gives the Legendre polynomial .


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.


open all close all

Basic Examples  (2)

Compute the 10^(th) Legendre polynomial:

Click for copyable input
Click for copyable input

Scope  (42)

Generalizations & Extensions  (3)

Applications  (3)

Properties & Relations  (4)

Possible Issues  (1)

Neat Examples  (2)

Introduced in 1988
Updated in 2003