Legendre polynomials

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

With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of

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

2, and equating the coefficients of powers of t in the resulting expansion gives Bonnet’s recursion formula

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.

A third definition is in terms of solutions to Legendre's differential equation: This differential equation has regular singular points at x = ±1 so if a solution is sought using the standard Frobenius or power series method, a series about the origin will only converge for |x| < 1 in general.

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

From this standpoint, the eigenfunctions of the angular part of the Laplacian operator are the spherical harmonics, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis.

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.

The last 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.

fixes the normalization of the Legendre polynomials (with respect to the L2 norm on the interval −1 ≤ x ≤ 1).

Since they are also orthogonal with respect to the same norm, the two statements[clarification needed] can be combined into the single equation,

This completeness property underlies all the expansions discussed in this article, and is often stated in the form

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, ∇2 Φ(x) = 0, 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).

Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):

The left-hand side of the equation is the generating function for the Legendre polynomials.

As an example, the electric potential Φ(r,θ) (in spherical coordinates) due to a point charge located on the z-axis at z = a (see diagram right) varies as

If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials

Conversely, if the radius r of the observation point P is smaller than a, the potential may still be expanded in the Legendre polynomials as above, but with a and r exchanged.

, can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation:

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.

is used to approximate a function or experimental data: the average of the series over the interval [−1, 1] is simply given by the leading expansion coefficient

The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using

where the unit vectors r and r′ have spherical coordinates (θ, φ) and (θ′, φ′), respectively.

As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by

These zeros play an important role in numerical integration based on Gaussian quadrature.