Mathematical Methods for Physics and Engineering : A Comprehensive Guide

18.1 LEGENDRE FUNCTIONS

Equation (18.16) can then be written, using (18.15), as

h

∑

Pnhn=(1− 2 xh+h^2 )

∑

Pn′hn,

and equating the coefficients ofhn+1we obtain the recurrence relation

Pn=Pn′+1− 2 xPn′+Pn′− 1. (18.18)

Equations (18.16) and (18.17) can be combined as

(x−h)

∑

Pn′hn=h

∑

nPnhn−^1 ,

from which the coefficent ofhnyields a second recurrence relation,

xPn′−Pn′− 1 =nPn; (18.19)

eliminatingPn′− 1 between (18.18) and (18.19) then gives the further result

(n+1)Pn=Pn′+1−xPn′. (18.20)
If we now take the result (18.20) withnreplaced byn−1andaddxtimes (18.19) to it
we obtain

(1−x^2 )Pn′=n(Pn− 1 −xPn). (18.21)

Finally, differentiating both sides with respect toxand using (18.19) again, we find

(1−x^2 )Pn′′− 2 xPn′=n[(Pn′− 1 −xPn′)−Pn]

=n(−nPn−Pn)=−n(n+1)Pn,

andsothePndefined by (18.15) do indeed satisfy Legendre’s equation.

The above example shows that the functionsPn(x) defined by (18.15) satisfy

Legendre’s equation with=n(an integer) and, also from (18.15), these functions

are regular atx=±1. ThusPnmust be some multiple of thenth Legendre

polynomial. It therefore remains only to verify the normalisation. This is easily

done atx=1,whenGbecomes

G(1,h)=[(1−h)^2 ]−^1 /^2 =1+h+h^2 +···,

and we can see that all thePnso defined havePn(1) = 1 as required, and are thus

identical to the Legendre polynomials.

A particular use of the generating function (18.15) is in representing the inverse

distance between two points in three-dimensional space in terms of Legendre

polynomials. If two pointsrandr′are at distancesrandr′, respectively, from

the origin, withr′<r,then

1

|r−r′|

=

1

(r^2 +r′^2 − 2 rr′cosθ)^1 /^2

=

1

r[1−2(r′/r)cosθ+(r′/r)^2 ]^1 /^2

=

1

r

∑∞

=0

(

r′

r

)

P(cosθ), (18.22)

whereθis the angle between the two position vectorsrandr′.Ifr′>r,however,