Special Functions

Description

Gamma and Beta Fuctions: Infinite Products, Wierstrass theorems. Analytic continuation of analytic functions. Euler, Gauss and Weierstrass definitions and Properties of Gamma and Beta functions on the complex plane. Stirling asymptotic formulas. Pochhammer symbols. Applications of Gamma and Beta functions. Hypergeometric functions:Frobenius series solution of second order differential equations. hypergeometric equations, representations of ordinary functions using
hypergeometric functions. Integral representations of hypergeometric functions.

Orthogonal Polynomials: Hermite, Lagueree and Legendre polynomials Definitions, generrating functions, Differential equations related to the orthogonal polynomials, Sturm-Liouville problems. Rodrigues formulae, integral representations. Orthocanonical relations and summation formulae. Series expansion of functions using bases of
orthogonal polynomials.

Cylindrical Functions:Bessel differential equation, power series solution. Bessel, Hermite and Neumann functions. Wronskians of cylindrical functions, recurrence relations. Bessel functions with integral index. Integral representation of Bessel functions.

Laplace and Poisson differential equtions in three dimensions. :Solution of Dirichlet
problems of Laplace equations. Separation of variables in spherical and cylidrical
coordinates. Applications in Electrostatics and Quantum Mechanics.

Course Coordinators

Books: 
Ηλεκτρονικές Σημειώσεις από την ιστοσελίδα του Κ. Δασκαλογιάννη.
Semester: 
Units: 
3
Credit Units (ECTS): 
5.0
Hours: 
3ώρες
ID: 
0264
Course Type: 
X