Periodic Differential Equations. An Introduction to Mathieu, Lamé, and Allied Functions

288 Pages·1964·13.48 MB·English

by  F. M. Arscott

PERIODIC DIFFERENTIAL
EQUATIONS
An Introduction to Mathieu, Lame,
and Allied Functions
by
F. M. ARSCOTT
Battersea College of Technology
London
THE MACMILLAN COMPANY
NEW YORK
1964
THE MACMILLAN COMPANY
60 Fifth Avenue
New York IL ΝΎ.
This book is distributed by
THE MACMILLAN COMPANY
pursuant to a special arrangement with
PERGAMON PRESS LIMITED
Oxford, England
Copyright © 1964
PERGAMON PRESS LTD.
Library of Congress Catalog Card Number 62-8703
Printed in Poland
PWN — DRP
P R E F A CE
OF RECENT years, the two subjects of special functions and ordinary
linear differential equations have (in this country at least) lain
somewhat in the penumbra of a partial eclipse. Many mathemati­
cians, not without reason, have come to regard the former as
little more than a haphazard collection of ugly and unmemorable
formulae, while attention in the latter has been directed mostly
to existence-theorems and similar results for equations of general
type. Only rarely does one find mention, at post-graduate level,
of any problems in connection with the process of actually solving
such equations. The electronic computer may perhaps be partly
to blame for this, since the impression prevails in many quarters
that almost any differential equation problem can be merely "put
on the machine", so that finding an analytic solution is largely
a waste of time. This, however, is only a small part of the truth,
for at the higher levels there are generally so many parameters
or boundary conditions involved that numerical solutions, even
if practicable, give no real idea of the properties of the equation.
Moreover, any analyst of sensibility will feel that to fall back on
numerical techniques savours somewhat of breaking a door with
a hammer when one could, with a little trouble, find the key.
In this book I have sought to give an account of half a dozen
important equations and the special functions which they generate,
ranging from Mathieu's equation, of which a good deal (though
still not enough) is now known, to the intractable ellipsoidal wave
equation which so far has yielded few of its secrets. As a pure
mathematician with an applied bias, I have concentrated on fun­
damental problems and techniques of solution rather than the
properties of particular functions; at the same time, however,
I have tried to keep an eye on the physical origins of the equations
vii
VIH PREFACE
and be mindful of the significance, in the material world, of their
solutions.
I hope the pure mathematician who ventures into this book may
be pleasantly surprised at the delicacy and subtlety of some of the
techniques which have proved successful in the field of periodic
differential equations, and also at the extent to which this subject
involves some of the most modern and fundamental ideas of analysis.
If this book serves only to persuade a few well-trained pure mathe­
maticians to turn their attention to this neglected field its purpose
will be half fulfilled. Functional analysts, for instance, could find
rich employment for their skill in a thorough study of two-parameter
eigenvalue and spectral problems*, or of the non-linear integral
equations which arise so naturally from some of the equations.
Nearer the field of classical analysis, the mathematician who
devises a practical treatment of third-order difference equations
will find in his hands a rod with which some big fish can be hooked.
Whatever hope I once had of making the treatment encyclopaedic
was soon abandoned, and I fear applied mathematicians will have
to wait longer for the book (which I understand they want) that
will do for Mathieu functions what Watson's treatise does for
Bessel functions. Nevertheless, I believe there is no important
technique which has not found its place in this book, and compa­
ratively few results whose existence has not been indicated, however
briefly, among the examples.
With these considerations in mind, I have started with an outline
of the physical origins of the equations, then given a fairly full
account of Mathieu's equation, the simplest and most typical
simply-periodic equation, using a treatment intermediate between
that of McLachlan's "Theory and Applications of Mathieu Func­
tions", written primarily for engineers, on the one hand, and
Meixner and Schäfke's "Mathieusche Funktionen und Sphäroid­
funktionen", which rests upon a fairly deep Banach space founda­
tion, on the other. At the same time, I have tried to make it as easy
as possible for the reader to pass on to a more detailed study of
either. Indeed, throughout the writing of this book I have endeav-
" See Additional Note A.7.
PREFACE IX
oured to smooth the path for the reader who (as will often be the
case) wants to use this as an introduction to the more advanced
literature; wherever possible I have used the same notation and
terminology as prevails elsewhere even when, as has sometimes
happened, this has led to inconsistencies of notation between one
chapter and another. To the reader who accuses me of writing
a series of even powers as Ó ^r^^'" in one place and Y^a^^z^" in
another, I can make only tliis defence.
It was with some surprise, in writing Chapter II, that I found
how far the theory of Mathieu functions could be developed from
the differential equation itself before solutions are actually construc­
ted, and although none of the material of this chapter is new the
presentation can fairly claim to be so. The rest of the section
on Mathieu's equation. Chapters III-VI, contains little that is
new, and I have leaned heavily on the writers mentioned, particu­
larly McLachlan, but I have been at some pains to explain
wherever possible the heuristic processes which led, or may have
led, to the various results.
Hill's general equation, treated in Chapter VII, is in many ways
only a natural extension of Mathieu's, but I have felt the particular
case of the three-term equation worth a special section. Space
forbade more than one chapter (Chapter VIII) on the important
spheroidal wave equation, and as there is a full and excellent expo­
sition available in Meixner and Schäfke's book, I have concentrated
rather on illustrating the technique of handling such an equation
in an algebraic form.
Finally, in the last two chapters, I deal with doubly-periodic
equations. Here, a new approach was inevitable; I hope it may
serve to unify the disjointed literature on Lame's equation and to
set the fascinatingly tricky ellipsoidal wave equation in perspec­
tive.
It was with regret that I decided against attempting to in­
clude an account of the modern stability theory relating to periodic
diflferential equations of very general type, associated with the
work of Liapunov and his successors; it was soon obvious that
one could not hope to do justice to so important a field of
study except at disproportionate length. Nevertheless, I hope the
÷ PREFACE
reader who intends to delve into this material may find Chap­
ter II of some use as a preliminary.
My debt to previous writers is quite uncountable, and I must
leave the references to indicate its full extent, but in addition to
those I have already mentioned I must record the valuable works
of Strutt and of Campbell. My sincere thanks are also due to Pro­
fessor I.N.Sneddon for guidance and help in the writing; to
Mrs. K. M. Urwin who has read and criticised most valuably the
entire manuscript; to Mrs. P.J. Goodwin who typed the bulk of
it, and to my daughter. Miss Elizabeth Arscott, who helped in
the preparation of the references.
The subject-matter of this book is a field in which British math­
ematicians, from G. H. Darwin and Sir Edmund Whittaker on­
wards, have played a leading part; it might seem invidious to single
out any one of them, but were I to venture on a dedication it would
be a double one; to the memory of the late Dr. E. L. Ince, whose
writings did so much to inspire the beginning of this work, and to
my wife whose encouragement has made possible its completion.
F. M. ARSCOTT
Battersea College of Technology
London
Note: The decimal system of paragraph numbering is used throughout the
book, and equations are numbered serially in each paragraph, thus **3.6(2)'*
refers to equation (2) in § 3.6. Similarly, **3.6 Ex. 2" refers to Example 2 of the
set immediately following § 3.6.
The miscellaneous examples at the end of each chapter are, by and large,
more difficult than those in the body of the chapter.
References in the text are generally to the place where the result quoted
can most conveniently be found in modern literature; this is not always
the original source.
CHAPTER I
F O R M A T I ON OF T HE E Q U A T I O N S:
T HE M A IN P R O B L E MS
1.1. Problems leading to periodic differential equations
The diíFerential equations with periodic coefficients considered
in this book arise in three main ways. In some practical problems
they occur naturally because some factor in the problem is itself
periodic; these are mainly problesm in connection with oscillations
or in electronic circuits, but a notable case is that of Hill's equation
which occurred in his investigation of the motion of the moon
(§ 1.5 (1)).* The main source of these equations, however, is the type
of problem in which we have to find a solution of a partial differen­
tial equation (for example, the wave equation) where the solution
has to be such as to satisfy given boundary conditions at certain
special surfaces, in particular elliptic cylinders, spheroids or ellip­
soids. This involves the introduction of new coordinates and
then the "separation" of the partial differential equation into two
or three (ordinary) equations in the new variables. This chapter
is concerned mainly with showing how these equations are formed.
A third source of periodic differential equations is of mainly
mathematical interest. Ordinary linear differential equations of
the second order are classified according to the number and com­
plexity of their singularities (Whittaker and Watson 1, § 10.6;
Ince 1, Chapter XX). It is found that when such an equation has
two (and only two) separate regular singularities (and no other
• It has also been remarked (Young 1) that Mathieu's equation gov­
erns the motion of the acrobat who holds an assistant poised on a pole
above his head while he himself stands on a spherical baH rolling on the
ground. There seems to be no record, however, of any experimental check
on the applicability of this theory.
1
2 PERIODIC DIFFERENTIAL EQUATIONS
singularities) in the 'finite part of the plane, it generally takes on
a more compact form when a trigonometric substitution is made
for the independent variable. Thus the equation
x{l - x )^ + (I- x)^ + (A + Bx)w = 0, (1)
dx^ dx
(which has regular singularities at Λ: = 0, 1 and oo) becomes, when
we put X = cos^z,
— + (4^ + 25 + 2B cos 2z) w = 0. (2)
dz^
Besides being more compact, the second form has the advantage
that there are now no finite singularities.
If the equation has three regular singularities in the finite part
of the plane, a similar transformation can be made using an elliptic
function instead of a trigonometric function (Ince 1 § 20.22). Fol­
lowing Ince's classification, the equations considered in this book
are those with formulae [2,0,1] (Mathieu's equation), [1,1,1]
(Spheroidal wave equation), [2,0, Ig], (Hill's equation with three
terms), [3, 1,0] (Lame's equation), and [3,0, 1] (Ellipsoidal wave
equation), besides Hill's general equation which is of a higher type.
Examples
1. Show that the substitution χ = sin ζ reduces the equation
to the form
^'"^ - 2v tan + (A + ^sin^z) w = 0.
dz^ dz
2. Show that the equation
d^w dw
dx^ dx
is transformed by the substitution χ = {CjBY e^'^ into
^ - 4{^ -f- 2 {BCf cos 2z} w = 0.
(Ince 1, § 20.32)
1.2. Separation of coordinates: examples
Before considering separation of coordinates in elliptic and
other systems, the reader may find it helpful to have first a simpler
example.
FORMATION OF THE EQUATIONS 3
Suppose* we wish to find a solution of the two-dimensional
wave equation
v v ^ -^ + - ^ - ^ = = i -^ (1)
such that for all values of r, y = 0 at all points of the boundary of
the circle + = a\
It is convenient to have the time-factor occurring in the formt
e~~'^^ and we accordingly set ψ = W(x, j)e~"^^^; this gives
where χ = pjc.
Since the boundary condition relates to the circle x^+y'^ = a^,
it is natural to go over to polar coordinates χ = gcos(^, y =
ρύηφ; then (2) becomes
Now we seek to find a solution in the form W = f(ρ)g(φ), where
/, g are functions of ρ only and <^ only respectively!:. Substitution
in (3) leads to
+ Qmim
QTWAQ) + XY = - ε"{ΦΜΦ)· (4)
But the left-hand side is a function of ρ only, while the right-hand
side is a function of φ only; consequently each must be a constant.
Let us denote this "separation constant", as it is called by a; then
(4) gives rise to the two equations
Qrwrn+Qmm+xv=a,
ϊ'(ΦΜΦ)=-α,
* This is the problem of the vibration of a circular membrane of radius
a clamped at its outer edge like a drum, ψ represents the displacement at the
point (x,y) at time r, and c is a constant depending on the material of
the membrane.
t Physically, this means that all the waves have the same fundamental
period. Alternatively, the forms exp (+ />/), cospt or sin/?/ may be used
according to circumstances; each gives the same equation (2), which is
called the "Helmholtz equation", **reduced wave equation", or simply
"wave equation'* when confusion is unlikely.
t This, of course, is far from being the general solution; it is merely
the type of solution needed in this particular problem.
4 PERIODIC DIFFERENTIAL EQUATIONS
that is,
ο'τί + ^Τ· + <^'ο''-«)/=0, (5a)
^ + . . = - 0, (5b)
Now, for physical applications, the function W(x,y) must be a
single-valued function of position—that is, of Λ: and y. But in the
equations χ = ρcosφ, y = ρ smφ, φ is arbitrary to the extent
of a multiple of 2π; that is, χ and y are periodic in φ with period 2π.
Consequently, to ensure that PF is a single-valued function of
position it is necessary not only to have / and g single-valued, but
also to have g a periodic function of φ with period 2π—that is,
g(φ+2π) = g(φ), The general solution of (5b) is, of course, ^(φ) =
^cos(0 ya)+^ sin (φ ]/α), but this will be periodic with period
2π if, and only if, j/a is an integer which we may denote by «. Then*
ΞίΦ) = A cos ηφ + Β sin ηφ.
A possible g(φ) is thus given if we take a = n\ η being any
integer; the particular value chosen must now be substituted in
(5a). If we then make the substitution QI = χρ, this reduces to
Bessel's equation
Ql^^ + Qi-y^ + (Ql-^')f=^^ (6)
άρΙ d^i
whose complete solution is /(ρ^) = CJJjo^+D Γ„(ρι), but the Γ„
term must be rejected because it is not single-valued. Thus a solution
of (2) in the "separated" form is
W=Jn (χρ) (A cos ηφ + Β sin ηφ\ (7)
in which η may be any integer and A, Β are arbitrary. Moreover,
any linear combination of such expressions will also satisfy (2).
It is now possible to apply the further condition that ψ (in (1))
is to be zero when ρ = α for all values of t and φ; this means that
W must be zero when ρ = α for all φ, and this implies, using (7),
* If Λ > 1 then the smallest period of g{^) is actually 2π//ι, a submultiple
of 2π, but this is immaterial, since 2π is still itself a period.