Engineering Mathematics, Volume 2

12

Special Functions

12.1 Introduction

In the study of Mathematics, we have so far dealt with algebraic and transcendental functions xⁿ, Σa_nxⁿ, e^x, log x, sin x, cos x, sinh x, cosh x and so on, which are called elementary functions.

In solving higher engineering problems, there arises a need to use higher transcendental functions which are of two types:

1. Functions defined by improper integrals and
2. Functions defined by convergent infinite series, which are solutions of linear differential equations.

A study of these functions comprises the theory of special functions and we undertake here a brief study of some of the important properties of the following special functions:

The gamma and the beta functions; and the Legendre's and the Bessel's ...