# Mathematical Analysis III

**Year**

2

**Academic year**

2022-2023

**Code**

01016318

**Subject Area**

Mathematics

**Language of Instruction**

Portuguese

**Mode of Delivery**

Face-to-face

**Duration**

SEMESTRIAL

**ECTS Credits**

6.0

**Type**

Compulsory

**Level**

1st Cycle Studies

## Recommended Prerequisites

Mathematical Analysis I, Mathematical Analysis II, Linear Algebra and Analytic Geometry.

## Teaching Methods

The teaching in this course assumes two formats: theoretical and example classes. During a theoretical class teaching is mostly expository. During an example class teaching consists of problem solving by the students under the guidance of the lecturer. A strong interaction between notions and their practical application is emphasized. In this task, the visualization and the analysis of concrete examples takes on a central role and prepares the way for the abstract definitions. Tutorial support is available to students to help them on the tasks assigned by the lecturers.

## Learning Outcomes

The student who successfully completes this course will be able to:

1. Compute the sum of geometric and telescoping series, real or complex;

2. Determine the nature of convergence of a series of real or complex numbers;

3. Compute the Fourier series of a periodic function;

4. Compute the Laurent series of an analytic function on an annulus;

5. Classify the singularities of a quotient of analytic functions;

6. Compute the residue of an analytic function on an annulus;

7. Compute the line integral of a complex function using the residue theorem.

## Work Placement(s)

No## Syllabus

1. Sequences and numerical series, real and complex - Convergence criteria

2. Fourier series

3. Complex functions; Limits e derivatives - Cauchy-Riemann equations; Analytic functions

4. Complex integration; Cauchy's Theorem

5. Power series; Taylor Series

6. Laurent series; Residue Theorem.

## Head Lecturer(s)

Ricardo Nuno Fonseca de Campos Pereira Mamede

## Assessment Methods

Continuous assessment

*2 or more midterm exams: 100.0%*

Final assessment

*Exam: 100.0%*

## Bibliography

[1] James Stewart: Cálculo, Volumes I e II. (Edição: Tradução da 8ª Edição Norte-americana, 2017)

[2] Ana d'Azevedo Breda, Joana Nunes da Costa: Cálculo com funções de várias variáveis. McGraw-Hill, Lisboa (1996).

[3] Zill, D. G., Shanahan, Patrick, D. (2003) - A first course in complex analysis with applications, Jones and Bartlett Publishers.

[4] Spiegel, M. (1977) – Análise de Fourier, Colecção Schaum.