The course consists of 9 UFC and is subdivided into two parts.

The first part is an introduction to analytic functions; in particular, to derivation and integration on the complex plane, to the Taylor and Laurent series, to residue theory and its applications.These techniques will be applied to Fourier and Laplace transformations and to a short introduction to distribution theory with particular focus upon the Dirac delta.

The second part of the course is an introduction to the theory of linear operators on Hilbert spaces. After defining scalar products and completeness in norm for linear spaces, the course will focus upon the notions of adjoint of an operator, of self adjointness, of projection operators, of spectrum and of spectral representation.

Program

1. Analytic functions

1.1 Complex plane

1.2 Analytic functions: Cauchy-Riemann Conditions

1.3 Complex integration: Cauchy integral representation

1.4 Taylor and Laurent series: zeroes and singularities of analytic functions

1.5 Residues: Jordan lemma and applications

1.6 Fourier and Laplace transforms

1.7 Introduction to distribution theory: Dirac delta

2. Linear operators on Hilbert spaces

2.1 Linear spaces: scalar products and norms

2.2 Banach spaces: norm completeness

2.3 Hilbert spaces: completeness and orthonormal bases

2.4 Bounded linear operators: adjointness and self-adjointness

2.5 Spectrum and spectral theorem for self-adjoint operators

Reference Books

E.B. Saff, A.D. Snider: Fundamental of Complex Analysis, Prentice-Hall 1976

F. Bagarello: Fisica Matematica, Zanichelli 2007