Mathematical Analysis III

Recommended Prerequisites

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

Teaching Methods

The teaching in this course will assume two formats: theoretical and example classes. During a theoretical class teaching will be mostly expository. During an example class teaching will consist 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 will be 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:

i)  classify the singularities of a quotient of analytic functions;

ii) compute the Laurent series of an analytic function on an annulus;

iii) compute the residue of an analytic function on an annulus;

iv) compute the integral of a function along a closed path using the Residue Theorem;

v) compute the Z-transform of a rational function using the method of decomposition into partial fractions;

vi) use the Z-transform to solve difference equations;

vii) compute the Fourier transform and its inverse;

viii) compute the Laplace transform and its inverse;

ix) solve a differential equation using the Laplace transform.

Work Placement(s)

No

Syllabus

I.   Complex functions

I.1 Algebra and topology of the complex plane

I.2 Elementary functions

I.3 Differentiation

I.4 Complex power series and Laurent series

I.5 Singularities, zeros and residues

I.6 Integration

 

II.   Z-transform and applications

II.1 Z-transform (definition and properties)

II.2 Inverse Z-transform

II.3 Application to solving difference equations

 

III.   Fourier transform and applications

III.1 Fourier series (revision) and its complex version.

III.2 Fourier transform and its inverse

III.3 Generalised Fourier transform (of Heaviside and Dirac's Delta functions)

III.4 Applications of the Fourier transform: Sampling Theorem

 

IV.   Laplace transform and applications

IV.1 Laplace transform (definition, existence and properties)

IV.2 Inverse Laplace transform

IV.3 Generalised Laplace transform (of Dirac's Delta function)

IV.5 Applications of the Laplace transform to solving ordinary linear differential equations.

Head Lecturer(s)

Júlio Severino das Neves

Assessment Methods

Continuous assessment
2 or more midterm exams : 100.0%

Final assessment
Exam: 100.0%

Bibliography

[1] Glyn James, Advanced Modern Engineering Mathematics, Prentice-Hall, 3ª edição, (2004).

[2] Ruel Vance Churchill, Complex variables and applications, McGraw-Hill, 4ª edição, (1986).

[3] Gueorgui V. Smirnov, Análise Complexa e Aplicações, Escolar Editora (2003).

[4] Natália Bebiano da Providência, Análise Complexa com aplicações e laboratórios de Mathematica, Gradiva, Colecção Trajectos Ciência, 2ª edição (2012).

[5] Júlio Severino das Neves, Apontamentos de Análise Matemática III, (distribuído online) (2018).