SPECIAL FUNCTIONS

Paper Code: 
MAT223
Credits: 
5
Contact Hours: 
75.00
Max. Marks: 
100.00
15.00

Gauss hypergeometric functions: Definition and its properties, Condition of convergence, Integral representation, Gauss theorem, Vandermonde’s theorem, Kummer’s theorem, Linear transformation, Differentiation formulae, Relations of contiguity.

15.00

Gauss’s hypergeometric differential equation and its solution, relation between the solutions of hypergeometric equation, Two summation theorems. Kummer’s confluent hypergeometric function: Definition and differential equation, Integral representation, Differentiation, Kummer’s first and second transformations, contiguous relations.

15.00

Legendre polynomials and functions: Definition, Solution of Legendre’s equation, Legendre functions of the first and second kind, Generating functions (first formula), Rodrigue 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), Laplace first and second integrals for Legendre polynomials.

15.00

Bessel Functions: Bessel equation and its solution, Recurrence relations, Generating function, Integral representations of Bessel function, Integrals involving Bessel’s functions.

 

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

Essential Readings: 
  1. E. D. Rainville, Special Functions, Macmillan, New York, 1989.
  2. J. L. Bansal, H.S Dhami, Differential Equations Vol-II, Pragati Prakashan, Meerut, 2008.
  3. Pathan, VBL.Chaurasia, J. Banerji , S.P.Goyal, Special Functions and Calculus of Variations, RBD, Jaipur, 2004.
  4. M. D. Raisinghania, Advanced Differential Equations, S. Chand & Co. New Delhi, 2012.
References: 
  1. I.N. Sneddon, Special Functions, Mc-Graw Hill, New Delhi, 1956.
  2. N.N. Lebedev, Richard A. Silverman, Special Functions and Their Application, Dover Publications INC, 1972.
  3. Z.X. Wang & Guo, Special Functions, World Scientific books, 1989.
  4. G.E. Andrews, Askey, Roy, Special Functions, Cambridge University, 2000.
Academic Year: 
2018-2019
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