Spherical Harmonics
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17 Claus MUlier Institut fur Reine und Angewandte Mathematik Technische Hochschule Aachen
Spherical Harmonics 1966
Springer-Verlag' Berlin· Heidelberg' New York
All rights, especially that of translation into foreign languages, reserved. It is also forbidden to reproduce this book, either whole or in part, by photomechanical means (photostat, microfilm and/or microcard) or by other procedure without written permission from Springer Verlag. C by Springer-Verlag Berlln : Heidelberg 1966 Ubrary of Congress Catalog Card Number 66-22467. Printed in Germany. Title No.7H7.
PREFACE
The subject of these lecture notes is the theory of regular spherical harmonics in any number of dimensions. The approach is such that the two- or three-dimensional problems do not stand out separately. They are on the contrary regarded as special cases of a more general structure. It seems that in this way it is possible to get a better understanding of the basic properties of these functions, which thus appear as extensions of well-known properties of elementary functions. One outstanding result is a proof of the addition theorem of spherical harmonics, which goes back to G. Herglotz. This proof of a fundamental property of the spherical harmonics does not require the use of a special system of coordinates and thus avoids the difficulties of representation, which arise from the singularities of the coordinate system. The intent of these lectures is to derive as many results as possible solely from the symmetry of the sphere, and to prove the basic properties which are, besides the addition theorem, the representation by a generating function, and the completeness of the entire system. The representation is self-contained. This approach to the theory of spherical harmonics was first presented in a series of lectures at the Boeing Scientific Research Laboratories. It has since been slightly modified. I am grateful to Dr. Theodore Higgins for his assistance in preparing these lecture notes and I should like to thank Dr. Ernest Roetman for a number of suggestions to improve the manuscript.
February 1966
Claus MUller
CONTENTS
General Background and Notation ••••••••••••••••••••••••••••••• 1 Orthogonal Transformations •• Addition Theorem
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Representation Theorem •••••••••••••••••••••••••••••••••••••••• 11 Applications of the Addition Theorem •••••••••••••••••••••••••• 14 Rodrigues Formula ••••••••••••••••••••••••••••••••••••••••••••• 16
Funk - Heeke Formula
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Integral Representations of Spherical Harmonic •••••••••••••••• 21 Associated Legendre Functions
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Properties of the Legendre Functions
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Differential Equations •••••••••••••••••••••••••••••••••••••••• 37 Expansions in Spherical Harmonics ••••••••••••••••••••••••••••• 40 Bibliography ••••••••••••••••••••••••••••••••••••••••••••••••••
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GENERAL BACKGROUND AND NOTATIO
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