An Elementary Treatise on Fourier's Series

and Spherical, Cylindrical, and Ellipsoidal Harmonics, with Applications to Problems in Mathematical

William Elwood Byerly

• Publisher: Dover Publications
• Year: 2014
• Print ISBN:
  9780486495460
• eBook ISBN:
  9780486159904
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

Originally published over a century ago, this work remains among the most useful and practical expositions of Fourier's series, and spherical, cylindrical, and ellipsoidal harmonics. The subsequent growth of science into a diverse range of specialties has enhanced the value of this classic, whose thorough, basic treatment presents material that is assumed in many other studies but seldom available in such concise form. The development of functions, series, and their differential equations receives detailed explanations, and throughout the text, theory is applied to practical problems, with the solutions fully worked out. In addition, 190 problems, many with hints, are included. 1893 edition. Appendix of 6 tables.

CHAPTER I.

INTRODUCTION.

1.In many important problems in mathematical physics we are obliged to deal with partial differential equations of a comparatively simple form.

For example, in the Analytical Theory of Heat we have for the change of temperature of any solid due to the flow of heat within the solid, the equation

where u represents the temperature at any point of the solid and t the time.

In the simplest case, that of a slab of infinite extent with parallel plane faces, where the temperature can be regarded as a function of one coördinate, [I] reduces to

a form of considerable importance in the consideration of the problem of the cooling of the earth’s crust.

In the problem of the permanent state of temperatures in a thin rectangular plate, the equation [I] becomes

In polar or spherical coördinates [I] is less simple, it is

In the case where the solid in question is a sphere and the temperature at any point depends merely on the distance of the point from the centre [IV] reduces to

In cylindrical coördinates [I] becomes

In considering the flow of heat in a cylinder when the temperature at any point depends merely on the distance r of the point from the axis [VI] becomes

In Acoustics in several problems we have the equation

for instance, in considering the transverse or the longitudinal vibrations of a stretched elastic string, or the transmission of plane sound waves through the air.

If in considering the transverse vibrations of a stretched string we take account of the resistance of the air [VIII] is replaced by

In dealing with the vibrations of a stretched elastic membrane, we have the equation

or in cylindrical coördinates

In the theory of Potential we constantly meet Laplace’s Equation

which in spherical coördinates becomes

and in cylindrical coördinates

In curvilinear coördinates it is

represent a set of surfaces which cut one another at right angles, no matter what values are given to ρ₁, ρ₂, and ρ₃; and where

and, of course, must be expressed in terms of ρ₁, ρ₂, and ρ₃.

If it happens that ∇²ρ₁ = 0, ∇²ρ₂ = 0, and ∇²ρ₃ = 0, then Laplace’s Equation [XV] assumes the very simple form

...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• Contents
• Chapter I. Introduction
• Chapter II. Development In Trigonometric Series
• Chapter III. Convergence of Fourier’s Series
• Chapter IV. Solution of Problems in Physics by the Aid of Fourier’s Integrals and Fourier’s Series
• Chapter V. Zonal Harmonics
• Chapter VI. Spherical Harmonics
• Chapter VII. Cylindrical Harmonics (Bessel’s Functions)
• Chapter VIII. Laplace’s Equation in Curvilinear Coordinates. Ellipsoidal Harmonics
• Chapter IX. Historical Summary
• Appendix