Page:EB1911 - Volume 25.djvu/675

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653
SPHERICAL HARMONICS


and let each side operate on i/r, then in virtue of (io), we have (rr')"P» ( xx '+yy' i + zz ' \ = P„( c os 9 cos 0'+sin sin 0' cos <j> -<)>') = P„(cose)P„(cose')+2S^^|p™(cos9)P^(cos9')cosm(0-«') (n)

which is known as the addition theorem for the function P». It has incidentally been proved that

p:(cos e) =^±^sin»e j cos--«e

_{n-m)(n-m-i) ^ ^^ ) _ ( , 2)

2.2OT+2 )

which is an expression for Pâ„¢ (cos 0) alternative to (4).

10. Legendre's Coefficients. — The reciprocal of the distance of a point (r, 0, $) from a point on the z axis distant r' from the origin is

(r 2 -2rr' M +r'T i which satisfies Laplace's equation, n denoting cos 0. Writing this expression in the forms

it is seen that when r < r' , the expression can be expanded in a convergent series of powers of r/r', and when r' < r in a convergent series of powers of r'/r. We have, when /j 2 (2/i — h) 2 < I

(l-2hn+Vy ! = l+h(2n-h)+±lh?{2n-hy+ . . .

I _ .3. 5 ..,2tt-I fr, ( _ h) n + _ _ ~ 2.4. . .2n v ^ ' ' and since the series is absolutely convergent, it may be rearranged as a series of powers of h, the coefficient of h" is then found to be I.3-S..-2B-I J - «(*-!) ._ , w(w-0(«-2)(»-3) „-< [

I.2.3. ..» < 2.2»-I M 2.4.2n— I.2B— 3 )

this is the expression we have already denoted by PnCu) I thus

(1 -2&m +&')"* = PoM+*PiG') + ...+^Pn(M) + .. •• (I3)

the function P„(ji) may thus be defined as the coefficient of h n in this expansion, and from this point of view is called the Legendre's coefficient or Legendre's function of degree n, and is identical with the zonal harmonic. It-may be shown that the expansion is valid for all real and complex values of h and m, such t hat mo d, h is less than the smaller of the two numbers mod. (m* V/* 2 — i)- We now see that

{r- — 2rr' ju-|-r' 2 )~<

is expressible in the form

From the identity

(1 -2h cos e+h 2 )-i = (1 -fce' 9 )-}(i -he-*} it can be shown that

i.n

2.4.6. . .2

Pn(cos 0) = I -, 3 - J 5 ft- 2 " w I j cos n0+

â– cos (n — 2)6 cos (« â€” 4)8 +

(14) when r < r', or

CO

2^ p - w

when r' < r; it follows that the two expressions r n P„0i), r~" 'P„Im) are solutions of Laplace's equation.

The values of the first few Legendre's coefficients are

Po(m) = i. PiW=M, P*(m) =5(3^-1), Ps(p)=|(5/^-3m)

P,W=§(35m 4 -30m 2 +3). P5(M)=g(63M 5 -70/x 3 + J5^)

P»(A>)=^(23iM«-3i3M* + i05/i ! -5), P7(m)=^(429m 7 -693m !

We find also

P„(l) = l,Pn(-i) = (-i)»


+3 1 5m 3 - 35*0 •

P»(0)=0, or (-i)*"^ 1 ^-

2.4. . .»

according as n is odd or even; these values may be at once obtained from the expansion (13), by putting p=*i, o, — 1.

11. Additional Expressions for Legendre's Coefficients. — The expression (3) for P„(>i) may be written in the form

P.O.) =_&£!. F (-2, 1=«, I-». 4) " w 2"n ! » r V 2 2 2 (j 2 /

with the usual notation for hypergeometric series. On writing this series in the reverse order

l.2« â€” I

â– i.3-n(n-i) " r i.2.(2n — l)(2« â€” 3)

By (13), or by the formula

which is known as Rodrigue's formula, we may prove that P»(cos9) = l- K ^ sin 2 -+' - ^ ,v Jy 'sm 4 -...

=f(« + i, -n, I, sin 2 ?). (15)

Also that P„(cos«)=cos 2 "|j x-Jtan^+^^tan^-... |

= cos 2 »^F l-n, -n, 1, -tan 2 |) . (16)

By means of the identity

(l-2ft M +fr)-»»(l-*/0- 1 | I + ^ I -7ff j~*.

it may be shown that

tw „.. „„ ( «(»— 1). ,„ 1 »(» — 1)(» — 2)(n — 3). ,„ ) P„(cos0) = cos"0 j 1 ~ 5 — 'tan 2 0+ — iTT2 ^ tan 4 — ... >

= cos n flF(-in, i-in, I, -tan 2 0). (17)

Laplace's definite integral expression (6) may be transformed into the expression

1 C* d4>

tJ (p. — V M 2 — 1 cos ^) n+l ' by means of the relation

(m+Vju 2 -i cos 0)0*— Vm 2 -i cos^) = i.

Two definite integral expressions for ?*(/«) given by Dirichlet have been put by Mehler into the forms

P„(ccs0) =2 r» cos(n + j)» ^ 2 p sin(n + j)» ^

y T./ 0V2 COS0 — 2COS9 T^ 9V2COSS — 2COS0

When re is large, and 9 is not nearly equal to o or to ir, an approximate value of P„(cos0) is |2/nx sin 8\i sin {(n-\-\)6-\-\i!\.

12. Relations between successive Legendre's Coefficients and their Derivatives. — If (1 — 2hii-\-h' i )~'h be denoted by u, we find

(i-2Am+A 2 )||+(A-m)« = 0;

on substituting SA"P„ for u, and equating to zero the coefficient of h", we obtain the relation

«P„-(2»-l)/iP»_i + (n-i)P n _j = 0.

From Laplace's definite integral, or otherwise, we find

(m'-i)^ =»G»P»-P-0 = -(« + i)(mP"-P" +1 )- We may also show that

dPn d?^ „

"â– ST"" 5T =mP "

( „ +l)P „ = _ M g. + ^ ( 2M + I )P„ = ^-^

C, 1 ,N ^P» I I ^Pn-1 1 rfPn+1 (2n + l)M _ =( „ + l)__-+*_^-

(2» + i)0 u 2 -i)^ = n(„ + i)(P„ +1 -P n _ 1 )

P-(m)=(-

2- (^J '.(-»)!

(_I) 2 2->«-l ,n-i^ F (~T- i + 1 ' 2 "V 2 2

according as n is even or odd.

dP n

dp

(2«-l)P^l + (2»~5)P»-J + (2»-9)P-«+- ■•

the last term being 3P! or P according as n is even or odd.

13. Integral ]*ropfrties of Legendre's Coefficients.— \t may be shown that if P»(?i) be multiplied by any one of the numbers I , /x. M 2 , • • .-ju" -1 and the product be integrated between the limits I,— I with respect to n, the result is zero, thus

rVP»fj*)^=0, a = 0, I, 2, . . .n-l. (18)

To prove this theorem we have