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 . That is, is a polynomial of degree , such that
With the additional standardization condition , all the polynomials can be uniquely determined. We then start the construction process: is the only correctly standardized polynomial of degree 0. must be orthogonal to , leading to , and is determined by demanding orthogonality to and , and so on. is fixed by demanding orthogonality to all with . This gives conditions, which, along with the standardization fixes all coefficients in . With work, all the coefficients of every polynomial can be systematically determined, leading to the explicit representation in powers of given below.
This definition of the '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, . Finally, by defining them via orthogonality with respect to the Lebesgue measure on , 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 with the weight , and the Hermite polynomials, orthogonal over the full line with weight .
Definition via generating function The Legendre polynomials can also be defined as the coefficients in a formal expansion in powers of of the generating function
The coefficient of is a polynomial in of degree with . Expanding up to gives
Expansion to higher orders gets increasingly cumbersome, but is possible to do systematically, and again leads to one of the explicit forms given below.
It is possible to obtain the higher 's without resorting to direct expansion of the Taylor series, however. Equation is differentiated with respect to on both sides and rearranged to obtain
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. We rewrite the differential equation as an eigenvalue problem,
with the eigenvalue in lieu of . This is a Sturm–Liouville equation with .
If we demand that the solution be regular at , the differential operator on the left is Hermitian. The eigenvalues are found to be of the form , with and the eigenfunctions are the . 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, 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 where 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:
This formula enables derivation of a large number of properties of the 's. Among these are explicit representations such as
Expressing the polynomial as a power series, , the coefficients of powers of can also be calculated using the recurrences
or
The Legendre polynomial is determined by the values used for the two constants and , where if is odd and if is even.
In the fourth representation, 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
for , where an empty product in the numerator (last factor less than the first factor) evaluates to 1.
The first few Legendre polynomials are: 0 1 2 3 4 5 6 7 8 9 10
The graphs of these polynomials (up to ) are shown below:
Main properties
Orthogonality and normalization
The standardization fixes the normalization of the Legendre polynomials (with respect to the norm on the interval ). Rodrigues' formula may be employed to give the normalization integral
The statements of normalization and orthogonality can then be compactly written in a single equation:
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 , the sequence of sums
converges in the mean to as , provided we take
This completeness property underlies all the expansions discussed in this article, and is often stated in the form
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 axis (the zenith angle), the solution for the potential will be
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
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
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 (see diagram right) varies as
If the radius of the observation point is greater than , the potential may be expanded in the Legendre polynomials
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 . The first several orders are as follows:
This can be summarized for as
where , , 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 , which is
In recurrent neural networks
A recurrent neural network that contains a -dimensional memory vector, , 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 across the past units of time is best approximated by a linear combination of the first shifted Legendre polynomials, weighted together by the elements of at time :
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, according to
Another useful property is
which follows from considering the orthogonality relation with . It is convenient when a Legendre series is used to approximate a function or experimental data: the average of the series over the interval is simply given by the leading expansion coefficient .
The antiderivative is
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 product expansion is
where .
The Askey–Gasper inequality for Legendre polynomials reads
The Legendre polynomials of a scalar product of unit vectors can be expanded with spherical harmonics using
where the unit vectors and have spherical coordinates and , respectively.
The product of two Legendre polynomials
where 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:Lord Rayleigh, Theory of sound, Volume II, p. 273where is the Bessel function of the first kind.
Recurrence relations As discussed above, the Legendre polynomials obey the three-term recurrence relation known as Bonnet's recursion formula given by
and
or, with the alternative expression, which also holds at the endpoints
Useful for the integration of Legendre polynomials is