Analytic functions of complex variable and Cauchy theorems. Taylor and Laurent series expansions. Analytic continuations. Residue theorem and its application to evaluation of integrals. Single-valued and multiple-valued functions. Laurent expansions for multiple-valued functions. Basic ideas and facts about distributions (generalized functions). Finite dimensional linear spaces: vectors and linear operators. Key inequalities in unitary linear spaces. Orthogonal polynomials. Eigenvalues and eigenvectors. Spectral representation and functions of linear operators. Adjoint operator. Self-adjoint, unitary and normal operators. Diagonalizability of operators. Baker-Campbell-Hausdorff formulae.
The course is taught by lectures and exercises, with wide room for questions from the students. It is expected an adequate knowledge of basic mathematics, i.e. math analysis, infinitesimal calculus, geometry and linear algebra.
The course aims at providing students with the mathematical tools that are needed for deep understanding and informed application of the fundamental laws of Modern and Contemporary Physics (from 20th century to date), in particular Quantum Mechanics, Special Relativity and General Relativity.
KNOWLEDGE AND UNDERSTANDING:
The course enables to extend the knowledge of basic mathematics at the undergraduate level as well as to learn the key elements of complex analysis and linear operator theory.
This course includes in particular activities aimed at strenghtening students' abilities in differential and integral calculus, which should be significantly improved and extended once they get familiar with the methods of complex analysis and linear operator theory.