Basic mathematical analysis: differential calculus for functions of one or several variables, ordinary and partial differential equations, integral calculus.
Written exam: exercises and problems with open questions.
Oral exam: discussion of the written exam; possible request for resolution of other exercises; questions on definitions, statements and proofs of theorems.
It is possible to take the oral exam even if the result of the written exam is not sufficient.
The aim of the course is to provide the basic tools of Mathematical Analysis necessary for the study of the differential equations of quantum and classical Mechanics and of Physics in general.
Complex analysis. Special functions. Fourier series. Convolution. Fourier transform. Distributions. Laplace transform. Elements of Calculus of Variations.
Holomorphic functions and harmonic functions. Cauchy's theorem. Laurent series. Residue theorem. Lemma of Jordan. Calculation of integrals applying the residual theorem.
Serial development compared to a complete orthonormal system. Parseval formula and inversion formula. Fourier series in real and complex form.
Fourier transform and applications
Parseval formula and inversion formula. Convolution of functions. Applications to the resolution of the heat equation and the wave equation. Calculation of Fourier transforms with the residual theorem. Gaussian function. Lorentzian function. Voigt function. Distribution of Fourier transforms. Approximation of the Dirac delta.
Functions of Laguerre, Legendre, Bessel. Spherical harmonics.
Elements of Calculus of Variations
Functional derivative. Euler-Lagrange equation.
Application of the Laplace transform to the resolution of ordinary differential equations with edge conditions.
K. F. Riley, M. P. Hobson and S. J. Bence. Mathematical Methods for Physics and Engineering, Cambridge University Press.
Lectures and exercises.