Mathematical Analysis III B
Code
5005
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Joaquim Eurico Anes Duarte Nogueira, João Pedro Bizarro Cabral
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
1. the student should understand the basic concepts and be able to compute the quantities presented in the exercises.
Prerequisites
The student should know the basic concepts of Calculus in one and several variables the he learns in the disciplines of Mathematical Analysis I and II.
Subject matter
I - Series
1. Numerical Series
Alternating series
Absolute convergence
Non-negative terms
Product of series
II – Complex analysis
1. Holomorphic functions
Complex numbers
Complex sequences and complex numerical series
Elementary and continuous function
Analytic functions
2. Cauchy’s Theorem
Integrals
Cauchy’s theorem: Green’s theorem
Cauchy integral formula
Maximum modulus theorem and Harmonic functions
3. Series representation of analytic functions
Pointwise and uniform convergence of function series
Convergent series of analytic functions
Power series and Taylor’s theorem
Laurent series and classification of singularities
4. Calculus of residues
Residue theorem
Evaluation of definite integrals
5. Conformal mappings
Bibliography
SAFF, E. B.; SNIDER, A. D. - Fundamentals of Complex Analysis with Applications to Engineering and Science - 3rd Edition, Pearson Education, 2003.
AHLFORS, L. V., Complex Analysis, McGraw-Hill, 1979.
ANTON, H.; BIVENS, I.; DAVIS, S. - Cálculo II; 8ª Edição, Bookman, 2007.
CAMPOS FERREIRA, J. - Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.
CARREIRA, M. A. e NÁPOLES, M. S., Variável complexa - Teoria elementar e exercícios resolvidos, McGraw-Hill.
DIAS AGUDO, F. R. - Análise Real, 2ª edição, Livraria Escolar Editora, 1994.
KREYSZIG, E., Advanced Engineering Methods, 8ª edição, John Wiley and Sons, Inc., 1999.
MARSDEN, J. E., e HOFFMAN, M. J., Basic Complex Analysis, 3ª edição, Freeman, 1999.
MARSDEN, J. e WEINSTEIN, A. - Calculus III; Springer, 2ªEdição, 1984.
SÁ, A. e LOURO, B. - Sucessões e Séries, Teoria e Prática. Escolar Editora, 2009.
SMIRNOV, Gueorgui V. - Análise Complexa e Aplicações. Escolar Editora, 2003.
Teaching method
The topics will be presented in lectures (two lectures of 1.5 hours per week), along with illustrative examples. Labs (a class of 2h per week), will be devoted to problem solving, assisted by the teacher.
Evaluation method
Important:
In order to be evaluated, the student must attend at least to 2/3 of all classes.
Evaluation Methods.
1-Evaluation by tests
This evaluation consists on three tests during the semester. The final grade is the average of the grades of the three tests. The student is approved if the final grade is greater or equal than 9,5.
Each test has a duration of 1h 30min.
2-Final exam evaluation.
The student is approved if the grade of the final exam is greater or equal than 9,5.
The final exam has a duration of 3h.