We prepared the following book in order to make several useful topics from the theory of special functions, in particular the spherical harmonics and Legendre polynomials of ℝp, available to undergraduates studying physics or mathematics. With this audience in mind, nearly all details of the calculations and proofs are written out, and extensive background material is covered before beginning the main subject matter. The reader is assumed to have knowledge of multivariable calculus and linear algebra (especially inner product spaces) as well as some level of comfort with reading proofs.
Literature in this area is scant, and for the undergraduate it is virtually nonexistent. To find the development of the spherical harmonics that arise in ℝ3, physics students can look in almost any text on mathematical methods, electrodynamics, or quantum mechanics (see [3], [5], [10], [12], [17], for example), and math students can search any book on boundary value problems, PDEs, or special functions (see [7], [20], for example). However, the undergraduate will have a very difficult time finding accessible material on the corresponding topics in arbitrary ℝp.
We have been greatly influenced by Hochstadt's The Functions of Mathematical Physics [9]. When this book was prepared and released as notes, Hochstadt's book was out of print. Fortunately, Dover reprinted it in the Spring of 2012. However, this book contains a more expanded and detailed point of view than in Hochstadt's book, as well additional information and a chapter with the solutions to all problems. There are several additional references (see [2], [6], [13], [21], [22], for example) where the reader can search for information on the topic of this book, but either the coverage is brief or the level of difficulty is considerably higher. In addition, the point of view is very different from the one adopted in this work. Therefore, we hope that the current book will become a useful supplement for any reader interested in special functions and mathematical physics, especially students who learn the topic.

Preface vii

Acknowledgements ix

List of Symbols xi

1 Introduction and Motivation 1

1.1 Separation of Variables 2

1.2 Quantum Mechanical Angular Momentum 9

2 Working in p Dimensions 13

2.1 Rotations in Ep 14

2.2 Spherical Coordinates in p Dimensions 15

2.3 The Sphere in Higher Dimensions 20

2.4 Arc Length in Spherical Coordinates 24

2.5 The Divergence Theorem in Ep 26

2.6 △p in Spherical Coordinates 30

2.7 Problems 34

3 Orthogonal Polynomials 39

3.1 Orthogonality and Expansions 39

3.2 The Recurrence Formula 42

3.3 The Rodrigues Formula 45

3.4 Approximations by Polynomials 48

3.5 Hilbert Space and Completeness 57

3.6 Problems 62

4 Spherical Harmonics in p Dimensions 63

4.1 Harmonic Homogeneous Polynomials 63

4.2 Spherical Harmonics and Orthogonality 71

4.3 Legendre Polynomials 78

4.4 Boundary Value Problems 105

4.5 Problems 116

5 Solutions to Problems 119

5.1 Solutions to Problems of Chapter 2 119

5.2 Solutions to Problems of Chapter 3 127

5.3 Solutions to Problems of Chapter 4 131

Bibliography 139

Index 141

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