Mathematical Analysis III B
Code
5005
Academic unit
Faculdade de Ciências e Tecnologia
Department
Departamento de Matemática
Credits
6.0
Teacher in charge
Ana Cristina Malheiro Casimiro, Joaquim Eurico Anes Duarte Nogueira
Weekly hours
5
Total hours
70
Teaching language
Português
Objectives
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
1 – Series
1.1 Numerical Series.
1.1.1 Convergence of Numerical Series. Necessary Condition for Convergence. Telescoping Series. Geometric Series.
1.1.2 Series with non-negative terms. Integral Test. Dirichlet Series. Comparison Tests.
1.1.3 Ratio Test. D’Alembert Test. Root Test. Cauchy Test. Kummer Test. Raabe Test.
1.1.4 Simple and Absolute Convergence. Alternating Series and Leibniz Test. Product of Series.
1.2 Series of Functions
1.2.1 Sequences of functions: Pointwise and Uniform Convergence.
1.2.2 Series of functions: Pointwise and Uniform Convergence. Weierstrass Test. Continuity. Term by term integration and differentiation.
1.2.3 Power Series. Radius of Convergence. Interval of Convergence. Uniform Convergence. Integrabilidade e diferenciabilidade termo a termo.
1.2.4 Taylor and MacLaurin Series.
2 Complex Analysis
2.1 Generalities about the field of complex numbers; conjugate, modulus and argument; polar form of a complex number. Roots of complex numbers. De Moivre’s Formula.
2.2 Polynomial functions of complex variable. Exponential function, circular and hyperbolic trigonometric functions, principal branch of the logarithm and inverse trigonometric functions;
2.3 Limits and Continuity of complex functions with complex variable.
2.4 Holomorphic Functions. Cauchy- Riemann Equations.
2.5 Integral of a complex function with complex variable along a piecewise smooth curve.
2.6 Cauchy Theorem.
2.7 Cauchy’s Integral Formula.
2.8 Analytic Functions. Taylor Series. Relation with holomorphic functions.
2.9 Essential Singularities, poles and removable singularities. Laurent Series.
2.10 Residue Theorem. Applications to improper integrals computation.
Bibliography
AHLFORS, L. V., Complex Analysis, McGraw-Hill, 1979.
AGARWAL, Ravi, PERERA, Kanisshka e PINELAS, Sandra - An Introduction to Complex Analysis, 2011, Springer
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.
MARSDEN, J., e HOFFMAN, M. J., Basic Complex Analysis, 3ª edição, Freeman, 1999.
MARSDEN, J. e WEINSTEIN, A. - Calculus III; Springer, 2ªEdição, 1984.
SAFF, E. B.; SNIDER, A. D. - Fundamentals of Complex Analysis with Applications to Engineering and Science - 3rd Edition, Pearson Education, 2003.
SÁ, A. e LOURO, B. - Sucessões e Séries, Teoria e Prática. Escolar Editora, 2009.
Teaching method
Teaching Method is based on conferences and problems solving sessions with the support of a personal attending schedule.
Evaluation method
Frequency
Frequency will be granted to any student who does not unjustifiably miss more than 2/3 of the practical classes taught. Students who have obtained it in one of the semesters of the 2016/2017 school year or who have any of the special statutes provided by law are exempt from attendance.
The evaluation is carried out through Continuous evaluation or Exam evaluation.
Continuous evaluation
During the semester two tests will be carried out with a duration of 1 hour 30 minutes and an evaluation of the practical classes (ap). Each test is rated up to a maximum of 20 values and the practical classes can be rated between 0 and 2 values.
1st Test, October 28, 2017, (t1): all students enrolled in the course may present themselves to the 1st test.
2nd Test, December 18, 2017, (t2): all the students enrolled in the course that have obtained a frequency or have a special status may submit to the 2nd test.
Evaluation of the practical classes: the teacher of the practical class in which the student is enrolled will provide at the end of the semester the classification between 0 and 2 values. This corresponds to the evaluation made by the teacher, through the student''s performance in solving the problems proposed in the classes.
The classification of continuous evaluation (CA) is obtained by the following formula:
AC = (t1 + t2) / 2 +ap
The student is approved in the course if AC is greater than or equal to 9.5 values. If AC is less than 16.5, the final grade of the course will be AC. If AC is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values or performing a supplementary examination.
Exam
All the students enrolled in the course that have obtained Frequency or have special status may submit to the Exam.
At the date and time scheduled for the Exam in January 2018, any student enrolled in the course that has obtained Frequency or has special status and who has not obtained approval in the Continuous Evaluation can take the exam for 3 hours or can opt for repeat one of the tests for 1 hour 30 minutes. If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation.
If the student performs the Exam (his or her classification is er) and
AE = er + ap
is greater than or equal to 9.5, the student is approved. If AE is less than 16.5 the final grade of the course will be AE. If AE is greater than or equal to 16.5 values, the student can choose between obtaining a final grade of 16 values or performing a supplementary examination.
Grade improvement
Students have the right to improve grade by enrollment within the established deadlines, at the time of the Exam. In this case, they may take the 3-hour Exam or repeat one of the 1-hour 30-minute Tests as described in the previous paragraph. In the case that a student wants to improve his grade, having obtained approval in a previous semester, he can only take the 3-hour Exam.
Logistics
Only those students who carry an official identification document with a photograph (for example, Citizen''s Card, Identity Card, Passport, some versions of Student Cards) can carry out any of the tests, blank examination notebook.
Final considerations
In all that this Regulation is missing, the FCT-UNL General Regulations are valid.