ON THE FUNCTIONS ASSOCIATED WITH THE ELLIPTIC
CYLINDER IN HARMONIC ANALYSIS
BY E. T.
1.
WHITTAKER.
Introduction.
It is well-known that the solution of the wave-equation
dx2
+
dy2
dt2
for circular bodies, or of the potential-equation
dx2
dy2
dz2
for circular-cylindrical distributions, leads to the functions of Bessel : in the same way,
the solution of these equations for elliptic bodies or elliptic-cylindrical distributions
leads to the " elliptic-cylinder functions," which are defined by the differential equation
| | + ( a + *» cos» *)# = <>
(1),
where a and k denote constants*.
One reason for the importance of these elliptic-cylinder functions lies in the fact
that they are not, like the functions of Legendre and Bessel, mere particular or
degenerate cases of the hypergeometric function : the differential equation (1) is,
indeed, the equation which most naturally presents itself for study, in the theory of
linear differential equations, when the hypergeometric equation has been disposed
of. It has a further practical interest in connexion with Hill's theory of the motion
of the moon's perigee.
The solutions of the above differential equation are not, in general, periodic
functions of z: but there are an infinite number of solutions which are periodic
functions of z, of period 27r. (This is analogous to the fact that the solutions of
Legendre's differential equation are not in general polynomials in z, although there
are an infinite number of them—namely Pn(z) when n is any integer—which are
* Mathieu, Liouville's Journal, (2), x m . (1868), p. 137. Recent papers by Butts, Amer. Journ. Math.
xxx. (1908), p. 129 and Marshall, ibid. xxxi. (1909), p. 311 : recent inaugural dissertations at Zurich by
Dannacher (1906) and Wiesmann (1909).
FUNCTIONS ASSOCIATED WITH THE ELLIPTIC CYLINDER IN HARMONIC ANALYSIS
367
polynomials in z.) It is these periodic solutions of the differential equation which
are required in mathematical physics, and it is with them that the present paper
is concerned.
2.
The elliptic-cylinder functions are the solutions of a certain integral-equation.
The starting-point of the present investigation is a result previously published
by the author*, that the general solution of the potential-equation is
rz-rr
V =
f(x cos 6 4- y sin 6 4- iz, 6) d9,
.'o
where / denotes an arbitrary function of its two arguments. In order to obtain the
solution in elliptic-cylinder functions, we replace x and y by the variables which are
appropriate to the elliptic cylinder, namely f and v, where
x = h cos £ cosh n,
y — h sin £ sinh v.
The potential-equation in terms of these variables becomes
d2V
2
2
+ d V + /l2(cosh2l, cos2d V = 0
W ^
" ^ '
and the general solution of the equation becomes
f27r
V = f(h cos £ cosh v cos 6 4- h sin £ sinh n sin 6 4- iz, 6) do.
Jo
The solutions required in the harmonic analysis appropriate to the elliptic
cylinder are of the form
V=eim°X(Ç)Y(v),
where X(£) denotes a function of £ alone and Y (n) denotes a function of n alone:
substituting, we have for the determination of X and Y the equations
-Ä + (A + ™a&a cos2 f) X = °>
^
4- ( - 4 - m2/i2 cosh2 *;) F = 0,
each of which is equivalent to the equation (1).
potential-equation evidently takes the form
The corresponding solution of the
C2ir
Y=
I
cm (h cos £wsh y ws 9+ h sin Ç sinh y sin 6+iz) A /û\ J g
Jo
where </> (6) is a function as yet undetermined. From this it follows that the
differential equation (1) must be satisfied by an integral of the form
y (£\ _ j
ßk (cos z coshi) cos 0+sin z sinh 17 sin 9) £ /g\
^g
Jo
where n is arbitrary. Taking n to be zero, we have the result that the differential
equation (1) must be satisfied by an integral of the form
[2TT
y(z)=\
e* cos * cos 0 (f)(0)
de.
Jo
* Mathematische Annalen, LVII. (1903), p. 333.
368
E. T. WHITTAKER
In order to determine the function (/> (0), we substitute this integral in the
differential equation, which gives
f27T
ehcoszCOSO
(jp
C0S2
g
+
£2 C 0 S 2 ^ s j n 2 g _ jc
cos
£ C 0 S # 4~ <^) (£ (6) dO = 0,
'0
or, integrating by parts, and supposing that $ (6) is periodic,
f27T
T27T
fc cos 0 cos1 9{</
: >"
((9) 4- a<f> (6) 4- &2 cos2 d<j> (6)} d6 = 0,
JO
which is evidently satisfied provided
• ^ + (a + ^
cos2 ö
) <£ = o.
Thus we see that (f)(0) must be a periodic elliptic-cylinder function of 0, formed with
the same constants a and k as y (z) itself ; but there does not exist more than one
distinct periodic solution of the equation (1) with the same constants a and & : so
that <jb (0) must be (save for a multiplicative constant) the same function of 0 as y (z)
is of z. Thus finally we have the result that the periodic solutions of the equation (1)
satisfy the homogeneous integral-equation
f2ir
y(z) + \
ekQOSZCO*6y(0)d0 = O
(2).
Jo
It is known from the general theory of integral-equations that this equation (2)
does not possess a solution except when X has one of a certain set of values X0, XL, X2,
\3, ... : and when X has one of these values, say Xr, there exists a corresponding solution yr(z).
This set of solutions yQ(z), yi(z), y^(z), ... are the periodic solutions of
the differential equation (1) ; that is to say, they are the elliptic-cylinder functions
required in mathematical physics*.
3.
Determination of the elliptic-cylinder junction of zero order, ce0 (z).
We shall now make use of the result just obtained in order to derive the
periodic solutions of the differential equation (1).
When k is zero, the solutions of the equation (1) with period 2ir are obtained
by taking a to be the square of an integer: the solutions are then
1,
cos z,
sin z,
cos 2z,
sin 2z,
cos 3^,
sin Sz,
As we shall see, the periodic solutions of the equation (1) when k is different
from zero correspond respectively to these, and reduce to them when k tends to
zero. We shall call these solutions
ceQ(z),
ceY(z),
se^(z),
ce2(z),
se2(z),
cez(z),
se%(z),
...,
cer (z) being the solution which reduces to cos rz when k is zero, and ser (z) being the
solution which reduces to sin rz.
* This integral-equation was mentioned, and ascribed to myself, by Mr H. Bateman in Trans. Camb.
Phil. Soc. xxi. p. 193 (1909): but I have not previously published anything on the subject, the theorem
having merely been communicated to Mr Bateman in conversation.
FUNCTIONS ASSOCIATED WITH THE ELLIPTIC CYLINDER IN HARMONIC ANALYSIS
369
The elliptic-cylinder functions cer (z) and ser (z) both reduce to the circularcylinder or Bessel function Jr (ik cos z) when the eccentricity of the elliptic cylinder
reduces to zero.
We shall first determine the elliptic-cylinder function of zero order, ce0 (z).
Since ceQ (z) reduces to unity when k reduces to zero, we see at once from the
integral-equation (2) that X must reduce to — 1/27T when k reduces to zero. So write
- TT-^ = l + ajc + a2k2 4- a3k3 -\- ...,
ce, (z) = 1 4-fc&i(z) + k% (z) 4- k% (z) 4- ...,
where bx (z), b2 (z), bz (z), ... are periodic functions of z, of period 27T, having no
constant term.
Substituting in the integral-equation (2), we have
{1 4- ajc 4- a2k2 4-...} {1 4- k\(z)
4- k%(z) 4-...}
= ^- [ln\l + kco8z cos 0+~
ATT Jo
{
cos 2 * cos2 0+..\{l+
l\
kb, (0) + k% (0) 4- ...} d0.
)
Equating the coefficients of k on both sides of this equation, we have
1 f277
h (z) 4- a1 = ^ {cos z cos 0 4- bx (0)} d0.
An Jo
Since the integral on the right-hand side vanishes, we have
b1(z) = 0, a 1= =0.
Next equating the coefficients of k2 on both sides of the equation, we have
1 f27r
b2 (z) 4 - ^ = 5 \b2 (0) 4-1 cos2 z cos2 0} d0
^7TJo
= \ cos2 z.
Since b2 (z) is to contain no constant term, we must have
b2(z) = £ cos 2z,
a2 = \.
z
Similarly by equating the coefficients of k and ¥, we find
h(z) = 0,
I
/ N
!
a3 = 0,
A
04(^) = 2 9 c o s 4 ^
7
a
4=2-9.
The first terms of the lowest-order solution of the integral-equation
|"2îT
"2îT
ce (z) = X
ek cos z cos Q ce (0) d9
Jo
are therefore given by the equation
k2
k4
ce0 (z) = l + — cos 2s 4--^ cos 4s* + ... ;
M.
c.
24
370
E. T. WHITTAKER
the corresponding value of X is at once obtained by writing z~\ir
equation, which becomes
ce0 (^7r) = X\
in the integral-
ce0 (0) d0 = 27rX,
Jo
f2ir
1
so we have
ce, (z) = ^ - ce0 QTT)
^7T
^ cos* cos e ceo (#) ^0.
Jo
By continuing the above procedure we could obtain as many terms as are
required of the expansions, but we should not obtain a formula for the general
term. I n order to obtain this, we write
00
ce, (z) = 1 4- S A2r (k) cos 2rz ;
r=l
then since it is known that
gwcoss = JQ (jm^ _ 2i Ji (im) cos z - 2J2 (im) cos 2z 4-...,
we can equate coefficients of cos 2rz in the integral-equation, and thus obtain
A2r
1
f27r
(k) = ~ ce0 (fw)
( - l ) r . 2J2r (ik cos 0) ce0 (0) d0,
(r = 1, 2, ... oo ).
When on the right-hand side we substitute the expansions, and use the formula
i rI 2ir nnvW tì nr^a V^fìrlfì —
2
r
Y~
:
2TTJ 0 cos ^ 0 cos 2p0d0
_ .
the last equation becomes
„ _
A
k2r
_Ül
2**q+p\q-p\'
_ r (3r 4- 4) k2r+"
We thus find for the elliptic-cylinder function of zero order the expansiion
, x -, , S f &2r
c,0 (,) = 1 4- ï
^
4.
/^+4r(3r4-4)
Ì
- 2 ^ - r 7 + T T r + T! + ' * j C0S ^
-
TAe o^er integral-equation satisfied by the elliptic-cylinder functions.
In the same way we can show that the elliptic-cylinder functions are the
solutions of the integral-equation
y(z) + xl "eiksmzsmey (0\.<jr0 = 0,
Jo
and in particular that
1
. f2?r
ce0 (z) = ^r~ ce0 (0) cos (k sin z sin 0) ce0 (0)d0.
lir
Jo
5.
The elliptic-cylinder functions of order unity, ce1(z) and sex(z).
The two elliptic-cylinder functions of order unity, which we denote by cex (z)
and se1 (z), may be obtained in the same way : the integral-equation for ce1 (z) is
found to be
1
C2w kcos cosö
e *
ceY (0) d0,
cei (z) = - j - ce; (\TT)
fCTT
J ()
FUNCTIONS ASSOCIATED WITH THE ELLIPTIC CYLINDER IN HARMONIC ANALYSIS
371
and from it we derive the expansion
oo
CO! (z) = COS Z 4- S
!
1,21'
y.l.2r-\-2
4r
, " ! |2 . r 4-1 ! r !
2
+
4r+4
.r + l ! r + l !
2^.r-l!TT2"! + -j C O S ( 2 r + 1 ) ^
Similarly
.
» f
se, (z) - sin z +*i\2«mr
&2r
+ l l r l
rk2r+2
+ 2"+*.r + 1 ! r + 1 !
+
2 4 ^ o . r , i ! r + 2 ! + - } ^ ( 2 r + l)^
The elliptic-cylinder functions of higher order can be determined by the same
method directly from the integral-equation: the method may indeed be applied to
solve a large class of integral-equations.
24—2