SPECIAL FUNCTIONS

Calculus of variation - Functionals, Variation of a functional and its properties, Variational problems with fixed boundaries, Euler's equation, Extremals, Functional dependent on several unknown functions and their first order derivatives, Functionals dependent on higher order derivatives, Functionals dependent on the function of more than one independent variable.

 

Gauss hypergeometric functions: Definition  and its properties, Integral representation, Gauss Theorem, Vandermonde’s Theorem, Kummer’s theorem, Linear transformation, Differentiation formulae, Gauss’s hypergeometric differential equation and it’s solution, relation between the solution of hypergeometric equation, Relations of contiguity, Kummer’s confluent hypergeometric function: Definition and it’s properties, Integral representation, Differentiation, Kummer’s first transformation. 

 

Legendre polynomials and functions: Definition, solution of Legendre’s equation, Legendre functions of the first and second kind, Generating functions(First formula), Rodrigues formula for Pn(x), Orthogonality of Legendre polynomials, Recurrence relations for Pn(x), Beltrami’s result, Christoffel expansion, Christoffel’s summation formula, Relation between Pn(x) and Qn(x).

 

Bessel Functions: Bessel equation and it’s solution, Recurrence relations, Generating function, Integral Representations of Bessel function.

 

Hermite polynomials: Definition, Generating function, Recurrence relations, Orthogonality of Hn(x), Rodrigues formula, Hermite’s differential equation and it’s solution. Legendre polynomials: Legendre’s differential equation and it’s solutions, Generating function, Rodrigue formula, Orthogonality and simple Legendre polynomials, Recurrence Relations.