Mathematical Analysis III B
Code
5005
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Rogério Ferreira Martins
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
2. Functions series
Pointwise and uniform convergence
Power series
Taylor and MacLaurin series
II – Complex analysis
1. Analytic functions
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
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
HUGHES-HALLET, GLEASON, MCCALLUM, et all, Calculus, 5th edition, Wiley, 2009.
AHLFORS, L. V., Complex Analysis, McGraw-Hill, 1979.
ANTON, H.; BIVENS, I.; DAVIS, S. - Calculus, 7ª edição, John Wiley and Sons, Inc., 2002.
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.
FIGUEIRA, M. - Fundamentos de Análise Infinitesimal, Textos de Matemática, vol. 5, Departamento de Matemática, Faculdade de Ciências da Universidade de Lisboa, 1996.
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.
SARRICO, C. - Análise Matemática, Leituras e Exercícios, Gradiva, 1997.
SPIVAK, M. - Calculus, World Student Series Edition, 1967.
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 2/3 of the labs.
The evaluation method consists on a three test evaluation during the semester or on a final exam evaluation. A complementary grade will be obtained in labs and theoretical lectures (up to 1 value for the labs and up to 1 value for the theoretical lectures) can improve the grade to a maximum of 2 values.
1-Two test evaluation
The student may realize two one and a half hours tests during the semester. We denote by N1 and N2 resp. their classification. At the final exam date, the student may choose to improve one of tests. Consequently, if the student is absent of one of the tests, he is not excluded of the three test evaluation method. The three test evaluation grade is AC=(N1+N2)/2.
2- Exam evaluation
Alternatively, the student may realize a 3 hour exam obtaining a grade AE.
3- Complementary grade
The teachers in the classroom in which the student is enrolled, at the end of the semester, provide a score bonus B, of 0 to 2 values.This last component corresponds to an impression of student performance and can be obtained in various ways, presentation of solutions of problems to the class, quizzes or by simple observation of student performance by direct observation.
4-Approval and final grade
The student obtains approval if AC+B (resp AE+B) is greater ou equal to 9,5 values. If AC+B (resp AE+B) is less than 16,5 , then the final grade NF is NF=AC+B (resp AE+B). In case AC+B (resp AE+B) is greater or equal to 16,5 values, the student may choose to obtain a 16 values final grade or to realize a complementary examination.