Page:EB1911 - Volume 25.djvu/681

From Wikisource
Jump to navigation Jump to search
This page needs to be proofread.
659
SPHERICAL HARMONICS


importance in connexion with the differential equations of physics. The two equations

are reducible by means of the substitutions u = e~ k 'v, u = e lk 'v to the form v*»+»=o. If we suppose v to be a function of r only, this last differential equation takes the form




0,


so that v has the values

sin r/r, cos rjr; in order to obtain more general solutions of the equation v*»+d=o, we may operate on

sin rjr, cos rjr with the operator

Y f±. ± -±) In Va*' dy' dzj '

where Y„ (x, y, z) is any spherical solid harmonic of degree n. The result of the operation may be at once obtained by taking Y„ {x, y, z) for /„(*, y, z) in the theorem (7'), we thus find as solutions, of V*v+f = o, the expressions

Y.(x, y. z)j^ By recurring to the definition of the function J m (r), we see that


sin r v , s d" cosy

-7—, '»ft y< z -><Z( r 2)» r '


j*wV£i'-


r 2 r

2.3 2-3-


4-5


-â– M


2 sin r


thus


r-ijl^


, /2 sin r


Using the relation between Bessel's functions whose orders differ by an integer, we have

Mr).(-2)^,^-(-.)VV' d " Sinr


<Z(r 2 )"


It may be shown at once that


that the function K„(z) has no real zeros unless «=2fe + l where k is an integer, when it has one real negative zero; and that K n (z) has no purely imaginary zeros, and no zero whose real part is positive, other than those at infinity. When l>re>0, K„(z) has no zeros other than those at infinity, when 2>n>l,it has one zero whose real part is negative, and when m+i>-n>m where m is ah integer, there are to zeros whose real parts are negative. When n is an integer, K„(z) has n zeros with negative real parts.

29. Spheroidal Harmonics. — For potential problems in which the boundary is an ellipsoid of revolution, the co-ordinates to be used are r, 0, <t> where in the case of a prolate spheroid

  • = c \V 2 — I sin cos <f>, y = c-Jr* — l sin sin <p, z = cr cos 0,

the surfaces r = r , 0=0o, 4>=<t><s are confocal prolate spheroids, confocal hyperboloids of revolution, and planes passing through the axis of revolution. We may suppose r to range from 1 to 00 , 8 from o to ir, and <j> from o to 2ir, every point in space has then unique co-ordinates r, 8, <j>.

For oblate spheroids, the corresponding co-ordinates are r, given by


8, 4,


»+i


d n cos r

d{r*)» r


a second solution of 4>) denotes a surface


x = c^r z + i sin 8cos<t>, y = c\V 2 + l sin 8 sin <t>, z = cr cos 8,


where


0< r < w , o < < ff, o < tfr < 2jt ;


these may be obtained from those for the prolate spheroid by chang- ing c into — ic, and r into ir.

Taking the case of the prolate spheroid, Laplace's equation becomes

3V\ , r 2 -cos'0 a 2 v


d { ,„ ,aV) . 1 a / . „av\ .


m<t>.


dr S ' sin 8 08 V*" " 68/ " r (^-i)sin»9 dtf and it will be found that the normal solutions are

p»)p;T(cos0);

Q:(r))Q:(cos8)

For the space inside a bounding spheroid the appropriate normal

m are positive integers, and for the external space


forms are P"(r)Vâ„¢ (cos 8)^?*m<t>, where n,


is a second solution of Bessel's equation of order n-f-j; thus the differential equation y*v+v = o is satisfied by the expression

y.(*. y, 3) J -^tV

and by the corresponding expression with Bessel's equation instead of J^+i (r) ; if S„(/j, harmonic of degree n, the expression

S.O., *)^ w l(r)

is a solution of the equation v 2 ti-{-v = o.

The Bessel's functions of degree half an odd integer are the only ones which are expressible in a closed form involving no trans- cendental functions other than circular functions. It will be observed that in this case the semi-convergent series for J m becomes a finite one as the expressions P, Q then break off after a finite number of terms.

28. The Zeros of Bessel's Functions. — The determination of the position of the zeros of the Bessel's functions, and the values of the argument at which they occur, have been investigated by Hurwitz {Math. Ann. vol. xxxiii.j, and more completely by H. M. Macdonald (Proc. Lond. Math. Soc. vols. xxix.,xxx.). It has been shown that the zeros of J»(z)/z* are all real and associated with the singular point at infinity when n is real and > — 1 , and that all the real zeros of J»(z)/z" when n is real and <— I, and not an integer, are associated with the essential singularity at infinity. When n is. a negative integer — to, Jr.(z)/z" has, in addition,, 2w real zeros co- incident at the origin. When n — — m— v, m being a positive integer, and i>f>o, LCzWz" has a finite number 2tn of zeros which are not associated with the essential singularity. If n is real, and starts with any positive value, the zeros nearest the origin approach it as n diminishes, two of them reaching it when n= -i-,i, and t\vo more reach it whenever n passes through a negative integral value; these zeros then become complex for values of n not integral. The zeros of J„(z)/z" are separated by those of J»+i(z)/z n , one zero of the latter, and one only, lies between two consecutive zeros of Jn(z),/z". When n is real and >— I, all the zeros of Jn(z)/z" are givenby a formula due to Stokes; the m th positive zero in order of magnitude is given by


q:wp„-(cos0)^>*.


5£r


For the case of an oblate spheroid, P"('f), Q7("')i take the place

of K(r), QT(r).

30. Toroidal Functions. — For potential problems connected with the anchor-ring, the following co-ordinates are appropriate: If A, B are points at the extremities of a diameter of a fixed circle, and P is any point in the plane PAB which is perpendicular to the plane of the fixed circle, let P = log(AP/BP), 0=/_APB, and let <t> be the angle the plane APB makes with a fixed plane through the axis of the circle. Let be restricted to lie between — ir and ir, a discontinuity in its value arising as we pass through the circle, so that within the circumference is ir on the upper side of the circle, and — ir on the lower side ; 8 is zero in the plane of the circle outside the circumference; p may have any value between —00 and co . and <j> any value between o and 2ir. The position of a point is then uniquely represented by the co-ordinates p, 8, <t>, which ate the parameters of a system of tores with the fixed circle as limiting circle, a system of bowls with the fixed circle as common rim, and a system of planes through the axis of the tores. If x, y r z are the co-ordinates of a point referred to axes, two of which x, y are in the plane of the circle and the third along its axis, we find that

a sinh p . , _ ,0. sin


,sin<p, Z =


cosh p— cos 8


4n»— 1 4(4^— i)(28w 8 — 31)


— &C.,


I


sinh p dtf " can be shown


that this


a sinh p x ~ cosh p - cos COS *' y = cosh p - cos C

where a is the radius of the fixed circle. ' Laplace's equation reduces to

_a ( sinhp aV ) _a \ sinhp 3V dp\ P 2 dp ) + d9\ P 2 d8 )

when P denotes V(cosh p— cos 0). It equation is satisfied by

1 1 u . P»_i(cosh p) cos „ cos ,

V (cosh p-cos ^Q-^cosh p) sin "" sin m *'

the functions P"_ } (cosh p), Qr_|(cosh p) required for the potential problems, are associated Legendre's functions of degree n — |, half an odd integer, of integral order «, and of argument real and greater than unity; these are known as toroidal functions. For the space

external to a boundary tore the function Qâ„¢_ i (cosh p) must be

where (i = iir(2»+4»»-i). It has been, shown by Macdonald | used > al » d for the internal space P'^cosh p).