## 1

## Special Functions

#### 1.1 Introduction

In the study of Mathematics, we have so far dealt with algebraic and transcendental functions *x*^{n}, ∑ *a*_{n}x^{n}, 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:

- Functions defined by improper integrals and
- 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 ...