Named in honor of Adrien-Marie (1752-1833) the mathematician, not Louis (1752-1797) the politician
A polynomial is a finite sum of terms like akxk, where k is a positive integer or zero. There are sets of polynomials such that the product of any two different ones, multiplied by a function w(x) called a weight function and integrated over a certain interval, vanishes. Such a set is called a set of orthogonal polynomials. Among other things, this property makes it possible to expand an arbitrary function f(x) as a sum of the polynomials, each multiplied by a coefficient c(k), which is easily and uniquely determined by integration. A Fourier series is similar, but the orthogonal functions are not polynomials. These functions can also be used to specify basis states in quantum mechanics, which must be orthogonal.
The Legendre polynomials Pn(x), n = 0, 1, 2 ... are orthogonal on the interval from -1 to +1, which is expressed by the integral . The Kronecker delta is zero if n ≠ m, and unity if n = m. In most applications, x = cos θ, and θ varies from 0 to π. In this case, dx = sin θ dθ, of course. The Legendre polynomials are a special case of the more general Jacobi polynomials P(α,β)n(x) orthogonal on (-1,1). By a suitable change of variable, the range can be changed from (-1,1) to an arbitrary (a,b). The weight function w(x) of the Legendre polynomials is unity, and this is what distinguishes them from the others and determines them.
The Lengendre polynomials are very clearly motivated by a problem that often appears. For example, suppose we have an electric charge q at point Q in the figure at the left, one of a group whose positions are referred to an origin at O, and we desire the potential at some point P. The distance PO is taken as unity for convenience; simply multiply all distances by the actual distance PO in any particular case. The potential due to this charge is q/R. We can find R as a function of r and θ by the Law of Cosines: R2 = 1 + r2 - 2r cos θ = 1 - 2rx + r2, where x = cos θ. Now we expand 1/R in powers of r, finding 1/R = Σ Pn(x)rn. The function 1/R is called the generating function of the Legendre polynomials, and can be used to investigate their properties. Generating functions are available for most orthogonal polynomials, but only in the Legendre case does the generating function have a clear and simple meaning.
If we let x = 1, we find that Pn(1) = 1, and Pn(-1) = (-1)n. By taking partial derivatives of 1/R with respect to x and r, and then considering the coefficients of individual powers of r, we can find a number of relations between the polynomials and their derivatives. These can be manipulated to find the recursion relation, (n + 1) Pn+1(x) = (2n + 1)x Pn(x) - n Pn-1(x), and the differential equation satisfied by the polynomials, (1 - x2) P"n(x) -2x P'n(x) + n(n + 1) Pn(x) = 0. The recurrence relation allows us to find all the polynomials, since it is easy to find that P0(x) = 1, P1(x) = x directly from the generating function, and this starts us off. The differential equation allows us to apply the polynomials to problems arising in mathematics and physics, among which is the important problem of the solution of Laplace's equation and spherical harmonics.
The recurrence relation shows that the coefficient An of the highest power of x satisfies the relation An+1 = (2k + 1)/(k + 1) An, and so from the known coefficients for n = 0, 1 we can find that the coefficient of the highest power of x in Pn is 1.3.5...(2n-1)/n!.
The polynomials can also be found by solving the differential equation by determining the coefficients of a power series substituted in the equation. This method was often used in quantum mechanics texts (see Reference 3), since the students were not usually acquainted with the mathematics of orthogonal polynomials. This method does not allow one to investigate the properties of the polynomials in any detail, however, yielding only the individual polynomials themselves.
Consider the polynomials Gn(x) = dn/dxn (x2 - 1)n. The quantity to be differentiated is indeed a polynomial, of degree 2n, and consisting of only even powers. When differentiated n times, it becomes a polynomial of order n consisting of either all odd or all even powers of x, as n is odd or even. The coefficient of the highest power of x is 2n(2n-1)(2n-2)...(n+1), and the first two polynomials are 1 and 2x. If G(x) is substituted in the recurrence relation for the Legendre polynomials, it is found to satisfy it. If we divide G(x) by the constant 2nn!, then the first two polynomials are 1 and x. Therefore, Pn(x) = (1/2nn!) dn/dxn (x2 - 1)n. This is called Rodrigues's formula; similar formulas exist for other orthogonal polynomials.
The great advantage of Rodrigues' formula is its form as an nth derivative. This means that in an integral, it can be used repeatedly in an integration by parts to evaluate the integral. The orthogonality of the Legendre polynomials follows very quickly when Rodrigues' formula is used. There is a Rodrigues' formula for many, but not all, orthogonal polynomials. It can be used to find the recurrence relation, the differential equation, and many other properties.
For finding solutions to Laplace's equation in spherical coordinates, the Legendre polynomials are sufficient so long as the problem is axially symmetric, in which there is no φ-dependence. The more general problem requires the introduction of related functions called the associated Legendre functions that are actually built up from Jacobi polynomials, and can also be expressed in terms of derivatives of the Legendre polynomials. Physics texts generally approached the problem from first principles, never mentioning Jacobi polynomials, and thereby losing valuable insight.
The Jacobi polynomials P(α,β)n(x) are orthogonal on (-1,1) with weight function w(x) = (1 - x)α(1 + x)β. Their Rodrigues' formula is P(α,β)n(x) = [(-1)n/2nn!] (1 - x)-α(1 + x)-β dn/dxn (1 - x)α+n(1 + x)β+n. The ordinary Legendre polynomial Pn(x) = P(0,0)n(x). They satisfy the differential equation (1 - x2)P"(α,β)n + [β - α - (α + β + 2)x] P'(α,β)n + n(α + β + n + 1) P(α,β)n = 0.
In solving Laplace's equation by the method of separation of variables, one obtains for the θ dependence T(x), x = cos θ, the differential equation
d/dx[(1 - x2)dT/dx] = [l(l+1) - m2/(1 - x2]T = 0
The substitution T(x) = (1 - x2)m/2y(x) now gives the equation
(1-x2)y" - 2(m + 1)xy' + [l(l+1) - m(m+1)]y = 0,
which we recognize as satisfied by the Jacobi polynomial P(m,m)l-m(x). Hence, T(x) = (1 - x2)m/2y(x) P(m,m)l-m(x). This is the associated Legendre function, often denoted Pml(x) in physics texts (e.g., Reference 4), and defined there as (-1)m(1 - x2)m/2 dm/dxm Pl. The subscript is no longer the degree of the polynomial.
All the above is for a positive m. Since the equation contains m2, the solution for negative m is essentially the same, except perhaps for a multiplicative factor. This is of little consequence for the traditional applications of spherical harmonics, but is critical for quantum mechanics, where relative phases matter. The choice in physics is that P-ml(x) = (-1)m[(l - m)!/(l + m)!] Pml(x), where m is always positive on the right. If you work the functions out explicitly, you will find that the functions for +m and -m are essentially the same, as might be expected, and differ at most by a factor of -1.
For the same m, Pml(x) and Pml'(x) are orthogonal, and the integral of the square of Pml(x) is the same as for Pl(x), multiplied by (l - m)!/(l + m)!. The functions are not orthogonal for different values of m; orthogonality of the spherical harmonics in this case depends on the φ functions.
Composed by J. B. Calvert
Created 15 November 2000