MATHEMATICS FIFTH SEMESTER
Mathematical Special F unctions 7A
Unit – 1: Beta and Gamma functions, Chebyshev polynomials
1. Euler’s Integrals-Beta and Gamma Functions, Elementary properties of Gamma Functions, Transformation of Gamma Functions.
2. Another form of Beta Function, Relation between Beta and Gamma Functions.
3. Chebyshev polynomials, orthogonal properties of Chebyshev polynomials, recurrence relations, generating functions for Chebyshev polynomials.
Unit – 2: Power series and Power series solutions of ordinary differential equations
1. Introduction, summary of useful results, power series, radius of convergence, theorems on Power series
2. Introduction of power series solutions of ordinary differential equation
3. Ordinary and singular points, regular and irregular singular points, power series solution.
Unit – 3: Hermite polynomials
1. Hermite Differential Equations, Solution of Hermite Equation, Hermite polynomials, generating function for Hermite polynomials.
2. Other forms for Hermite Polynomials, Rodrigues formula for Hermite Polynomials, to find first few Hermite Polynomials.
3. Orthogonal properties of Hermite Polynomials, Recurrence formulae for Hermite Polynomials.
Unit – 4: Legendre polynomials
1. Definition, Solution of Legendre’s equation, Legendre polynomial of degree n, generating function of Legendre polynomials.
2. Definition of Pn x and Qn x, General solution of Legendre’s Equation (derivations not required) to show that Pn xis the coefficient of hn, in the expansion of (1 − 2𝑥ℎ + ℎ2)−1 2 3. Orthogonal properties of Legendre’s polynomials, Recurrence formulas for Legendre’s Polynomials.
Unit – 5: Bessel’s equation
1. Definition, Solution of Bessel’s equation, Bessel’s function of the first kind of order n, Bessel’s function of the second kind of order n.
2. Integration of Bessel’s equation in series form=0, Definitionof Jn xrecurrence for mulae for Jn x 3. Generating function for Jn x, orthogonally of Bessel functions.