# 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.

- In the second part we intend the students use the Laplace Transforms for solving differential and integral 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 - 6

Total - 40

##### 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)

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

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

S. Lang, *Complex Analysis*, Springer (1999), ISBN 0-387-98592-1

J. E. Marsden and M. J. Hoffman, *Basic Complex Analysis - Third Edition*, Freeman (1999), ISBN 0-7167-2877-X

Zill, D. G. ; Cullen, M.R. - Differential equations with boundary value problems; 6th edition.

### 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:

Attendance

In the case of a repeating UC in its edition outside the curriculum, the maximum number of unexcused absences fixed for a student to obtain attendance is five. Students with a status that exempts them are exempt from attendance.

Evaluation

The assessment of the discipline is carried out by performing two written tests (theoretical and practical evaluation) during the semester, each lasting two hours. 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

(t1 + t2) / 2

to the units, by the usual conventions.

The student is approved if he has obtained frequency according to the above rules if C ≥10.

Exam

Students who have been disqualified for assessment with attendance or with a status that exempts them of it may take the 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.

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.

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.

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

## Subject matter

**1. Complex Plane Geometry**

1.1 Introduction: Generalities about the field of complex numbers; conjugate, modulus and argument; polar form of a complex number, nth roots of complex numbers. De Moivre''''''''s formulas and applications to the linearization of trigonometric polynomials.

1.2 Isometries and Magnifications: Classification of plane isometries (translations, rotations and reflections) and study of their representations by functions of complex variable. Magnifications and transformations of the complex plane of the form f (z) = wz + z0 and f (z) = wz + z0; inversions of the form f (z) = 1 / z.

1.3 Elementary functions: Complex variable polynomial functions. Exponential function, circular and hyperbolic trigonometric functions, main branch of the logarithm and inverse trigonometric functions.

1.4 Conformal mappings: Holomorphic functions, Cauchy-Riemann conditions and their geometric interpretation. The notion of conformal mapping. Transformation of elementatary regions (delimited by straight line segments and / or conic curves) by elementary conformal mappings (z2, z3, ez, sin z, sinh z, 1 / z, etc.).

**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 The Navier-Stokes equation. Notion of conservation law; notion of material derivative. The Euler equation for incompressible liquids; conservation of mass, momentum and energy. Case of the Navier-Stokes equation: study of exact solutions in different one-dimensional contexts.

## Programs

Programs where the course is taught: