chapter 2, Special Functions

Excerpt

Historical Comments: The study of special functions involves too many names to mention them all, so we concentrate on a few of the more well known people. The famous mathematician Leonhard Euler (1707–1783) was concerned with a generalization of the factorial function now called the gamma function. Other mathematicians involved with the gamma function were Adrien M, Legendre (1752–1833) and Karl Weierstrass (1815–1897). Legendre is also credited with developing the polynomial set now bearing his name in connection with his development of the gravitational potential into a power series. Additional polynomial sets with similar properties carry the name of Charles Hermite (1822–1901) and Edmund Laguerre (1834–1886), among others.

The German astronomer Friedrich W. Bessel (1784–1846) first achieved fame by computing the orbit of Halley's comet. In addition to many other accomplishments in connection with his studies of planetary motion, he is credited with deriving the differential equation now bearing his name and carrying out the first systematic study of the general properties of Bessel functions in his famous memoir of 1824. Nonetheless, Bessel functions were first (unknowingly) discovered in 1732 by Daniel Bernoulli (1700–1782) who provided a series representation. There now exists a wide variety of types of Bessel functions that also bear such famous names as Hankel, Kelvin, Lommel, Struve, Airy, Anger, and Weber.

The major development of the properties of the hypergeometric function was carried out in 1812 by Carl F. Gauss (1777–1855), and a similar analysis involving the confluent hypergeometric functions was carried out in 1836 by Ernst E. Kummer (1810–1893). Generalized hypergeometric functions were developed in the late 1800s and early 1900s by people such as Clausen, Appel, Lauriceila, Whittaker, MacRobert, and Miejer.

Our objective in this chapter is to familiarize the reader with many of the special functions that arise in advanced engineering applications. In addition to defining these functions, we provide lists of their most important properties for easy reference. We start, however, with a review of several of the standard engineering-type functions that are widely used in practice, like the step function and Dirac delta function.

2.1 Introduction

Functions introduced in introductory calculus and engineering courses are called elementary functions—i.e., algebraic functions, trigonometric functions, exponential functions, logarithm functions, and so on.