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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 Differentialgleichungen,” to accompany a short course on The Potential Function, given by Professor B. O. Peirce. My course has been gradually modified and extended until it has become an introduction to Spherical Harmonics and Bessel’s and Lam´e’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 Potential Function” (Boston, Ginn & Co.), and the two sets of lectures form a course (Math. 10) given regularly at Harvard, and intended as a partial introduction to modern Mathematical Physics. Students taking this course are supposed to be familiar with so much of the infinitesimal calculus as is contained in my “Differential 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 “differential coefficient,” and δ for “partial derivative with respect to x.” use the notation Dx instead of δx The course was at first, as I have said, an exposition of Riemann’s “Partielle Differentialgleichungen.” 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 Lam´e”), Lord Rayleigh (“Theory of Sound”), and Forsyth (“Diff