Wolfram Language & System 10.3 (2015)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

LegendreP

LegendreP[n,x]
gives the Legendre polynomial .

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

DetailsDetails

• 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.

ExamplesExamplesopen allclose all

Basic Examples  (2)Basic Examples  (2)

Compute the Legendre polynomial:

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