This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
 LegendreP 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, Section 3.2.10. See also: SphericalHarmonicY. Further Examples Here are the first 4 Legendre polynomials. In[1]:= Out[1]//TableForm= Here are the first four associated Legendre polynomials. In[2]:= Out[2]//TableForm= This is the derivative. In[3]:= Out[3]= This is the integral. In[4]:= Out[4]= This is a series expansion around . In[5]:= Out[5]=