# Mathematical Analysis II

**Year**

1

**Academic year**

2015-2016

**Code**

01000068

**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

Mathematic Analysis I and Linear Algebra.

## Teaching Methods

The classes are of theoretical and theoretical-practical type, so that they are of expository nature and should include examples and exercises that lead the students to understanding and applying the material being taught.

The teaching methods in the theoretical components will be predominantly expository. In the practical components problems will be solved under the guidance of the teacher. The independent resolution of problems must be encouraged.

The strong interaction between concepts and their practical application must be discussed, giving as much as possible a central role in the visualization and analysis of particular situations before making a progressive abstraction of the concepts being introduced. The transformation of concepts into working tools will be achieved by encouraging personal work.

Some tutorial support will be available to help the students solving the proposed tasks.

## Learning Outcomes

The main goal of the course is to introduce the basic tools from Differential and Integral Calculus for functions of several real variables and Complex Analysis. To learn how to solve linear difference and differential equations of constant coefficients with the help of Z- transform and Laplace transform, respectively, and apply this knowledge to model and solve problems. It is also intended that students acquire knowledge of the concepts in order to assess the scope and limitations of the materials studied and their applications.

The course aims at developing the following skills: analysis and synthesis, organization and planning, oral and written communication, problem-solving skills and computational ability. On the personal level it also allows to develop self-learning skills and independent thinking.

## Work Placement(s)

No## Syllabus

I- Real functions of several real variables

Limits. Partial and Directional Derivatives. Linear Approximations and Differentials. Extremes. Integral Calculus over R2 and R3

II- Complex functions of complex variable.

Complex numbers. Differentiation of complex functions. Power and Laurent series. Singularities, zeros and residues. Integration of complex functions. Applications.

III- Z-transform and applications. Z-transform. Inverse Z-transform. Resolution of linear difference equations.

IV- Laplace transform and applications. Laplace transform. Inverse Laplace transform. Heaviside theorem. Dirac delta function and its (generalized) Laplace. Laplace transform for the convolution. Resolution of ordinary linear differential equations.

V. Fourier transform and applications

Fourier transform. Inverse Fourier transform. Fourier transform for the convolution. Dirac delta function, Heaviside function and theirs (generalized) Fourier transform. Shannon Sampling Theorem.

## Head Lecturer(s)

Júlio Severino das Neves

## Assessment Methods

Final assessment

*Exam: 100.0%*

Continuous assessment

*Resolution Problems: 50.0%*

*
Mini Tests: 50.0%*

## Bibliography

- James Stewart,
*Cálculo*, Volume II, 4a ed., Pioneira, São Paulo, 2001. - Glyn James,
*Advanced Modern Engineering Mathematics*, Prentice -Hall, 2ª ed., 1999. - Ruel Vance Churchill,
*Complex variables and applications*, McGraw-Hill, 1986, 4th ed. - Gueorgui V. Smirnov,
*Análise Complexa e Aplicações*, Escolar Editora (2003). - Natália Bebiano da Providência,
*Análise Complexa com aplicações e laboratórios de Mathematica*, Gradiva, Colecção Trajectos Ciência, 2009.