LegendreP
 Usage
• LegendreP[n, x] gives the Legendre polynomial  .
• LegendreP[n, m, x] gives the associated Legendre polynomial  .
Notes
• 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. • New in Version 1; modified in 5.
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