Special Functions for Scientists and Engineers

W. W. Bell

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486435213
• eBook ISBN: 9780486317564
• Topic: Technology & Engineering
• Subtopic: Engineering General
• Index: Technology & Engineering

Clear and comprehensive, this text provides undergraduates with a straightforward guide to special functions. It is equally suitable as a reference volume for professionals, and readers need no higher level of mathematical knowledge beyond elementary calculus. Topics include the solution of second-order differential equations in terms of power series; gamma and beta functions; Legendre polynomials and functions; Bessel functions; Hermite, Laguerre, and Chebyshev polynomials; Gegenbauer and Jacobi polynomials; and hypergeometric and other special functions. Three appendices offer convenient tabulation of principal results, and a generous supply of worked examples and problems includes some hints and solutions. 1968 edition. 25 figures.

1

SERIES SOLUTION
OF DIFFERENTIAL EQUATIONS

1.1 METHOD OF FROBENIUS

Many special functions arise in the consideration of the solutions of equations of the form

We shall restrict ourselves to equations of the type

where q(x) and r(x) may be expanded as power series in x,

convergent for some range of x including the point x = 0.

The basis of Frobenius’ method is to try for a solution of equation (1.2) of the form

with a₀ ≠ 0. (We may always take a₀ ≠ 0, since otherwise we should just have another series of the type (1.5) with a different value of s, the first coefficient of which we could now call a₀.)

From equation (1.5) we have

and

so that if we require z to satisfy equation (1.2) we must have

and this reduces to

which, on using equations (1.3) and (1.4) for q(x) and r(x) and cancelling a common factor of x⁸, becomes

For the infinite series on the left-hand side of equation (1.6) to be zero for all values of x in some range, we must have the coefficient of each power of x equal to zero. This requirement gives rise to a set of equations as follows.

Requiring that the coefficient of x⁰ be zero, and noting that ...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• List of Symbols
• Contents
• 1 Series Solution of Differential Equations
• 2 Gamma and Beta Functions
• 3 Legendre Polynomials and Functions
• 4 Bessel Functions
• 5 Hermite Polynomials
• 6 Laguerre Polynomials
• 7 Chebyshev Polynomials
• 8 Gegenbauer and Jacobi Polynomials
• 9 Hypergeometric Functions
• 10 Other Special Functions
• Appendices
• Hints and solutions to problems
• Bibliography
• Index