ADVANCED STUDIES OF SPECIAL FUNCTIONS AND INTEGRAL TRANSFORMS (Optional Paper)
Associated Legendre polynomials of first and second kind: Differential equation, Relation between solutions of associated Legendre equation, Recurrence relation, Orthogonal properties, Hyper geometric forms.
Chebyshev polynomials: Chebyshev equation and its solutions, Expansions, Generating relations and orthogonal property.
Generalized Hypergeometric Function: Definition, Special cases, Series, integral and contour representations, Convergence conditions of these representations, Saalssutz, Whipple theorems, Contiguous function relations, Differentiation and integral formulas.
Laplace Transforms: Complex inversion formula, Use of residue theorem in calculation of inverse Laplace transform including the functions with branch points and infinitely many singularities, Solution of Heat conduction and Wave problems by using complex inversion formula for Laplace transform.
Z-Transforms: Definition, Inverse, Images of elementary functions, Basic operational properties, Partial derivatives, Initial and Final value theorems and applications.
2. K P Gupta, J K Goyal, Integral Transforms, Pragati Prakashan, New Delhi,2015
3. Z.X. Wang, D.R. Guo, Special Functions, World Scientific publishing Ltd., 1989.
1. George E. Andrews, Richard Askey, Ranjan Roy, Special Functions, Cambridge University, 2000.
2. N.N. Lebedev, Richard A. Silver Man, Special functions and their application, Dover Publications INC, 1972.
3. I.N. Sneddon, Special Functions, TMH, New Delhi, 1956.
4. Mohamed F. EL-Hewie, Laplace Transform, Create space Independent Publication, 2013.
5. Joel L. Schiff, The Laplace transform: Theory and application, Springer Science & Business Media, 1999.
6. John Miles, Integral Transform in Applied Mathematics, Cambridge University Press, 1971.
