L:G	7	P.E - estudantes com 1ª matricula anterior a 09/10	3	-	7,5	-
		P.E - estudantes com 1ª matricula em 09/10	3	-	7,5	-
L:M	82	Plano de estudos a partir de 2009	2	-	7,5	-
L:Q	0	Plano de estudos Oficial	3	-	7,5	-

Teaching language

Suitable for English-speaking students

Objectives

Students are expected to learn the basic concepts of the theory of functions of one complex variable, with particular emphasis on power series expansions and Cauchy's theory. They will thereby acquire better skills in dealing with the main objects and techniques of mathematical analysis.

Learning outcomes and competences

See above paragraph.

Working method

Presencial

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

Calculus of one or several variables

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. Integrals over paths. Homotopy. Cauchy formula. Liouville, Goursat and Morera theorems. Analyticity of holomorphic functions. Singularities and meromorphic functions. Laurent representation. Riemann extension theorem. Casorati-Weierstrass theorem. Residue theorem. Principle of the argument. Rouché theorem. Calculation of integrals using residues.Introdução às séries trigonométricas de Fourier: desigualdade de Bessel, igualdade de Parseval, convergência pontual e convergência uniforme de séries de Fourier. Trigonometric Fourier series: Bessel inequality, Parseval identity, pointwise and uniform convergence of Fourier series.

Mandatory literature

Figueiredo Djairo Guedes; Análise de Fourier e equações diferenciais parciais. ISBN: 85-244-0026-9
Matos Aníbal Coimbra A. de; Curso de análise complexa. ISBN: 9789725921159
Neto Alcides Lins; Funcoes de uma variavel complexa. 2ªed. ISBN: 85-244-0087-0
Stein Elias / Shakarchi Rami; Complex Analysis, Princeton University Press, 2003. ISBN: 0-691-11385-8

Teaching methods and learning activities

Lectures and practical classes

Evaluation Type

Distributed evaluation with final exam

Assessment Components

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

Calculation formula of final grade

The regular evaluation will be based on one test and a final examination, the classification distributed over 10 points in the test and 10 in the final exam. The final grade is the sum of the grades of the test and exam.

All registered students are admitted to the test and exam.

There will be an extra exam available to any student who does not obtain approval according to the above scheme.