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Wikipedia / Legendre polynomials

In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications.

Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions.

Definition and representation

Definition by construction as an orthogonal system

In this approach, the polynomials are defined as an orthogonal system with respect to the weight function over the interval

given below.

. Finally, by defining them via orthogonality with respect to the Lebesgue measure, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, and the Hermite polynomials.

Definition via generating function

of the generating function

Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.

's without resorting to direct expansion of the Taylor series, however. Equation is differentiated with respect to Replacing the quotient of the square root with its definition in Eq., and equating the coefficients of powers of in the resulting expansion gives Bonnet's recursion formula This relation, along with the first two polynomials and , allows all the rest to be generated recursively.

The generating function approach is directly connected to the multipole expansion in electrostatics, as explained below, and is how the polynomials were first defined by Legendre in 1782.

Definition via differential equation

A third definition is in terms of solutions to Legendre's differential equation:

This differential equation has regular singular points at so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for in general. When is an integer, the solution that is regular at is also regular at , and the series for this solution terminates (i.e. it is a polynomial). The orthogonality and completeness of these solutions is best seen from the viewpoint of Sturm–Liouville theory.

, the differential operator on the left is Hermitian. The eigenvalues are found to be of the form . The orthogonality and completeness of this set of solutions follows at once from the larger framework of Sturm–Liouville theory.

The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind. A two-parameter generalization of (Eq.) is called Legendre's general differential equation, solved by the Associated Legendre polynomials. Legendre functions are solutions of Legendre's differential equation (generalized or not) with non-integer parameters.

In physical settings, Legendre's differential equation arises naturally whenever one solves Laplace's equation (and related partial differential equations) by separation of variables in spherical coordinates. From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics is the polar angle. This approach to the Legendre polynomials provides a deep connection to rotational symmetry. Many of their properties which are found laboriously through the methods of analysis — for example the addition theorem — are more easily found using the methods of symmetry and group theory, and acquire profound physical and geometrical meaning.

Rodrigues' formula and other explicit formulas

An especially compact expression for the Legendre polynomials is given by Rodrigues' formula

can also be calculated using the recurrences

is even.

stands for the largest integer less than or equal to. The fifth representation, which is also immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the generalized form of the binomial coefficient.

The reversal of the representation as a power series is

, where an empty product in the numerator (last factor less than the first factor) evaluates to 1.

The first few Legendre polynomials are:The graphs of these polynomials (up to ) are shown below:

Main properties

Orthogonality and normalization

fixes the normalization of the Legendre polynomials (with respect to the L2 norm on the interval where denotes the Kronecker delta.

Completeness

That the polynomials are complete means the following. Given any piecewise continuous function with finitely many discontinuities in the interval

with and .

Applications

Expanding an inverse distance potential

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre as the coefficients in the expansion of the Newtonian potential where and are the lengths of the vectors and respectively and is the angle between those two vectors. The series converges when . The expression gives the gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.

Legendre polynomials occur in the solution of Laplace's equation of the static potential, , in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where is the axis of symmetry and is the angle between the position of the observer and the

and are to be determined according to the boundary condition of each problem.

They also appear when solving the Schrödinger equation in three dimensions for a central force.

In multipole expansions

which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential (in spherical coordinates) due to a point charge located on the -axis at

If the radius of the observation point is greater than where we have defined and . This expansion is used to develop the normal multipole expansion.

Conversely, if the radius of the observation point is smaller than , the potential may still be expanded in the Legendre polynomials as above, but with and exchanged. This expansion is the basis of interior multipole expansion.

In trigonometry

The trigonometric functions , also denoted as the Chebyshev polynomials , can also be multipole expanded by the Legendre polynomials

, and where the products with the steps of two in the numerator and denominator are to be interpreted as 1 if they are empty, i.e., if the last factor is smaller than the first factor.

Another property is the expression for

In recurrent neural networks

A recurrent neural network that contains a , can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation

units of time is best approximated by a linear combination

When combined with deep learning methods, these networks can be trained to outperform long short-term memory units and related architectures, while using fewer computational resources.

Additional properties

Legendre polynomials have definite parity. That is, they are even or odd,

is used to approximate a function or experimental data: the average of the series over the interval .

The antiderivative is

The product expansion is

The Askey–Gasper inequality

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics where the unit vectors and have spherical coordinates and , respectively.

The product of two Legendre polynomials is the complete elliptic integral of the first kind.

The formulas of Dirichlet-Mehler:which has generalizations for associated Legendre polynomials.

The Fourier-Legendre series: is the Bessel function of the first kind.

Recurrence relations

where is the norm over the interval More generally, all orders of derivatives are expressible as a sum of Legendre polynomials:

Asymptotics

, the Legendre polynomials can be written as the Hilb's formula: and for arguments of magnitude greater than 1 where , , and are Bessel functions.

Zeros

. These zeros play an important role in numerical integration based on Gaussian quadrature's is known as Gauss-Legendre quadrature.

. This can be proved by looking at the first formula of Dirichlet-Mehler.

Pointwise evaluations

Variants with transformed argument

Shifted Legendre polynomials

The shifted Legendre polynomials Here the "shifting" function is an affine transformation that bijectively maps the interval to the interval , implying that the polynomials are orthogonal on

The analogue of Rodrigues' formula

The first few shifted Legendre polynomials are:

Legendre rational functions

The Legendre rational functions are a sequence of orthogonal functions on [0, ∞). They are obtained by composing the Cayley transform with Legendre polynomials.