Mathematical Analysis III

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.