MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY, JAIPUR.
DEPARTMENT OF MATHEMATICS
SYLLABUS
MAT-667
Special Functions
4 Credits (3L+1T+0P)
The Gamma and Beta Functions: Eulers’ integral forГ(z), the beta function, factorial
function , Legendre’s duplication formula, Gauss’s multiplication theorem, summation
formula due to Euler, behaviour of log Г(z) for large |z |
The Hypergeometric function: An integral representation. Its differential equation and
solutions. , F(a,b,c;1) as a function of the parameters, evaluation of F(a,b,c;1), contiguous
function relations, the hypergeometric differential equation, logarithmic solutions of the
hypergeometric equation, F(a,b,c;z) as a function of its parameters, Elementary series
manipulations, simple transformations, relation between functions of Г(z) and, Г(1-z)
quadratric transformations, theorem due to Kummer, additional properties
The Confluent Hypergeometric function: Basic properties of 1F1, Kummer’s first formula.
Kummer’s second formula,
Generalized Hypergeometric Series: The function pFq, the exponential and binomial
functions, differential equation, contiguous function relations, integral representation pFq,
with unit argument, Saalshutz’ theorem, Whipple’s theorem, Dixon’s theorem, Contour
integrals of Barnes’ type.
Bessel Functions: Definition, Differential equation, differential recurrence relations, pure
recurrence relation, generating function, Bessel’s Integral, index half an odd integer,
modified Bessel functions
Introduction to Legendre function, Meijer G-function and some basic properties.
Books Recommended:
1. Earl. D. Ranvillie, Special Functions , Macmillan, 1960.
2.L.C. Andrews ,Special Functions of Mathematics for Engineers, SPIE Press, 1992.
3. Gabor Szego, Orthogonal Polynomials, American mathematical society, 1939.
4. L.J. Slater,Generalized Hypergeometric Functions , Cambridge University Press; Reissue
edition ,2008.