# Page:EB1911 - Volume 25.djvu/675

653
SPHERICAL HARMONICS

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

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

p:(cos e) =^Â±^sinÂ»e j cos--Â«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. â€” 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 â€” h) 2 < I

(l-2hn+Vy ! = l+h(2n-h)+Â±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 - Â«(*-!) ._ , w(w-0(Â«-2)(Â»-3) â€ž-< [

I.2.3. ..Â» < 2.2Â»-I M 2.4.2nâ€” I.2Bâ€” 3 )

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

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

the function Pâ€ž(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 â€” i)- We now see that

{r- â€” 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 â€” 2)6 cos (Â« â€” 4)8 +

(14) when r < r', or

CO

2^ p - w

when r' < r; it follows that the two expressions r n Pâ€ž0i), r~" 'Pâ€žIm) 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=Â§(35m 4 -30m 2 +3). P5(M)=g(63M 5 -70/x 3 + J5^)

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

We find also

Pâ€ž(l) = l,Pn(-i) = (-i)Â»

+3 1 5m 3 - 35*0 â€¢

PÂ»(0)=0, or (-i)*"^ 1 ^-

2.4. . .Â»

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

11. Additional Expressions for Legendre's Coefficients. â€” The expression (3) for Pâ€ž(>i) may be written in the form

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

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

l.2Â« â€” I

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

By (13), or by the formula

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

=f(Â« + i, -n, I, sin 2 ?). (15)

Also that Pâ€ž(cosÂ«)=cos 2 "|j x-Jtan^+^^tan^-... |

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

By means of the identity

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

it may be shown that

tw â€ž.. â€žâ€ž ( Â«(Â»â€” 1). ,â€ž 1 Â»(Â» â€” 1)(Â» â€” 2)(n â€” 3). ,â€ž ) Pâ€ž(cos0) = cos"0 j 1 ~ 5 â€” 'tan 2 0+ â€” iTT2 ^ tan 4 â€” ... >

= 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. â€” V M 2 â€” 1 cos ^) n+l ' by means of the relation

(m+Vju 2 -i cos 0)0*â€” Vm 2 -i cos^) = i.

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

Pâ€ž(ccs0) =2 rÂ» cos(n + j)Â» ^ 2 p sin(n + j)Â» ^

y T./ 0V2 COS0 â€” 2COS9 T^ 9V2COSS â€” 2COS0

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

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

(i-2Am+A 2 )||+(A-m)Â« = 0;

on substituting SA"Pâ€ž for u, and equating to zero the coefficient of h", we obtain the relation

Â«Pâ€ž-(2Â»-l)/iPÂ»_i + (n-i)P n _j = 0.

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

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

dPn d?^ â€ž

"â– ST"" 5T =mP "

( â€ž +l)P â€ž = _ M g. + ^ ( 2M + I )Pâ€ž = ^-^

C, 1 ,N ^PÂ» I I ^Pn-1 1 rfPn+1 (2n + l)M _ =( â€ž + l)__-+*_^-

(2Â» + i)0 u 2 -i)^ = n(â€ž + i)(Pâ€ž +1 -P n _ 1 )

P-(m)=(-

2- (^J '.(-Â»)!

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

according as n is even or odd.

dP n

dp

(2Â«-l)P^l + (2Â»~5)PÂ»-J + (2Â»-9)P-Â«+- â– â€¢

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

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

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

To prove this theorem we have