# Mathematics Applied to Electrotechnics

**Electronics and Computer Engineering**

6 ECTS; 2º Ano, 1º Semestre, 28,0 T + 14,0 PL + 28,0 TP + 5,0 OT

**Lecturer**

- Luis Miguel Lindinho da Cunha Mendes Grilo

- Ricardo Jorge Viegas Covas

- Carlos Filipe Perquilhas Baptista

- Maria Manuela Morgado Fernandes Oliveira

**Prerequisites**

Not applicable.

**Objectives**

1. Acquisition of knowledge in Statistics, Complex Analysis and Transformed.

2. Provide students with the tools they need to develop analytical and reasoning skills that enable them to design and implement solutions to different problems and facilitate decision making.

**Program**

1. Probability Reviews

1.1. Probability basics;

1.2. Random variables;

1.3. Discrete and Continuous Theoretical Distributions: the Normal distribution;

1.4. Approximation of Binomial and Poisson distributions to Normal;

1.5. The exponential distribution.

2. Sampling Distributions

2.1. Statistical inference. Random sampling;

2.2. Central Limit Theorem;

2.3. Chebychev Inequality;

2.4. Theoretical sample distributions;

2.5. Distribution of the sample mean in a normal population;

2.6. Distribution of sample variance in a normal population;

2.7. Distribution of the sample proportion.

3. Parametric Estimation

3.1. Point estimation. Estimators and Estimates;

3.2. Properties of estimators;

3.3. Maximum Likelihood Estimation;

3.4. Interval estimation;

3.5 Confidence intervals for the average of a normal population;

3.6. Confidence interval for standard deviation and variance of a normal population;

3.7. Confidence interval for a ratio;

3.8. Choice of sample size.

4. Hypothesis Tests

4.1. Basic concepts;

4.2. Hypothesis tests for the average of a normal population;

4.3. Hypothesis tests for the variance of a normal population;

4.4. Hypothesis tests for a ratio.

5. Introduction to Simple Linear Regression

5.1. Regression models;

5.2. Least squares method in simple linear regression;

5.3. Analysis of variance: ANOVA table;

5.4. Correlation and determination coefficients;

5.5. Inferences in the simple linear regression model.

6. Complex Numbers

6.1. Algebraic, trigonometric and polar form;

6.2. Powers and roots;

6.3. Geometry in the complex plane.

7. Analytical Functions

7.1. Complex variable functions;

7.2. Limits and continuity;

7.3. Analyticality;

7.4. Cauchy-Riemann equations;

7.5. Harmonic functions.

8. Elementary Functions

8.1. Exponential, trigonometric and hyperbolic functions;

8.2. Logarithmic function;

8.3. Complex power and inverse trigonometric functions;

8.4. Application to oscillatory systems.

9. Complex Integration

9.1. Contours;

9.2. Contour integrals;

9.3. Cauchy's integral theorem;

9.4. Integration of analytical functions.

10. Serial Development of Analytical Functions

10.1. Taylor series;

10.2. Power series;

10.3. Laurent Series;

10.4. Zeros and singularities.

11. Residue Theory

11.1. Residue theorem;

11.2. Application to the calculation of trigonometric and improper integrals.

12. Differential and Transformed Equations

12.1. Fundamental Types of Differential Equations;

12.2. Fourier Series;

12.3. Fourier Transforms: definition, properties and their use in solving some types of differential equations;

12.4. Laplace transforms: definition, properties and their use in solving some types of differential equations.

**Evaluation Methodology**

Continuous assessment: two written tests during the semester, classified from 0 to 10, with the minimum classification restriction of 3 values in both. The syllabus is equally divided by the tests to be performed. The student is exempt from examination, ie, it is approved by continuous assessment if the sum obtained from the written tests, rounded to the units, is equal to or greater than 10.

Assessment by exam (normal time): written test with all the subjects taught in the course (rated from 0 to 20). The student is approved to the course if the final grade, rounded to the units, is equal to or higher than 10.

Other times: written test with all the subjects taught (rated from 0 to 20). The student is approved to the curricular unit if the classification of this test, rounded to the units, is equal or higher to 10 values.

**Bibliography**

- , .(1993). *Fundamentals of Complex Analysis for Mathematics, Science and Enginnering*. New Jersey: Prentice-Hill

- , .(2004). *Introdução Computacional à Probabilidade e Estatística*. : Porto Editora

- , .(2007). *Estatística*. : McGraw-Hill

- , .(1998). *Variável Complexa*. Lisboa: McGraw-Hill

- , .(1993). *Fundamentals of Complex Analysis for Mathematics, Science and Enginnering*. New Jersey: Prentice-Hill

- , .(2004). *Introdução Computacional à Probabilidade e Estatística*. : Porto Editora

- , .(2007). *Estatística*. : McGraw-Hill

- , .(1998). *Variável Complexa*. Lisboa: McGraw-Hill

- Novais, J. (2008). *Programação de Autómatos, Método GRAFCET*. (Vol. 1). Portugal: Fundação Calouste

- Siemens, S. (2003). *Simatic, S7-200 Programmable Controller.*. (Vol. 1). Alemanha: Siemens

- Morriss, S. (2000). *Programmable Logic Controllers*. (Vol. 1). US: Prentice-Hall

**Method of interaction**

Theoretical and theoretical-practical classes, which expose and exemplify the subjects related to each of the syllabus. Laboratory practical classes, using specific software, and tutorial classes.

**Software used in class**

EXCEL Spreadsheet and Statistical Package for Social Science (SPSS) Package.