LegendreP

LegendreP[n,x]

gives the Legendre polynomial .

LegendreP[n,m,x]

gives the associated Legendre polynomial .

Details

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 $TemplateBox[{n, m, x}, LegendreP3]=(-1)^m(1-x^2)^(m/2)(d^m/dx^m)TemplateBox[{n, x}, LegendreP]$ .
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.
LegendreP can be used with Interval and CenteredInterval objects. »

Examples

open allclose all

Basic Examples (6)

Evaluate numerically:

Compute the Legendre polynomial:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Asymptotic expansion at Infinity:

Scope (50)

Numerical Evaluation (7)

Evaluate numerically at fixed points:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Evaluate for complex orders and arguments:

Evaluate LegendreP efficiently at high precision:

LegendreP can deal with real-valued intervals:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix LegendreP function using MatrixFunction:

Specific Values (5)

Legendre polynomial for symbolic :

Find a local maximum as a root of :

Compute the associated Legendre polynomial :

Compute an associated Legendre polynomial for half-integer and :

Different LegendreP types give different symbolic forms:

Visualization (3)

Plot the LegendreP function for various orders:

Plot the real part of :

Plot the imaginary part of :

Type 2 and 3 of Legendre functions have different branch cut structures:

Function Properties (12)

is defined for all for integer and for for noninteger :

In the complex plane, it is defined for when is not an integer:

The associated Legendre function is additionally undefined at when is not an even integer:

The range for Legendre polynomials of integer orders:

The range for complex values is the whole plane:

Legendre polynomial of an odd order is odd:

Legendre polynomial of an even order is even:

Legendre polynomials have the mirror property :

is an analytic function of for integer :

It is neither analytic nor meromorphic for noninteger :

The associated Legendre function is analytic as long as is also an even integer:

is neither non-decreasing nor non-increasing for integers :

is surjective for positive odd integer values of but not even values:

LegendreP is neither non-negative nor non-positive:

has no singularities or discontinuities when is an integer:

The associated Legendre function has additional singularities when is not an even integer:

is neither non-decreasing nor non-increasing for integers :

Differentiation (3)

First derivative:

Higher derivatives:

Plot higher derivatives for :

Formula for the derivative:

Integration (3)

Indefinite integral of LegendreP:

Indefinite integral of an algebraic function with :

Definite integral of :

Series Expansions (4)

Taylor expansion for :

Plot the first three approximations for at :

General term in the series expansion of :

Taylor expansion for the associated Legendre polynomial :

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:

Associated Legendre polynomials in terms of ordinary Legendre polynomials:

Sum of Legendre polynomials:

Recurrence relation:

Function Representations (5)

Representation in terms of MeijerG:

LegendreP can be expressed as a DifferentialRoot:

SphericalHarmonicY uses associated Legendre function in its definition:

Associated Legendre polynomials in terms of the angular spheroidal function:

TraditionalForm formatting:

Generalizations & Extensions (3)

LegendreP can deal with real-valued intervals:

Different LegendreP types give different symbolic forms:

Types 2 and 3 have different branch cut structures:

Applications (5)

Angular momentum eigenfunctions:

The Pöschl–Teller potential is a special class of potentials for which the one-dimensional Schrödinger equation can be solved in terms of special functions.

Find quantum eigenfunctions for the modified Pöschl–Teller potential:

A -analog of the Legendre polynomial can be defined in terms of QHypergeometricPFQ:

Recover the Legendre polynomial as :

Generalized Fourier transform for functions on the interval -1 to 1:

An n-point Gaussian quadrature rule is based on the roots of the n-order Legendre polynomial. Compute the nodes and weights of an n-point Gaussian quadrature rule:

Use the n-point Gaussian quadrature rule to numerically evaluate an integral:

Compare the result of the Gaussian quadrature with the result from NIntegrate:

Properties & Relations (4)

Use FunctionExpand to expand into simpler functions:

LegendreP can be expressed as a DifferenceRoot:

The generating function for LegendreP:

The exponential generating function for LegendreP:

Possible Issues (1)

Cancellations in the polynomial form may lead to inaccurate numerical results:

Evaluate the function directly:

Neat Examples (3)

Visualize distribution of zeros:

Generalized Lissajous figures:

An expression for the Legendre polynomial in terms of the Hilbert matrix:

Verify the expression for the first few cases:

Top