The course will cover the following topics, not necessarily in the given order, with applications to problems and differential equations of interest in physics:

Complex analysis: The complex plane.  Complex functions of complex variable. The derivative of a function in C. Cauchy-Riemann conditions.  Integration on the complex plane.  Cauchy theorem.   The behaviour of a complex function close to an isolated singularity.  Laurent series expansion.  Residue theorem. Computational techniques for integrals along the real axis by using the analytic continuation in C.  Analytic continuation and multivalued functions.

Funtional spaces: Summary of the main properties of topological spaces, metric spaces, Banach spaces. Hilbert spaces. Orthonormal basis. Fisher-Riesz theorem. Lp spaces. Important examples of orthonormal basis:  Fourier series, Hermite, Legendre, Laguerre polynomials. Linear operators in Hilbert spaces and their properties. Continuous and bounded operators. The norm of an operator. Spectral problem, classification of the eigenvalues. Definition of eigenfunctions. Self-adjoint and unitary operators. Eigenvalues and eigenvectors of self-adjoint operators. Theorem of spectral decomposition. Fourier transform in L1 and L2 its properties. Lapace Transform

Distributions: brief introduction to the theory of distributions. Examples of distributions. Operations on distributions.