In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 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.
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
This determines the polynomials completely up to an overall scale factor, which is fixed by the standardization . That this is a constructive definition is seen thus: 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 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 , and the Hermite polynomials, orthogonal over the full line , with weight functions that are the most natural analytic functions that ensure convergence of all integrals.
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 . 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. Eq. is differentiated with respect to t 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 t in the resulting expansion gives Bonnet’s recursion formula
This relation, along with the first two polynomials P0 and P1, 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 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. When n is an integer, the solution Pn(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,
with the eigenvalue in lieu of . 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 n(n + 1), 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.
Orthonormality and completeness
The standardization 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 can be combined into the single equation,
That the polynomials are complete means the following. Given any piecewise continuous function with finitely many discontinuities in the interval [−1,1], 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 −1 ≤ x ≤ 1 and −1 ≤ y ≤ 1.
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
where the last, 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:
The graphs of these polynomials (up to n = 5) are shown below:
Applications of Legendre polynomials
Expanding a 1/r potential
where r and r′ are the lengths of the vectors x and x′ respectively and γ is the angle between those two vectors. The series converges when r > r′. 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, ∇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
Al and Bl 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.
Legendre polynomials in multipole expansions
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
If the radius r of the observation point P is greater than a, the potential may be expanded in the Legendre polynomials
where we have defined η = a/ < 1 and x = cos θ. This expansion is used to develop the normal multipole expansion.
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 .
Legendre polynomials in trigonometry
The trigonometric functions cos nθ, also denoted as the Chebyshev polynomials Tn(cos θ) ≡ cos nθ, can also be multipole expanded by the Legendre polynomials Pn(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
A recurrent neural network that contains a d-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 :
Additional properties of Legendre polynomials
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 [−1, 1] is simply given by the leading expansion coefficient .
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
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
or, with the alternative expression, which also holds at the endpoints
Useful for the integration of Legendre polynomials is
From the above one can see also that
where ||Pn|| is the norm over the interval −1 ≤ x ≤ 1
and for arguments of magnitude greater than 1
where J0 and I0 are Bessel functions.
All zeros of are real, distinct from each other, and lie in the interval . Further, if we regard them as dividing the interval into subintervals, each subinterval will contain exactly one zero of . This is known as the interlacing property. Because of the parity property it is evident that if is a zero of , so is . These zeros play an important role in numerical integration based on Gaussian quadrature. The specific quadrature based on the 's is known as Gauss-Legendre quadrature.
From this property and the facts that , it follows that has local minima and maxima in . Equivalently, has zeros in .
The parity and normalization implicate the values at the boundaries to be
At the origin one can show that the values are given by
Legendre polynomials with transformed argument
Shifted Legendre polynomials
The shifted Legendre polynomials are defined as
An explicit expression for the shifted Legendre polynomials is given by
The analogue of Rodrigues' formula for the shifted Legendre polynomials is
The first few shifted Legendre polynomials are:
Legendre rational functions
A rational Legendre function of degree n is defined as:
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