*The prerequisite(s) for this Teaching Unit (Unité d’enseignement – UE) for the programmes/courses that offer this Teaching Unit are specified at the end of this sheet.*

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1 |
1.1, 1.3, 1.4, 2.1, 3.1, 3.2, 3.3, 3.4, 3.5, 3.6
¿ find the solutions of the classical partial differential equations of physics in simple geometries; ¿ determine the Fourier series of a given function; ¿ apply the abstract theory of Fourier series in Hilbert spaces; ¿ construct the classical orthogonal polynomials and apply them to the solution of differential equations; ¿ apply the Fourier transform to solve partial differential equations. |

*The contribution of this Teaching Unit to the development and command of the skills and learning outcomes of the programme(s) can be accessed at the end of this sheet, in the section entitled “Programmes/courses offering this Teaching Unit”.*

**Fourier series :**periodic functions, trigonometric polynomials, Fourier series, Bessel's inequality, Parseval's theorem, convergence and Dirichlet's theorem, applications.

2.

**Partial differential equations :**classification of linear partial differential equations of second order, heat equation, wave equation, Laplace equation, existence and uniqueness of solutions, solution methods.

3.

**Hilbert spaces :**pre-Hilbert spaces, completeness and Hilbert spaces, Hilbert bases, examples (sequence and function spaces), abstract theory of Fourier series.

4.

**Orthogonal polynomials :**definition on finite and infinite intervals, recurrence relations, Rodriguez' formula and the classical orthogonal polynomials (Jacobi, Chebyshev, Legendre, Laguerre, Hermite), second order differential equations, application of Legendre polynomials and spherical harmonics in physics.

5.

**The Fourier transformation :**definition and properties, convolution product, Poisson summation formula, applications to the solution of linear differential equations, distributions and their Fourier transformation.

The lectures introduce the concepts and ideas of mathematical methods that are necessary for understanding modernphysics (such as quantum physics), establish rigorous results and present computational techniques and strategies. Furthermore, the connection with other teaching unitsof the Bachelor's programme in physics are emphasized.

The main objective of the exercisesessions is the application of the theory to concrete examples.

' C. Aslangul 'Des mathématiques pour les sciences, De Boeck (2011).

**PHYS**