LegendreP[n, x] gives the Legendre polynomial .
LegendreP[n, m, x] gives the associated Legendre polynomial .
Mathematical function (see Section A.3.10).
Explicit formulas are given for integer n and m.
The Legendre polynomials satisfy the differential equation .
The Legendre polynomials are orthogonal with unit weight function.
The associated Legendre polynomials are defined by .
For arbitrary complex values of n, m and z, LegendreP[n, z] and LegendreP[n, m, z] give Legendre functions of the first kind.
LegendreP[n, m, a, z] gives Legendre functions of type a. The default is type 1.
The symbolic form of type 1 involves , of type 2 involves and of type 3 involves .
Type 1 is defined only for within the unit circle in the complex plane. Type 2 represents an analytic continuation of type 1 outside the unit circle.
Type 2 functions have branch cuts from to and from to in the complex plane.
Type 3 functions have a single branch cut from to .
LegendreP[n, m, a, z] is defined to be Hypergeometric2F1Regularized[-n,n+1,1-m,(1-z)/2] multiplied by for type 2 and by for type 3.
See The Mathematica Book: Section 3.2.9 and Section 3.2.10.
See also: SphericalHarmonicY.