LegendreP[n, x]
gives the Legendre polynomial .

LegendreP[n, m, x]
gives the associated Legendre polynomial .


  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • Explicit formulas are given for integers 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.
  • For certain special arguments, LegendreP automatically evaluates to exact values.
  • LegendreP can be evaluated to arbitrary numerical precision.
  • LegendreP automatically threads over lists.
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