# Mathematical Analysis III B

## Objectives

The student should understand the basic concepts and be able to compute the quantities presented in the exercises.

## General characterization

##### Code

5005

##### Credits

6.0

##### Responsible teacher

Paulo José Fernandes Louro Ribeiro Doutor

##### Hours

Weekly - 4

Total - 59

##### Teaching language

Português

### Prerequisites

The student should know the basic concepts of Calculus in one and several variables that he learnt in the disciplines of Mathematical Analysis I and II.

### Bibliography

AGARWAL, R., PERERA, K., and PINELAS, S., An Introduction to Complex Analysis, 2011, Springer

ANTON, H., BIVENS, I., and DAVIS, S., Cálculo II; 8ª Edição, Bookman, 2007.

ASMAR, N. H. and GRAFAKOS, L., Complex Analysis with Applications, Springer, 2018.

CAMPOS FERREIRA, J., Introdução à Análise Matemática, Fundação Calouste Gulbenkian, 1982.

CARREIRA, M. A. ande NÁPOLES, M. S., Variável complexa - Teoria elementar e exercícios resolvidos, McGraw-Hill.

DIAS AGUDO, F. R., Análise Real, 2ª edição, Livraria Escolar Editora, 1994.

SAFF, E. B. and SNIDER, A. D., Fundamentals of Complex Analysis with Applications to Engineering and Science - 3rd Edition, Pearson Education, 2003.

SÁ, A. and LOURO, B., Sucessões e Séries, Teoria e Prática. Escolar Editora, 2009.

STEIN, E. M. and SHAKARCHI, R., Complex Analysis, Princeton Lectures in Analysis, 2003.

### Teaching method

Teaching Method is based on conferences and problems solving sessions with the support of a personal attending schedule.

### Evaluation method

**Frequency**

Students with two or more enrollments in the course or who have any of the special statutes provided for by law are exempt from frequency.

Frequency will be granted to any first-time student enrolled in the course that does not unjustifiably miss more than 1/3 of the face-to-face practical classes taught, corresponding to the practical class in which they are enrolled in CLIP.

The justifications must be delivered to the teacher of the pratical course in which the student is enrolled in CLIP, within a maximum period of 14 days from the first presence (in the pratical course in which the student is enrolled in CLIP) after the absences to which the justification refers to. If the consecutive period of absences extends after the end of classes, this must be delivered until the end of the day of the assessment test that the student presents (test or exam).

Knowledge assessment is carried out through Continuous Evaluation or Exam Evaluation Examination, presential. The Continuous Assessment consists of two tests and a grade in class.

**Grade In Class (AEA)**

For each week, a list of proposed exercises for students to work on before practical classes (either in person or online) will be made available in advance. At the end of the semester, the teacher of the practical class in which the student is enrolled in CLIP will assign a grade of 0 or 1, based on the work done by the student on the list of exercises proposed throughout the semester.

**Continuous evaluation**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

During the semester two tests will be carried out with a duration of 1 hour 30 minutes. Each test is rated up to a maximum of 20 values.

1st Test: all students enrolled in the course may present themselves to the 1st test.

2nd Test: all students enrolled in the discipline who have obtained frequency or are exempt from it may appear at the 2nd test.

Let *CT* be the simple arithmetic mean of the two tests rounded up to the units and *NF *the minimum between 20 and CT + AEA. The student obtains approval in the course if NF≥10. If NF≤16, the student is approved with the final classification NF. If NF≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**Exam**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

All students enrolled in the discipline who have obtained Frequency or are exempt from it, and who have not obtained approval in the Continuous Evaluation, can take the appeal exam. Students can choose to repeat one of the 1 hour and 30 minute tests. If the student chooses to repeat one of the tests, the classification is calculated as in the case of Continuous Evaluation. If the student takes the appeal exam, let ER be the exam classification, rounded to the nearest integer, and NR the minimum between 20 and ER+ AEA. The student obtains approval in the course if NR≥10. If NR≤16, the student is approved with the final classification NR. If NR≥17, the student can choose to stay with the final classification of 16 values or take a complementary test to defend his grade.

**Grade improvement**

It is forbidden to use graphical calculating machines, or any calculation support instruments during evaluation moments.

Students have the right to improve their grades, upon enrollment within the deadlines, at the time of exam. In that case, they can take the 3 hour Exam or repeat one of the 1 hour and 30 minute tests as described in the previous paragraph. In the event that a student makes a grade improvement having obtained approval in a previous semester, he can only take the 3 hour exam.

**Logistics**

In order to rationalize the resources of FCT (facilities, teaching staff and non-teaching staff), only students who register for the purpose through CLIP, during the period stipulated therein, may take any of the tests.

If, at the time of the exam, the student chooses to repeat one of the tests, he / she must register for this test, otherwise he / she will take an appeal exam.

Only students who, at the time of the exam, carry an official identification document, containing a photograph (for example, Citizen Card, Identity Card, Passport, some versions of Student Card) and blank exam notebook.

**Final considerations**

In any omitted situation, the Knowledge Assessment Regulation of the FCT-UNL applies.

## Subject matter

**1. Series**

1.1 Number series

1.1.1 Convergence of Numeric Series. Telescopic Series. Geometric Series.

1.1.2 Series of non-negative terms. Criterion of integral. Dirichlet series. Criteria for comparison. Ratio Criterion. Criterion of d''Alembert. Root Criterion. Cauchy Root Criterion. Kummer Criterion. Raabe Criterion.

1.1.3 Simple and absolute convergence. Alternate series and Leibniz criteria. Multiplication of series.

**1.2 Series of Functions**

1.2.1. Sequences of functions. Point convergence and uniform convergence.

1.2.2 Function series: point convergence and uniform convergence. Weierstrass criterion. Continuity. Integrability and differentiability term by term.

1.2.3 Power series. Radius of convergence. Interval of convergence. Uniform convergence. Integrability and differentiability term by term.

1.2.4 Series of Taylor and MacLaurin.

**2. Complex analysis**

2.1 Generalities about the field of complex numbers; conjugate, module and argument; polar form of a complex number. N-th roots of complex numbers. Formulas of De Moivre.

2.2 Complex variable polynomial functions. Exponential function, circular and hyperbolic trigonometric functions, main branch of the logarithm and inverse trigonometric functions.

2.3 Limits and continuity of complex functions of complex variable.

2.4 Holomorphic Functions. Cauchy-Riemann equations.

2.5 Integral of a complex complex variable function along a sectionally regular curve.

2.6 Cauchy''s theorem. Cauchy integral formulas.

2.7 Analytical functions. Taylor series. Relation with holomorphic functions.

2.8 Essential singularities, poles and removable singularities. Laurent series.

2.9 Residue Theorem. Applications to the calculation of improper integrals.

## Programs

Programs where the course is taught: