00709 - Mathematical Methods Of Physics

Learning outcomes

To provide the basic mathematical concepts and methods needed for the study and the understanding of modern phyics. In the first partof the course the students will learn the theory of ofholomorfic functions, and will be able to compute contour integrals in the complex plane. In the second part of the course the students will learn the mathematical formalism of infinite dimensional linear spaces, in particular the Hilbert spaces. They will be able to understand the mathematical method needed for the study of quantum mechanic.

Course contents

Complements on analitical functions: differentiability and holomorfism, contour integrals, Cauchy theorem, residuals theorem, computation of residuals. Metric and normed spaces: basic topological concepts, algebric structures, norms and Banach spaces, examples. Boundedoperators: strong convergency, norm convergency, operatorial series. Hilbert spaces: scalar products and the geometry of Hilbert spaces, orthonormal systems,Fourier series, orthogonal polynomials, adjoint of a bounded operator, examples. Convolution product.
Fourier transform. Rapidly decreasing functions.

Readings/Bibliography

H. Cartan: Elementary theory of analitical functions in one or several complex variables, Addison Wesley T. Kato: Perturbation theory for linear operators, Springer. Notes of the teacher.

Teaching methods

Lectures and tutorials.

Assessment methods

Oral test, with resolution of an exercise.

Office hours

See the website of Alexandr Kamenchtchik