# Mathematical Analysis III D

## Objectives

- Basic features of the theory of complex variable functions.

- Study of ordinary differential equations. We hope that the students be able to determine the general solution and particular solutions of various types of ordinary differential equations.

- The student is also supposed to learn the essential about Fourier series and their application to the resolution of partial differential equations.

## General characterization

##### Code

7544

##### Credits

6.0

##### Responsible teacher

Luís Manuel Trabucho de Campos

##### Hours

Weekly - 4

Total - 56

##### Teaching language

Português

### Prerequisites

Background in Mathematical Analysis I and II.

### Bibliography

**M. A. Carreira e M. S. Nápoles, Variável complexa - teoria elementar e exercícios resolvidos, McGraw-Hill (1998).**

**N.H. Asmar e L. Grafakos, Complex Analysis with Applications. Undergraduate Texts in Mathematics, Springer 2018.**

**M. Braun. Differential Equations and their applications (4th edition). Springer-Verlag, 1993.**

E. Kreyszig. Advanced engineering mathematics (8th edition). John Wiley & Sons, 1999.

### Teaching method

Theoretical and Practical cllasses.

Students can ask for any questions either in class or in the professor''''''''''''''''s office ours.

### Evaluation method

**Assessment Method - Mathematical Analysis III-D**

In accordance with the Regulation of Knowledge Evaluation of the Faculdade de Ciências e Tecnologia da Universidade Nova de Lisboa, the discipline of Mathematical Analysis III-D has the following method of evaluation:

The assessment of the discipline is carried out by performing two written tests (theoretical and practical evaluation), in person, during the semester, each lasting two hours. **Students must declare their willing to take the test.**

**If a student does not enroll inthe test and even so whishes to take it, then the nº9 of Art.º5 of Regulamento de Avaliação de Conhecimentos da FCT applies:**

**If 10 minutes after the beginning of the test the set of students not enrolled but whishing to take the test will have a seat in a specific room for them, they will be admited to the test without the right of recuperating the time lost in the process. If there will not be enough seats in that room then, none of the students will take the test.**

Each written test will be assigned a rating** (t1, and t2)** on the scale from 0 to 20 rounded to tenths.

The final grade of the Assessment "**C**" is calculated by

**C=(t1 + t2) / 2**

to the units, by the usual conventions.

The student is approved if **C ≥10**.

If C ≥ 16 an oral examination may be required.

If for health reasons the student will not be able to show up at the written test, he must offficialy notice FCT (send a medical certificate to the Academic Division and to the Course Coordinator) and the test will be replaced by an oral examination which must take place before the date and time of the above mentioned test.

**Exam**

Students who have not been approved, during the semester, may take the** final exam**.

The exam consists of a written test lasting 3 hours on the entire contents of the course.

The exam will be assigned an entire classification between 0 and 20 values, being the student approved to the discipline, with this classification, if it is greater than or equal to 10 values.

If the classification is greater than or equal to 16 an oral examination may be required.

**The use of calculating machines or any other calculation support instruments or consultation devices is prohibited at all times of evaluation.**

**Rating Improvement**

Students who are approved may, through compliance with all the provisions imposed by FCT-UNL, apply for an **improvement in the classification**. In that case, they may take the exam. The final classification will be the maximum between the marks obtained in the semester evaluation and in the examination. If the classification is greater than or equal to 16 an oral examination is required.

**Conditions for carrying out the written tests (tests and examination)**

**Each of the Tests and the Examination may be admistered to students regularly enrolled in the Clip. Registration must be made until at least one week before the test (test or examination).** On the day of the test each student must have a blank examination notebook that will be delivered at the beginning of the exam to the teacher who is carrying out the surveillance. During the test the student must carry an official identification document with a recent photograph.

If due to circumstances outside the FCT it is not possible to carry out the tests or the exams, these may, exceptionally, be replaced by online tests and oral examinations.

In all that this Regulation is missing, the FCT-UNL General Regulations are valid.

## Subject matter

**1. Functions of a Complex Variable**

1. Complex Functions. Algebra of complex numbers.Definition of the elementary complex functions. Limits and continuity. Differentiability - holomorfic functions. Harmonic functions. Differentiability of the elementary functions. Conformal mappings; fractional linear transformations.

**2. ORDINARY DIFFERENTIAL EQUATIONS (EDO)**

2.1 First Order Differential Equations: Field of directions associated with a first order ODE; integral curves and solutions. Some results of existence and uniqueness of solutions: the theorems of Picard and Peano. Notion of implicit solution of a differential equation. Autonomous equations and equilibrium solutions. Linear equations, separable variable equations and Bernoulli equations. Exact differential equations and notion of integrating factor.

2.2 Second order differential equations. Homogeneous equations: characteristic polynomial and base of the solution vector space. Generalization to homogeneous linear differential equations of order greater than or equal to three. The Wronskian and linear independence of solutions. The structure of the set of solutions of a second order linear ODE. The method of d '''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''' Alembert. Method of the variation of the parameters. Method of indeterminate coefficients. Notion of resonance.

2.3 Systems of linear differential equations of constant coefficients: General and structure of the solutions. Base space vector solution; relationship between the spectrum of the associated linear system and the stability of the solutions.

**3. Partial differential equations (EDP)**

3. 1 Representation of periodic functions in Fourier series . Generalities about periodic functions. Modes sin (2πt / n) and cos (2πt / n); Fourier series associated to a sufficiently regular periodic function (real and complex formalism); Study of the convergence of a Fourier series; discontinuity points and Gibbs phenomenon. Representation of a regular function in series of sines / cosines in a given interval.

3.2 Applications of Fourier series to EDP: General about EDP. The method of separation of variables. Applications to the parabolic case (heat equation), hyperbolic (wave equation) and elliptic (Laplace equation).

3.3 Some examples of application to the study of Navier-Stokes equation.

## Programs

Programs where the course is taught: