Differential equations Complex variables

Study special functions relevant in applications from Physics to Finance, their properties and examples. Study methods of asymptotic analysis for differential equations and integrals: method of Laplace, method of stationary phase and method of steepest descent. Know basic properties (algebraic and analytical) of orthogonal polynomials on the real line, as well as some of their applications. CB6, CB7, CB8, CB9, CB10 CG2, CG4, CG5, CG6, CG7 CE1, CE2, CE3, CE4, CE8, CE14

1. Special Functions of Mathematical Physics. Hypergeometric functions. 1.1. Euler's Gamma and Beta functions. Riemann¿s zeta function. 1.2. Hypergeometric functions. 1.3. Integral representations. 2. Asymptotic methods for ODE and integrals. Laplace and saddle point methods. Steepest descent paths. 2.1. Method of Laplace. 2.2. Method of stationary phase. 2.3. Method of steepest descent. 2.4. Liouville-Green transformations of ODEs. 3. Orthogonal polynomials in the real line. Algebraic and analytic properties. Computational problems. 3.1. Motivation and examples. 3.2. Algebraic and analytical properties. 3.3. Computational problems. 4. Introduction to Painlevé equations. Continuous and discrete cases.

Learning activities: - Theoretical lectures - Exercise sessions - Tutorials - Individual or group work Methodology: - Exposition in class of the main concepts of the course by the lecturer. - Critical reading of recommended bibliography to consolidate the main topics of the course, and to complement them in those directions that the students are more interested in. - Problem solving sessions, individually or in groups. Two hours per week will be allocated to tutorials, to solve questions related to the syllabus of the course and the proposed exercises.