# 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., Calculus; 10th Edition, Wiley, 2012.

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

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

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**

In this edition of the curricular unit there will be no evaluation of attendance.

**Knowledge assessment**

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.

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

**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. Every fortnight, in the last minutes of the practical class, the students will solve an exercise from the list and give it to the teacher. At the beginning of the next practical class, they will correct the resolution given by one of their colleagues in the previous class. At the end of the semester, the teacher of the practical class in which the student is enrolled in CLIP will assign a grade from 0 to 4 values, based on the resolutions and corrections made by the student throughout the semester.

**Tests**

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 8 values.

**Continuous evaluation**

The continuous assessment (*NF*) classification is obtained by adding the test scores and the classroom assessment, in case it is the student''s first enrollment in the curricular unit. Otherwise, NF is obtained by rounding to the nearest units: 1) the sum of the test scores and the classroom assessment and 2) the sum of the test scores multiplied by 20/16.

The student passes the course if NF≥10. If NF≤16, the student is approved with the final classification NF. If NF≥17, the student can choose between keeping the final classification of 16 values or taking a complementary test to defend the grade.

**Exam**

On the exam date, students who have not yet passed may choose to retake one of the 1 hour 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 exam, which is graded up to 20, NF is obtained by rounding up to units the maximum between: 1) the exam grade and 2) the sum of the exam grade and the classroom assessment multiplied by 20/24.

The student passes the course if NF≥10. If NF≤16, the student is approved with the final classification NF . If NF≥17, the student can choose between keeping the final classification of 16 values or taking a complementary test to defend the grade.

**Grade improvement**

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: