Complements of complex variable theory: Analytic and multivalued functions. Complex integrals. Pole expansion of meromorphic functions. Infinite product representation of complex functions. Local invertibility and the reciprocal of analytic functions.
Asymptotic expansions: Integration by parts. Laplace method and Watson lemma. Stirling’s formula. Stokes phenomenon and analytic continuation. Stationary phase, steepest descent and saddle point techniques.
Ordinary differential equations: Distribution theory. Green's functions. Second order linear equations. Cauchy and Sturm-Liouville problems. Differential operators in Hilbert spaces.
Equations in complex space. Power series method.
Fourier and Laplace transforms: Discrete and integral transforms. Multidimensional cases.
Special functions of physical science: Gamma, Diagamma, Polygamma, Beta and Zeta functions. Hypergeometric, confluent hypergeometric, Bessel functions. Legendre functions and spherical harmonics. Orthogonal polynomials.
Partial differential equations: Classification, physical motivation and notable examples.
Separation of variables and integral transform methods. Boundary value problems.
Sound understanding and ability in the use of mathematical methods that are the foundation to courses of modern physics and to current research in all branches of physics, both at the theoretical and experimental level.
KNOWLEDGE AND UNDERSTANDING:
Understanding and mastery of advanced mathematical methods that are of use in other courses and in research contexts.
APPLYING KNOWLEDGE AND UNDERSTANDING:
Ability to prove theorems, derive mathematical properties, and make complex calculations. Moreover, ability to identify the areas of applicability of the mathematical methods proposed in class, especially for the resolution of complex problems, even when concerning new topics.
Ability to integrate competences and assess autonomously the efficiency, adequacy, and correctness of the different mathematical methods in the resolution of problems concerning different fields in physics.
Ability in clearly and correctly presenting the topics of the program. Additionally, ability to discuss the formulation, the hypotheses and the demonstration of theorems, and to communicate without ambiguity the logical steps and the conclusions of the analysis of a problem.
Capability of searching and integrating information present on different sources, like different textbooks and the WEB. It is also crucial the ability of elaborating and extending the examples and the applications presented in class.