Course: 2022/2023

Math Extension

(15382)

Requirements (Subjects that are assumed to be known)

Linear Algebra, Calculus I, Calculus II

A. Learning objectives (PO: a)
A.1. To understand the concept of complex analyticity.
A.2. To be able o compute the Laurent or Taylor series expansions associated to a function which is analytic in part of the complex plane, and to determine the region of convergence of such series.
A.3. To acquire the basic concepts related to the elementary complex functions.
A.4. To compute definite integrals by means of the residue calculus.
A.5. To understand and solve first and second order linear homogeneous and non-homogeneous differential equations.
A.6. To solve second order equations using power series methods.
A.7. To recognisee classical PDEs describing physical processes such as diffusion, wave propagation and electrostatics.
A.8. To solve analytically, using the method of separation of variables, the heat and wave equations (in one space variable).
B. Specific skills (PO: a)
B.1. To understand the concept of complex differentiation and its practical applications.
B.2. To be able to handle functions given in terms of series.
B.3. To understand the concept of concept of complex integration and its practical applications.
B.4. To be able to solve first and second order linear homogeneous and non-homogeneous ODEs.
B.5. To be able to solve second order ODEs using power series methods.
B.5. To be able to model real-world problems using PDEs, and solve them using Fourier techniques.
C. General skills (PO: a)
C.1. To be able to think abstractly, and to use induction and deduction.
C.2. To be able to communicate in oral and written forms using appropriately mathematical language.
C.3. To be able to model a real situation using differential equation techniques.
C.4. To be able to interpret a mathematical solution of a given problem, its accuracy, and its limitations.

Skills and learning outcomes

Description of contents: programme

1. COMPLEX ANALYSIS.
Analytic functions and singularities. Laurent series. Contour integration and Cauchy's integral formula. The residue theorem and its applications.
2. ORDINARY DIFFERENTIAL EQUATIONS.
First order equations. Second order linear equations. Power series solutions and special functions. Fourier series solutions of ODEs. The Laplace transform: Applications to differential equations.
3. PARTIAL DIFFERENTIAL EQUATIONS.
Heat, wave, and Laplace equations. Fourier's method of separation of variables.

Learning activities and methodology

Lecture sessions: 3 credits (PO: a).
Problem sessions: 3 credits (PO: a).

Assessment System

- % end-of-term-examination 60
- % of continuous assessment (assigments, laboratory, practicals...) 40

Basic Bibliography

- G. F. Simmons. Differential equations with applications and historical notes . McGraw-Hill. 1991
- P. J. Hernando. Clases de Ámpliación de Matemáticas para Ingeniería. Revisión 3.2 - 2019.
- R. Haberman. Elementary applied partial differential equations : with Fourier series and boundary value problems. Prentice Hall. 1998
- R. V. Churchill. Complex variables and applications. McGraw-Hill. 1992

Additional Bibliography

- M. R. Spiegel. Variable compleja. McGraw-Hill. México, 1991
- William E. Boyce, Richard C. DiPrima . Elementary differential equations and boundary value problems. John Wiley & Sons. 2013

The course syllabus may change due academic events or other reasons.

**More information: **https://aplicaciones.uc3m.es/cpa/generaFicha?&est=371&plan=456&asig=15382&idioma=1