SpecialFuncs.LegendreP Method

Legendre polynomial P(l,x) = P(l,0,x) (order m = 0).

Computes the ordinary Legendre polynomial of degree l, i.e. the associated Legendre function at order m = 0:

P_l(x)=1/(2^l l!)d^l/dx^l(x^2-1)^l, P_0=1, P_1=x, (l+1)P_(l+1)=(2l+1)x P_l-l P_(l-1).

Domain: integer l\ge 0 and real x on the closed interval [-1,1]. Defined behavior: returns NaN when |x|>1. Known values: P_l(1)=1 and P_l(-1)=(-1)^l.

Vector of Legendre polynomials P(0,x), P(1,x), ..., P(n,x) (order m = 0).

Fills result with the sequence of ordinary Legendre polynomials evaluated at x:

result[l]=P_l(x), l=0,1,...,n,

using the three-term recurrence (l+1)P_{l+1}=(2l+1)x\,P_l-l\,P_{l-1} with P_0=1, P_1=x. The result operand is auto-sized to length n+1 and its precision matches the operand (single or double). Domain: integer n\ge 0 and real x on the closed interval [-1,1]. Defined behavior: an exception is raised if result is complex or if n<0; for |x|>1 each element follows the scalar contract (NaN).

Associated Legendre function P(l,m,x) (Condon-Shortley phase).

Computes the associated Legendre function of degree l and order m, including the Condon-Shortley phase factor (-1)^m, by the standard upward recurrence in l. With P_l^m derived from the Legendre polynomial P_l by

P_l^m(x)=(-1)^m (1-x^2)^(m/2) d^m/dx^mP_l(x), m=0,1,...,l,

the function uses P_m^m(x)=(-1)^m(2m-1)!!\,(1-x^2)^{m/2}, P_{m+1}^m(x)=x\,(2m+1)\,P_m^m(x) and the recurrence (l-m)\,P_l^m=x(2l-1)P_{l-1}^m-(l+m-1)P_{l-2}^m.
Domain: integer l\ge 0, integer m on [0,l], and real x on the closed interval [-1,1]. Defined behavior on invalid input: if m<0, m>l, or |x|>1 the function returns NaN. The order-zero case P(l,0,x) equals the ordinary Legendre
polynomial P_l(x).