Course: 2020/2021

Advanced Mathematics

(15943)

Students are expected to have completed

Calculus I, Calculus II and Linear Algebra.

Competences and skills that will be acquired and learning results. Further information on this link

The student should be familiar with the most important techniques in complex variable functions. Specifically, he/she should understand and manage the following basic concepts:
1. Elementary functions of one complex variable.
2. Integration in the complex plane.
3. Power series developments.
4. Aplications of the residue theorem.
The course is complemented with some basic topics in ordinary differential equations:
1. Solution of first order differential equations.
2. Solution of higher order linear differential equations.
3. Use of Laplace transform to solve linear equations and systems with constant coefficients.

Description of contents: programme

1. ORDINARY DIFFERENTIAL EQUATIONS
1.1. Initial and boundary value problems.
1.2. Existence and uniqueness.
1.3. Elementary solution methods.
1.3.1. Separable differential equations.
1.3.2. Homogeneous differential equations.
1.3.3. Exact differential equations.
1.3.4. Integrating factor.
1.3.5. Linear differential equations.
1.3.6. Bernoulli equations.
1.3.7. Reduction of order.
1.4. Linear equations and systems.
1.4.1. Characteristic polynomial.
1.4.2. Laplace Transform and applications.
2. FUNCTIONS OF ONE COMPLEX VARIABLE
2.1. Complex numbers.
2.1.1. Operations with complex numbers.
2.1.2. Absolute value and argument.
2.2. Holomorphic functions.
2.2.1. Limits and continuity.
2.2.2. Complex derivative.
2.2.3. Cauchy-Riemann equations.
2.2.4. Harmonic functions.
2.3. Analytic functions.
2.3.1. Power series.
2.3.2. Elementary functions.
2.4. Complex integration.
2.4.1. Cauchy's theorem and applications.
2.4.2. Laurent series.
2.4.3. Calculus of residues.
2.4.4. The residue theorem and applications.
2.4.5. Computation of real integrals.

Learning activities and methodology

The docent methodology will include:
1. MASTER CLASSES, where the knowledge that the students must acquire will be presented. To make
easier the development of the class, the students will have written notes and also will have the
basic texts of reference that will facilitate their subsequent work.
2. RESOLUTION OF EXERCISES by the student that will serve as self-evaluation and to acquire the
necessary skills.
3. PROBLEM CLASSES, in which the proposed problems are discussed and developed.
4. PARTIAL CONTROLS.
5. FINAL EXAM.
6. TUTORIALS.

Assessment System

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

Basic Bibliography

- LEVINSON, N., REDHEFFER, R. M. . Curso de Variable Compleja . Ed. Reverté, Madrid . 2003
- LÓPEZ-GÓMEZ, J.. Ecuaciones diferenciales y variable compleja : problemas y ejercicios resueltos. Prentice Hall . 2002
- PESTANA, D., RODRÍGUEZ, J. M. Y MARCELLÁN, F.. Curso práctico de variable compleja y teoría de transformadas. Pearson Educación, S. A.. 2014
- SIMMONS, G.F. y KRANTZ, S.G. . Ecuaciones Diferenciales, Teoría, técnica y práctica . Ed. McGraw-Hill, México . 2007

- Herbert Gross · Complex Variables, Differential Equations and Linear Algebra : https://ocw.mit.edu/resources/res-18-008-calculus-revisited-complex-variables-differential-equations-and-linear-algebra-fall-2011/part-i/

Additional Bibliography

- EDWARDS, C. H. Jr., PENNEY, D. E. . Elementary Differential Equations with Boundary Value Problems . Ed. Prentice Hall Inc. . 1993
- NAGLE, R.K. y SAFF, E.B. . Fundamentals of Differential Equations, second edition . Ed. The Benjamin/Cummings Publishing Company Inc., Redwood City, California, U.S.A..
- PESTANA, D., RODRÍGUEZ, J. M., MARCELLÁN, F.. Variable compleja, un curso práctico. Editorial Síntesis. 1999
- VOLKOVYSKII, L.I., LUNTS, G.L. y ARAMANOVICH, I.G. . A collection of problems in complex analysis . Ed. Dover, N.Y., U.S.A. . 1991
- WUNSCH, A. D.. Complex Variables with Applications . Ed. Addisson-Wesley Publishing Company Inc. Reading, Massachusetts . 1994
- ZILL, D. G. . Differential Equations with Modeling Applications . Ed. Brookes/Cole Publishing, 6th. ed. .

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