ABSTRACT
If n = 0, 1, 2, . . . , a solution of this equation is the Legendre polynomials Pn(x) given by Rodrigue’s formula
Pn(x) = 1
2n n! dn
dxn ( x2 − 1). (C.2)
C.1.1. Special Legendre polynomials are: P0(x) = 1, P4(x) = 18
( 35x4 − 30x2 + 3) ,
P1(x) = x, P5(x) = 18 ( 63x5 − 70x3 + 15x) ,
P2(x) = 12 ( 3x2 − 1) , P6(x) = 116
( 231x6 − 315x4 + 105x2 − 5) ,
P3(x) = 12 ( 5x3 − 3x2) , P7(x) = 116
( 429x7 − 693x5 + 315x3 − 35x).
C.1.2. If we set x = cos θ, then some of the above Legendre polynomials become P0(cos θ) = 1,
