18.1 LEGENDRE FUNCTIONS
Q 0
Q 1
Q 2
x
− 1− 1
− 0. 5
− 0. 5
0. 5
0. 5
1
1
Figure 18.2 The first three Legendre functions of the second kind.the coefficientc 2 of the functionQ(x) in (18.7) to be zero, sinceQ(x) is singular
atx=±1, with the result that the general solution is simply some multiple of the
relevant Legendre polynomialP(x). In this section we will study the properties
of the Legendre polynomialsP(x) in some detail.
Rodrigues’ formulaAs an aid to establishing further properties of the Legendre polynomials we now
develop Rodrigues’ representation of these functions. Rodrigues’ formula for the
P(x)is
P(x)=1
2 !d
dx(x^2 −1). (18.9)To prove that this is a representation we letu=(x^2 −1),sothatu′=2x(x^2 −1)−^1
and
(x^2 −1)u′− 2 xu=0.If we differentiate this expression+ 1 times using Leibnitz’ theorem, we obtain
[
(x^2 −1)u(+2)+2x(+1)u(+1)+(+1)u()]
− 2 [
xu(+1)+(+1)u()]
=0,