The Legendre differential equation may be solved using the standard power series method. The equation has regular singular points at x = ±1 so, in general, a series solution about the origin will only converge for |x| < 1. 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).
These solutions for n = 0, 1, 2, ... (with the normalization Pn(1) = 1) form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula:
That these polynomials satisfy the Legendre differential equation (Eq. 1) follows by differentiating n + 1 times both sides of the identity
Expanding the Taylor series in Eq. 2 for the first two terms gives
for the first two Legendre polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, Eq. 2 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. 2, 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 the Legendre polynomials to be generated recursively.
Explicit representations include
where the latter, which is immediate from the recursion formula, expresses the Legendre polynomials by simple monomials and involves the multiplicative formula of the binomial coefficient.
The first few Legendre polynomials are:
The graphs of these polynomials (up to n = 5) are shown below:
Orthogonality
An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 norm on the interval −1 ≤ x ≤ 1:
(where δmn denotes the Kronecker delta, equal to 1 if m = n and to 0 otherwise).
In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram–Schmidt process on the polynomials {1, x, x2, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are eigenfunctions of a Hermitiandifferential operator:
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, ∇2Φ(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.[4]
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/r < 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 interior multipole expansion.
Legendre polynomials in trigonometry
The trigonometric functions cos nθ, also denoted as the Chebyshev polynomialsTn(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
Since the differential equation and the orthogonality property are independent of scaling, the Legendre polynomials' definitions are "standardized" (sometimes called "normalization", but note that the actual norm is not 1) by being scaled so that
The derivative at the end point is given by
As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula
and
Useful for the integration of Legendre polynomials is
From the above one can see also that
or equivalently
where ‖Pn‖ is the norm over the interval −1 ≤ x ≤ 1
From Bonnet’s recursion formula one obtains by induction the explicit representation
Here the "shifting" function x ↦ 2x − 1 (in fact, it is an affine transformation) is chosen such that it bijectively maps the interval [0,1] to the interval [−1,1], implying that the polynomials P̃n(x) are orthogonal on [0,1]:
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 functions of the second kind (Qn)
As well as polynomial solutions, the Legendre equation has non-polynomial solutions represented by infinite series. These are the Legendre functions of the second kind, denoted by Qn(x).
The differential equation
has the general solution
,
where A and B are constants.
Legendre functions of fractional degree
Legendre functions of fractional degree exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into Rodrigues' formula. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional degree Legendre function Pn agrees with the associated Legendre polynomialP0 n.