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Legendre polynomials

Inmathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a vast 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 by construction as an orthogonal system[edit]

In this approach, the polynomials are defined as an orthogonal system with respect to the weight function ${\displaystyle w($x)=1} over the interval ${\displaystyle [-1,1]}$. That is, ${\displaystyle P_{n}($x)} is a polynomial of degree ${\displaystyle n}$, such that

With the additional standardization condition ${\displaystyle P_{n}($1)=1} , all the polynomials can be uniquely determined. We then start the construction process: ${\displaystyle P_{0}($x)=1} is the only correctly standardized polynomial of degree 0. ${\displaystyle P_{1}($x)} must be orthogonal to ${\displaystyle P_{0}}$, leading to ${\displaystyle P_{1}($x)=x} , and ${\displaystyle P_{2}($x)} is determined by demanding orthogonality to ${\displaystyle P_{0}}$ and ${\displaystyle P_{1}}$, and so on. ${\displaystyle P_{n}}$ is fixed by demanding orthogonality to all ${\displaystyle P_{m}}$ with ${\displaystyle m<n}$. This gives ${\displaystyle n}$ conditions, which, along with the standardization ${\displaystyle P_{n}($1)=1} fixes all ${\displaystyle n+1}$ coefficients in ${\displaystyle P_{n}($x)} . With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of ${\displaystyle x}$ given below.

This definition of the ${\displaystyle P_{n}}$'s is the simplest one. It does not appeal to the theory of differential equations. Second, the completeness of the polynomials follows immediately from the completeness of the powers 1, ${\displaystyle x,x^{2},x^{3},\ldots }$. Finally, by defining them via orthogonality with respect to the most obvious weight function on a finite interval, it sets up the Legendre polynomials as one of the three classical orthogonal polynomial systems. The other two are the Laguerre polynomials, which are orthogonal over the half line ${\displaystyle [0,\infty )}$, and the Hermite polynomials, orthogonal over the full line ${\displaystyle (-\infty ,\infty )}$, with weight functions that are the most natural analytic functions that ensure convergence of all integrals.

Definition via generating function[edit]

The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of ${\displaystyle t}$ of the generating function^[1]

$${\displaystyle {\frac {1}{\sqrt {1-2xt+t^{2}}}}=\sum _{n=0}^{\infty }P_{n}($$x)t^{n}\,.}

(2)

The coefficient of ${\displaystyle t^{n}}$ is a polynomial in ${\displaystyle x}$ of degree ${\displaystyle n}$ with ${\displaystyle |x|\leq 1}$. Expanding up to ${\displaystyle t^{1}}$ gives

It is possible to obtain the higher ${\displaystyle P_{n}}$'s without resorting to direct expansion of the Taylor series, however. Equation 2 is differentiated with respect to t on both sides and rearranged to obtain

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[edit]

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

$${\displaystyle (1-x^{2})P_{n}''($$x)-2xP_{n}'(x)+n(n+1)P_{n}(x)=0.}

(1)

This differential equation has regular singular pointsatx = ±1 so if a solution is sought using the standard Frobeniusorpower series method, a series about the origin will only converge for |x| <1 in general. When n is an integer, the solution P_n(x) that is regular at x = 1 is also regular at x = −1, 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. We rewrite the differential equation as an eigenvalue problem,

${\displaystyle \lambda }$

${\displaystyle n(n+1)}$

${\displaystyle x=\pm 1}$

${\displaystyle n=0,1,2,\ldots }$

${\displaystyle P_{n}($x)}

The differential equation admits another, non-polynomial solution, the Legendre functions of the second kind ${\displaystyle Q_{n}}$. A two-parameter generalization of (Eq. 1) 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, of which the Legendre polynomials are (up to a multiplicative constant) the subset that is left invariant by rotations about the polar axis. The polynomials appear as ${\displaystyle P_{n}(\cos \theta )}$ where ${\displaystyle \theta }$ 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.

Orthogonality and completeness[edit]

The standardization ${\displaystyle P_{n}($1)=1} 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,

That the polynomials are complete means the following. Given any piecewise continuous function ${\displaystyle f($x)} with finitely many discontinuities in the interval [−1, 1], the sequence of sums

${\displaystyle f($x)}

${\displaystyle n\to \infty }$

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

Rodrigues' formula and other explicit formulas[edit]

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

This formula enables derivation of a large number of properties of the ${\displaystyle P_{n}}$'s. Among these are explicit representations such as

${\textstyle P_{n}($x)=\sum a_{k}x^{k}}

${\displaystyle x}$

${\textstyle a_{0}}$

${\textstyle a_{1}}$

${\textstyle a_{0}=0}$

${\displaystyle n}$

${\textstyle a_{1}=0}$

${\displaystyle n}$

In the fourth representation, ${\displaystyle \lfloor n/2\rfloor }$ stands for the largest integer less than or equal to ${\displaystyle n/2}$. 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.

The first few Legendre polynomials are:

${\displaystyle n}$	${\displaystyle P_{n}($x)}
0	${\textstyle 1}$
1	${\textstyle x}$
2	${\textstyle {\tfrac {1}{2}}\left(3x^{2}-1\right)}$
3	${\textstyle {\tfrac {1}{2}}\left(5x^{3}-3x\right)}$
4	${\textstyle {\tfrac {1}{8}}\left(35x^{4}-30x^{2}+3\right)}$
5	${\textstyle {\tfrac {1}{8}}\left(63x^{5}-70x^{3}+15x\right)}$
6	${\textstyle {\tfrac {1}{16}}\left(231x^{6}-315x^{4}+105x^{2}-5\right)}$
7	${\textstyle {\tfrac {1}{16}}\left(429x^{7}-693x^{5}+315x^{3}-35x\right)}$
8	${\textstyle {\tfrac {1}{128}}\left(6435x^{8}-12012x^{6}+6930x^{4}-1260x^{2}+35\right)}$
9	${\textstyle {\tfrac {1}{128}}\left(12155x^{9}-25740x^{7}+18018x^{5}-4620x^{3}+315x\right)}$
10	${\textstyle {\tfrac {1}{256}}\left(46189x^{10}-109395x^{8}+90090x^{6}-30030x^{4}+3465x^{2}-63\right)}$

The graphs of these polynomials (up to n = 5) are shown below:

Applications of Legendre polynomials[edit]

Expanding a 1/r potential[edit]

The Legendre polynomials were first introduced in 1782 by Adrien-Marie Legendre^[3] as the coefficients in the expansion of the Newtonian potential

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). Where ẑ is the axis of symmetry and θ is the angle between the position of the observer and the ẑ axis (the zenith angle), the solution for the potential will be

A_l and B_l are to be determined according to the boundary condition of each problem.^[4]

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

Legendre polynomials in multipole expansions[edit]

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

As an example, the electric potential Φ(r,θ) (inspherical 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. This expansion is the basis of interior multipole expansion.

Legendre polynomials in trigonometry[edit]

The trigonometric functions cos nθ, also denoted as the Chebyshev polynomials T_n(cos θ) ≡ cos nθ, can also be multipole expanded by the Legendre polynomials P_n(cos θ). The first several orders are as follows:

Another property is the expression for sin (n + 1)θ, which is

Legendre polynomials in recurrent neural networks[edit]

Arecurrent neural network that contains a d-dimensional memory vector, ${\displaystyle \mathbf {m} \in \mathbb {R} ^{d}}$, can be optimized such that its neural activities obey the linear time-invariant system given by the following state-space representation:

In this case, the sliding window of ${\displaystyle u}$ across the past ${\displaystyle \theta }$ units of time is best approximated by a linear combination of the first ${\displaystyle d}$ shifted Legendre polynomials, weighted together by the elements of ${\displaystyle \mathbf {m} }$ at time ${\displaystyle t}$:

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.^[5]

Additional properties of Legendre polynomials[edit]

Legendre polynomials have definite parity. That is, they are even or odd,^[6] according to

Another useful property is

${\displaystyle P_{0}($x)=1}

${\textstyle \sum _{i}a_{i}P_{i}}$

${\displaystyle a_{0}}$

Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but the actual norm is not 1) by being scaled so that

The derivative at the end point is given by

The Askey–Gasper inequality for Legendre polynomials reads

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

The product of two Legendre polynomials ^[7]

${\displaystyle K(\cdot )}$

Recurrence relations[edit]

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

Useful for the integration of Legendre polynomials is