Page:EB1911 - Volume 25.djvu/678

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656
SPHERICAL HARMONICS


From this form it can be shown that

Q.G0 =i P.(m) log K±|- W^Cm),

where W»_i(/x) is a rational integral function of degree n — I in m; it can be shown that this form is in agreement with the definition of Qnto by series, for the case mod /*>i. In case mod /n x . but differs from it by an imaginary multiple of P„(m). It will be observed that Q n (i), Q n ( — i) are infinite, and Qn(x) =o. The function Wn-i(ju) has been expressed by Christoffel in the form

2 "-?P„_ lW + - 2 "-

•w+|^ p — w+-

I . n ^' . ' 3 . n — I

and it can also be expressed in the form

~P (p.)P^ 1 (p.)+~P l (p)?^. 2 (p.) + . . . +P„-,(m)Po(m).

It can easily be shown that the formula (28) is equivalent to

which is analogous to Rodrigue's expression for PnM- Another expression of a similar character is

It can be shown that under the condition mod \u — V (u- — 1)\ >mod \ti — V (v?— 1)|, the function il(n — u) can be expanded in the form 2(2»-|-i)P«(tt)Q„(tt) ; this expansion is connected with the definite integral formula for Q n (v) which was used by F. Neumann as a definition of the function Q»(p.), this is

which holds for all values of \i which are not real and between ± I. From Neumann's integral can be deduced the formula

d4>

••00 Qn(M) = J ,

iM+vc^-o.cosh^r'

which holds for all values of p. which are not real and between ="= 1 , provided the sign of V (/* 2 — 1 ) is properly chosen ; when p is real and greater than 1, V (p 2 — 1) has its positive value. By means of the substitution.

Jm + V(m 2 — i).cosh \fr\\ii — V(m 2 — l).cosh x! = i,

the above integral becomes

Q„0) = J *V~V (m 2 - 1) cosh xNx. where xa = ^og c j~. '

This formula gives a simple means of calculating Q„(p) for small values of n; thus

  • w-/>-->*e±j.

If, in Legendre's equation, we differentiate m times, we find

. . n d m **u , . . d M+1 u , , ., , , .d m u

• c 11 1 d m u , m , , ,v m d m u

it follows that u = ^'hence «„ = 0* 2 -i)- - a -^-

2m'"

The complete solution of (26) is therefore

when m is real and lies between ± 1 , the two functions

U-M - ) 4u'" ' U_M - ) C^" 1 are called Legendre's associated functions of degree <n, and ordur m, of the fhrt and second kinds respectively. When /j is not real and between =*=i, the same names are given to the two functions

Qi(p) =/»X«-V (m 2 - I) -sinh Xo = M-^oSj^~ I.

Neumann's integral affords a means of establishing a relation between successive Q functions, thus

»Q»-(2rt-i)/«Q»-i + (n-l)(?»_2

"1 «P,(tt) + (n- i)P„- 2 (h.) - (2w- p — u

pP n -i{u)du

-ITJ

= -|J _,(2»- l)P»-l(") =0.

Again, it may similarly be proved that

19. Legendre Associated Functions. — Returning to the equation (26) satisfied by «™ the factor in the normal forms „_, ™ mtfr.u™,

we shall consider the case in which n, m are positive integers, and

n^m. Let « = (m 2 — i)* m v, then it will be found that v satisfies the equation

(m 2 -0

dp'" '

(V-i)

U d m Q*W

dp m

in either case the functions may be denoted by P n (m), Qn (m). It can be shown that, when p is real and between ± 1

p:m =5^11 (S3 im ^io.-i)"^o.+i)-i

2»(n-m)l\l+p) dp" 1 ^ V " +l ' <• In the same case, we find

Pr +2 (cos 8)-2(m + i) cot $ Pâ„¢ +1 (cos 0)

+ (n — m)(n+m + l)P™(cos 8)=o, (ra-«+2)Pr +2 (cos 0) - (2n+3) M Pr + i(cos 0)

+ (»+m + l)Pr(cos0)=o.

20. Bessel's Functions.— -If we take for three orthogonal systems of surfaces a system of parallel planes, a system of co-axial circular cylinders perpendicular to the planes, and a system of planes through the axis of the cylinders, the parameters are a, p, $, the cylindrical co-ordinates; in that case Hi = i, Hi = i, H 3 = i/p, and the equation (25) becomes

d"v yy 1 av 1 a-v d~ 2 + <v +p dp v a<^> 2 ~ '

To find the normal functions which satisfy this equation, we put V = ZR, when Z is a function of z only, R of p only, and * of <f>, the equation then becomes

I ^?j_± /^P:j_I ^E\ j.1 I 4!?_ Z dz- + K Up 2 + p dp/ V* d^~°-

1 <i 2 Z That this may be satisfied we must have y~r^ constant, say =k',

tt yj conetant, say = — m-, and R, for which we write u, must satisfy the differential equation d 2 u , I dti

dp'

. 1 du /,„ pi-\

it follows that the normal forms arc e ^ z '-m<p.u(kp), where u(p) satisfies the equation

",7/ T n ii t n'"\

(29)

d 2 u , I du , / n7 ! \

d?+pd7 + { l -7) u=0 -

This is known as Bessel's equation of order m\ the particular case

d"-u , I du

d9 + pT P + u=0 '

(30)

corresponding to m = o. is known as Bessel's equation.

If we solve the equation (29) in series, we find by the usual process that it is satisfied by the series

+0

2.2OT+2 r 2.4.2/W-(-2.2W-|-4

the expression

P" \, P 2 ,

2"*II(m) ( 2.2m+2 2.4.2m+2.2m+4

...| s

or

(— i)V +2n

/ |) 2"' +2 "n(m+»)n(w' l

is denoted by J m (p). """i When m=o, the solution

i _pJ + _A_...

2 2 ' 2 2 . 4 2

of the equation (30) is denoted by Jo(p) or by J(p).