Spheroidal Wave Functions

By Carson Flammer

Intended to facilitate the use and calculation of spheroidal wave functions, this applications-oriented text features a detailed and unified account of the properties of these functions. Addressed to applied mathematicians, mathematical physicists, and mathematical engineers, it presents tables that provide a convenient means for handling wave problems in spheroidal coordinates.
Topics include separation of the scalar wave equation in spheroidal coordinates, angle and radial functions, integral representations and relations, and expansions in spherical Bessel function products. Additional subjects include recurrence relations of Whittaker type, asymptotic expansions for large values of c, and vector wave functions. The text concludes with an appendix, references, and tables of numerical values.

• Language: English
• Publisher: Dover Publications
• Release date: Mar 5, 2014
• ISBN: 9780486158518

Book preview

Spheroidal Wave Functions

Spheroidal Wave Functions

Carson Flammer

Dover Publications, Inc.

Mineola, New York

DOVER PHOENIX EDITIONS

Bibliographical Note

This Dover edition, first published in 2005, is an unabridged republication of a Stanford Research Institute monograph, originally published by Stanford University Press, Stanford, California, 1957. Page 141 of this edition was intentionally left blank to allow the next table to spread across two pages.

International Standard Book Number: 0-486-44639-5

eISBN-13: 978-0-486-15851-8

Manufactured in the United States of America

Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

preface

The purpose of this monograph is to facilitate the use and calculation of spheroidal wave functions by the presentation of a detailed and unified account of the properties of these functions as they are known to date, along with some useful tables.

From the beginning, the development of the spheroidal functions has been incited, for the most part, by investigations of physical problems. The present work is no exception. While treating the electromagnetic problems of the prolate spheroidal monopole antenna, on the one hand, and the scattering of electromagnetic waves by a perfectly conducting circular disk, on the other, the author was lead in turn to the prolate and oblate spheroidal wave functions. In each case, further development of the functions in certain directions seemed desirable for a better treatment of the physical problem. The great interest expressed by many people in a resulting report on the prolate functions has inspired the preparation of this more comprehensive treatment of both the prolate and oblate functions.

The notation (with a slight change in designation) and normalization of the functions originally proposed by J. A. Stratton and L. J. Chu have been adopted in this book. In much of the development, the analyses of the properties of the prolate and of the oblate functions proceed along parallel lines. The formulas for the one type are then obtained from the other by a simple transformation. But in some of the development there is considerable asymmetry. This is especially true in the case of asymptotic solutions, where satisfactory solutions have been obtained for the oblate, but not for the prolate functions. The included tables of numerical values of the various quantities are also very unsymmetrical and, regrettably, scanty. In regard to tables of spheroidal wave functions, it is a pleasure to note that a really significant contribution has been made in the book with the same title as the present one by J. A. Stratton, P. M. Morse, L. J. Chu, J. D. C. Little, and F. J. Corbat1ó, which has come to hand after the completion of the present text. The book by Stratton, et al, has five hundred and fifty pages of tables of eigenvalues and expansion coefficients!

In consonance with its express purpose, the present monograph is addressed to the applied mathematician, the mathematical physicist, and the mathematical engineer, and not to the pure mathematician. The existence of solutions and the completeness of sets of functions are presumed, not demonstrated, in the text that follows. To the pure mathematician the recent book, Mathieusche Funktionen und Sphäroidfunktionen, by J. Meixner and F. W. Schäfke, is heartily recommended.

I am greatly indebted to Miss Mary Humphrey for her assistance in the preparation of the tables. I should also like to thank Professor R. D. Spence for making available his computations of the prolate spheroidal functions. To Dr George W. Evans, II, I express my gratitude for his careful reading and criticism of the manuscript. Finally, it is a pleasure for me to express my appreciation to Stanford Research Institute for instituting the series of monographs of which this is the first.

The manuscript was prepared with the support of the United States Air Force, and I wish to acknowledge my obligation to this support.*

Carson Flammer

Menlo Park, California

February 1956

* The ideas expressed in this book represent the personal views of the author and are not necessarily those of the United States Air Force.

contents

preface

chapter 1 introduction

1.1 Historical Survey

1.2 Survey of Applications

Spheroidal antennas. Acoustic diffraction by circular disks and apertures. Acoustic radiation from a vibrating circular disk. Acoustic scattering by spheroids. Steady flow of a viscous fluid past a spheroid. Spheroidal potential well. Spheroidal cavity systems. Electromagnetic diffraction by circular disks and apertures. Electromagnetic scattering by spheroids

chapter 2 separation of the scalar wave equation in spheroidal coordinates

2.1 Spheroidal Coordinates

2.2 The Spheroidal Differential Equations

2.3 General Properties

table I Notations for Prolate Spheroidal Wave Functions

table II Notations for Oblate Spheroidal Wave Functions

chapter 3 the angle functions

3.1 The Functions Smn

Expansions in associated Legendre functions of the first kind. Power-series developments of the eigenvalues. Power-series developments of the expansion coefficients. Bouwkamp’s method of approximation. Normalization. The case of negative values of m

3.2 Expansions in Powers of 1 — η²

3.3 The Special Case of m = 1, c =

3.4 Other Expansions

3.5 The Functions

table III λon and λin in Powers of c²

chapter 4 the radial functions

4.1 Expansions in Spherical Bessel Functions

4.2 Relations Between the Radial and Angle Functions

4.3 The Special Case of m — 1, c =

4.4 Power-series Expansions

The prolate radial functions. The oblate radial functions

4.5 Other Expansions

4.6 Special Values

Special values of the prolate functions. Special values of the oblate functions table IV The Coefficients

chapter 5 integral representations and relations

5.1 A Theorem

5.2 Expansions of the Green’s Functions

5.3 Various Integral Relations

5.4 Generalization of F. Neumann’s Integral

chapter 6 expansions in spherical bessel function products

6.1 The Prolate Functions

6.2 The Oblate Functions

chapter 7 recurrence relations of whittaker type

7.1 Recurrence Relations for the Radial Functions

Relations between functions of order m. Relations between functions of order m and order m ± 1

chapter 8 asymptotic expansions for large values of c

8.1 The Prolate Functions

8.2 The Oblate Functions

The angle functions. The radial functions

chapter 9 vector wave functions

9.1 Definition of the Functions

9.2 Expansions in Vector Wave Functions

Expansions of plane polarized waves. Expansions of the radiation from Hertzian dipoles

table V Expressions of the Spheroidal Vector Wave Functions

appendix

references

tables of numerical values

Introduction

Comparison with the Tables of Stratton et al

Power-Series Expansion Coefficients for λmn

Power-Series Expansion Coefficients for

Power-Series Expansion Coefficients for

Prolate Spheroidal Functions:

Eigenvalues λmn

Expansion coefficients and

Expansion coefficients

Expansion coefficients

Joining factors

Angle functions Smn(c, η)

Normalization constants Nmn

Radial functions of the first kind

Derivative of the radial functions of the first kind

Radial functions of the second kind

Derivative of the radial functions of the second kind

Oblate Spheroidal Functions:

Eigenvalues λmn

Expansion coefficients

Angle functions Smn(–ic, η)

Radial functions of the first and second kinds

1

introduction

1.1 Historical Survey

The differential equations resulting from the separation of the scalar Helmholtz differential equation

in general ellipsoidal coordinates, were first obtained in the case of k² = 0, i.e., for the Laplace equation, by G. Lamé [l]¹ in 1837. This was three years after Lamé had for the first time effected the transformation of to general orthogonal coordinates [2]. Consequently the designation of ‘Lamé’s equation’ came into use, and the integrals of the separated differential equations became known as ‘Lamé’s functions.’ Later, as the wave equation with k² > 0 came under consideration, the designations ‘Lamé’s potential functions,’ or ellipsoidal harmonics, and ‘Lamé’s