18.1 LEGENDRE FUNCTIONS
P 0
P 1
P 2
P 3
− 1
− 1
− 0. 5 0.^5
1
1
x− 2
2
Figure 18.1 The first four Legendre polynomials.The first four Legendre polynomials are plotted in figure 18.1.
Although, according to whetheris an even or odd integer, respectively, eithery 1 (x) in (18.3) ory 2 (x) in (18.4) terminates to give a multiple of the corresponding
Legendre polynomialP(x), the other series in each case does not terminate and
therefore converges only for|x|<1. According to whetheris even or odd, we
defineLegendre functions of the second kindasQ(x)=αy 2 (x)orQ(x)=βy 1 (x),
respectively, where the constantsαandβare conventionally taken to have the
values
α=(−1)/^22 [(/2)!]^2
!foreven, (18.5)β=(−1)(+1)/^22 −^1 {[(−1)/2]!}^2
!forodd. (18.6)These normalisation factors are chosen so that theQ(x) obey the same recurrence
relations as theP(x) (see subsection 18.1.2).
The general solution of Legendre’s equation forintegeris thereforey(x)=c 1 P(x)+c 2 Q(x), (18.7)