MATHEMATICAL METHODS FOR PHYSICS

1. Differential Equations in the Complex Field
1.1 - Regular and singular points of equations. Uniformity.
1.2 - Fuchs’ Theorem
1.3 - Equations with three singular points. Riemann’s symbol.
1.3 - Gauss Equation. Hypergeometric function and series.
1.5 - Confluent hypergeometric equantion.
1.6 - Legendre Equation and polynomials. Spherical harmonics.
1.7 - Hermite and La guerre Equations and polynomials.
1.8 - Bessel Equations and functions.
1.9 - Applications to the Schroedinger’s equations.
 
2. Approximation by functions
2.1 - Approximation criteria.
2.2 - Orthogonalization.
2.3 - Classical polynomials.
2.4 - Trigonometric and exponential series.
2.5 - L1 and L2 spaces, general properties.
 
3. Integral Representations.
3.1 - The Fourier Integral and its L1 properties.
3.2 - Plancherel Thgeorem, L2 properties.
3.3 - The Laplace Integral and its properties.
3.4 - Inversion of the Laplace Transform.
3.5 - Applications to partila derivatives equations.
4. Outlines in Measure Theory and Integration.
4.1 – Families of sets, rings, algebras and sigma algebras.
4.2 – Lebesgue Measure
4.3 – Lebesgue Integral.
4.4 – Absolutely continuous and singular functions.