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.
Examplesopen allclose all
Basic Examples (6)
Asymptotic expansion at Infinity:
Numerical Evaluation (6)
Evaluate LegendreP efficiently at high precision:
LegendreP can deal with real-valued intervals:
LegendreP threads elementwise over lists:
Specific Values (5)
Different LegendreP types give different symbolic forms:
Plot the LegendreP function for various orders:
Function Properties (5)
LegendreP has the mirror property :
Indefinite integral of LegendreP:
Series Expansions (4)
LegendreP can be applied to a power series:
Integral Transforms (4)
The Fourier transform of a Legendre polynomial with order using FourierTransform:
The Laplace transform of a Legendre polynomial with order using LaplaceTransform:
The Mellin transform of a Legendre polynomial with order using MellinTransform:
The Hankel transform of a Legendre polynomial with order using HankelTransform:
Function Identities and Simplifications (4)
LegendreP may reduce to simpler functions: