The Project Gutenberg EBook of An Elementary Treatise on Fourier’s Series and Spherical

The Project Gutenberg EBook of An Elementary Treatise on Fourier’s Series and Spherical, Cylindrical, and Ellipsoidal Harmonics, by William Elwood Byerly This eBook is for the use of anyone anywhere at no cost and with almost no restrictions whatsoever. You may copy it, give it away or re-use it under the terms of the Project Gutenberg License included with this eBook or online at www.gutenberg.org Title: An Elementary Treatise on Fourier’s Series and Spherical, Cylindrical, and Ellipsoidal Harmonics With Applications to Problems in Mathematical Physics Author: William Elwood Byerly Release Date: August 19, 2009 [EBook #29779] Language: English Character set encoding: ISO-8859-1 *** START OF THIS PROJECT GUTENBERG EBOOK TREATISE ON FOURIER’S SERIES *** Produced by Laura Wisewell, Carl Hudkins, Keith Edkins and the Online Distributed Proofreading Team at http://www.pgdp.net (The original copy of this book was generously made available for scanning by the Department of Mathematics at the University of Glasgow.) AN ELEMENTARY TREATISE ON FOURIER’S SERIES AND SPHERICAL, CYLINDRICAL, AND ELLIPSOIDAL HARMONICS, WITH APPLICATIONS TO PROBLEMS IN MATHEMATICAL PHYSICS. BY WILLIAM ELWOOD BYERLY, Ph.D., PROFESSOR OF MATHEMATICS IN HARVARD UNIVERSITY. GINN & COMPANY BOSTON NEW YORK CHICAGO LONDON Copyright, 1893, By WILLIAM ELWOOD BYERLY. ALL RIGHTS RESERVED. Transcriber’s Note: A few typographical errors have been corrected - these are noted at the end of the text. i PREFACE. About ten years ago I gave a course of lectures on Trigonometric Series, following closely the treatment of that subject in Riemann’s \Partielle Dieren-tialgleichungen," to accompany a short course on The Potential Function, given by Professor B. O. Peirce. My course has been gradually modied and extended until it has become an introduction to Spherical Harmonics and Bessel’s and Lame’s Functions. Two years ago my lecture notes were lithographed by my class for their own use and were found so convenient that I have prepared them for publication, hoping that they may prove useful to others as well as to my own students. Meanwhile, Professor Peirce has published his lectures on \The Newtonian Po-tential Function" (Boston, Ginn & Co.), and the two sets of lectures form a course (Math. 10) given regularly at Harvard, and intended as a partial intro-duction to modern Mathematical Physics. Students taking this course are supposed to be familiar with so much of the innitesimal calculus as is contained in my \Dierential Calculus" (Boston, Ginn & Co.) and my \Integral Calculus" (second edition, same publishers), to which I refer in the present book as \Dif. Cal." and \Int. Cal." Here, as in the \Calculus," I speak of a \derivative" rather than a \dierential coecient," and use the notation Dx instead of for \partial derivative with respect to x." The course was at rst, as I have said, an exposition of Riemann’s \Partielle Dierentialgleichungen." In extending it, I drew largely from Ferrer’s \Spherical Harmonics" and Heine’s \Kugelfunctionen," and was somewhat indebted to Todhunter (\Functions of Laplace, Bessel, and Lame"), Lord Rayleigh (\Theory of Sound"), and Forsyth (\Dierential Equations"). In preparing the notes for publication, I have been greatly aided by the criticisms and suggestions of my colleagues, Professor B. O. Peirce and Dr. Maxime Bo^cher, and the latter has kindly contributed the brief historical sketch contained in Chapter IX. W. E. BYERLY. Cambridge, Mass., Sept. 1893. ii ANALYTICAL TABLE OF CONTENTS. CHAPTER I. pages Introduction 1{29 Art. 1. List of some important homogeneous linear partial dierential equa-tions of Physics.|Arts. 2{4. Distinction between the general solution and a particular solution of a dierential equation. Need of additional data to make the solution of a dierential equation determinate. Denition of linear and of linear and homogeneous.|Arts. 5{6. Particular solutions of homogeneous lin-ear dierential equations may be combined into a more general solution. Need of development in terms of normal forms.|Art. 7. Problem: Permanent state of temperatures in a thin rectangular plate. Need of a development in sine series. Example.|Art. 8. Problem: Transverse vibrations of a stretched elastic string. A development in sine series suggested.|Art. 9. Problem: Potential function due to the attraction of a circular ring of small cross-section. Surface Zonal Har-monics (Legendre’s Coecients). Example.|Art. 10. Problem: Permanent state of temperatures in a solid sphere. Development in terms of Surface Zonal Harmonics suggested.|Arts. 11{12. Problem: Vibrations of a circular drum-head. Cylindrical Harmonics (Bessel’s Functions). Recapitulation.|Art. 13. Method of making the solution of a linear partial dierential equation depend upon solving a set of ordinary dierential equations by assuming the dependent variable equal to a product of factors each of which involves but one of the inde-pendent variables. Arts. 14{15 Method of solving ordinary homogeneous linear dierential equations by development in power series. Applications.|Art. 16. Application to Legendre’s Equation. Several forms of general solution obtained. Zonal Harmonics of the second kind.|Art. 17. Application to Bessel’s Equa-tion. General solution obtained for the case where m is not an integer, and for the case where m is zero. Bessel’s Function of the second kind and zeroth order.|Art. 18. Method of obtaining the general solution of an ordinary lin-ear dierential equation of the second order from a given particular solution. Application to the equations considered in Arts. 14{17. CHAPTER II. Development in Trigonometric Series 30{55 Arts. 19{22. Determination of the coecients of n terms of a sine series so that the sum of the terms shall be equal to a given function of x for n given val-ues of x. Numerical example.|Art. 23. Problem of development in sine series treated as a limiting case of the problem just solved.|Arts. 24{25. Shorter TABLE OF CONTENTS iii method of solving the problem of development in series involving sines of whole multiples of the variable. Working rule deduced. Recapitulation.|Art. 26. A few important sine developments obtained. Examples.|Arts. 27{28. Develop-ment in cosine series. Examples.|Art. 29. Sine series an odd function of the variable, cosine series an even function, and both series periodic functions.| Art. 30. Development in series involving both sines and cosines of whole mul-tiples of the variable. Fourier’s series. Examples.|Art. 31. Extension of the range within which the function and the series are equal. Examples.|Art. 32. Fourier’s Integral obtained. CHAPTER III. Convergence of Fourier’s Series 56{69 Arts. 33{36. The question of the convergence of the sine series for unity considered at length.|Arts. 37{38. Statement of the conditions which are sucient to warrant the development of a function into a Fourier’s series. His-torical note. Art. 39. Graphical representation of successive approximations to a sine series. Properties of a Fourier’s series inferred from the constructions.| Arts. 40{42. Investigation of the conditions under which a Fourier’s series can be dierentiated term by term.|Art. 43. Conditions under which a function can be expressed as a Fourier’s Integral. CHAPTER IV. Solution of Problems in Physics by the Aid of Fourier’s Inte-grals and Fourier’s Series 70{135 Arts. 44{48. Logarithmic Potential. Flow of electricity in an innite plane, where the value of the Potential Function is given along an innite straight line; along two mutually perpendicular straight lines; along two parallel straight lines. Examples. Use of Conjugate Functions. Sources and Sinks. Equipotential lines and lines of Flow. Examples.|Arts. 49{52. One-dimensional ow of heat. Flow of heat in an innite solid; in a solid with one plane face at the tempera-ture zero; in a solid with one plane face whose temperature is a function of the time (Riemann’s solution); in a bar of small cross section from whose surface heat escapes into air at temperature zero. Limiting state approached when the temperature of the origin is a periodic function of the time. Examples.|Arts. 53{54. Temperatures due to instantaneous and to permanent heat sources and sinks, and to heat doublets. Examples. Application to the case where there is leakage.|Arts. 55{56. Transmission of a disturbance along an innite stretched elastic string. Examples.|Arts. 57{58. Stationary temperatures in a long rectangular plate. Temperature of the base unity. Summation of a Trigonometric series. Isothermal lines and lines of ow. Examples.|Art. 59. ... - tailieumienphi.vn