Canonical solutions of the general Legendre equation
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation
or equivalently
where the indices ℓ and m (which are integers) are referred to as the degree and order of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on [−1, 1] only if ℓ and m are integers with 0 ≤ m ≤ ℓ, or with trivially equivalent negative values. When in addition m is even, the function is a polynomial. When m is zero and ℓ integer, these functions are identical to the Legendre polynomials. In general, when ℓ and m are integers, the regular solutions are sometimes called "associated Legendre polynomials", even though they are not polynomials when m is odd. The fully general class of functions with arbitrary real or complex values of ℓ and m are Legendre functions. In that case the parameters are usually labelled with Greek letters.
These functions are denoted , where the superscript indicates the order and not a power of P. Their most straightforward definition is in terms
of derivatives of ordinary Legendre polynomials (m ≥ 0)
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle P_\ell^{m}(x) = (-1)^m (1-x^2)^{m/2} \frac{d^m}{dx^m} \left( P_\ell(x) \right), }
The (−1)m factor in this formula is known as the Condon–Shortley phase. Some authors omit it. That the functions described by this equation satisfy the general Legendre differential equation with the indicated values of the parameters ℓ and m follows by differentiating m times the Legendre equation for Pℓ:[1]
Moreover, since by Rodrigues' formula,
the Pm ℓ can be expressed in the form
This equation allows extension of the range of m to: −ℓ ≤ m ≤ ℓ. The definitions of Pℓ±m, resulting from this expression by substitution of ±m, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of
then it follows that the proportionality constant is
so that
The associated Legendre polynomials are not mutually orthogonal in general. For example, is not orthogonal to . However, some subsets are orthogonal. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:
The integral over the product of three associated Legendre polynomials (with orders matching as shown below) is a necessary ingredient when developing products of Legendre polynomials into a series linear in the Legendre polynomials. For instance, this turns out to be necessary when doing atomic calculations of the Hartree–Fock variety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula [3]
This formula is to be used under the following assumptions:
the degrees are non-negative integers
all three orders are non-negative integers
is the largest of the three orders
the orders sum up
the degrees obey
Other quantities appearing in the formula are defined as
The integral is zero unless
the sum of degrees is even so that is an integer
the triangular condition is satisfied
Dong and Lemus (2002)[4] generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.
These functions are most useful when the argument is reparameterized in terms of angles, letting :
Using the relation , the list given above yields the first few polynomials, parameterized this way, as:
The orthogonality relations given above become in this formulation:
for fixed m, are orthogonal, parameterized by θ over , with weight :
Also, for fixed ℓ:
In terms of θ, are solutions of
More precisely, given an integer m0, the above equation has
nonsingular solutions only when for ℓ
an integer ≥ m, and those solutions are proportional to
.
In many occasions in physics, associated Legendre polynomials in terms of angles occur where sphericalsymmetry is involved. The colatitude angle in spherical coordinates is
the angle used above. The longitude angle, , appears in a multiplying factor. Together, they make a set of functions called spherical harmonics. These functions express the symmetry of the two-sphere under the action of the Lie group SO(3).[citation needed]
What makes these functions useful is that they are central to the solution of the equation
on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is
is solved by the method of separation of variables, one gets a φ-dependent part or for integer m≥0, and an equation for the θ-dependent part
for which the solutions are with
and .
Therefore, the equation
has nonsingular separated solutions only when ,
and those solutions are proportional to
and
For each choice of ℓ, there are 2ℓ + 1 functions
for the various values of m and choices of sine and cosine.
They are all orthogonal in both ℓ and m when integrated over the
surface of the sphere.
The functions are the spherical harmonics, and the quantity in the square root is a normalizing factor.
Recalling the relation between the associated Legendre functions of positive and negative m, it is easily shown that the spherical harmonics satisfy the identity[5]
The spherical harmonic functions form a complete orthonormal set of functions in the sense of Fourier series. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see spherical harmonics).
When a 3-dimensional spherically symmetric partial differential equation is solved by the method of separation of variables in spherical coordinates, the part that remains after removal of the radial part is typically
of the form
The Legendre polynomials are closely related to hypergeometric series. In the form of spherical harmonics, they express the symmetry of the two-sphere under the action of the Lie group SO(3). There are many other Lie groups besides SO(3), and analogous generalizations of the Legendre polynomials exist to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces. Crudely speaking, one may define a Laplacian on symmetric spaces; the eigenfunctions of the Laplacian can be thought of as generalizations of the spherical harmonics to other settings.
^From John C. Slater Quantum Theory of Atomic Structure, McGraw-Hill (New York, 1960), Volume I, page 309, which cites the original work of J. A. Gaunt, Philosophical Transactions of the Royal Society of London, A228:151 (1929)
^This identity can also be shown by relating the spherical harmonics to Wigner D-matrices and use of the time-reversal property of the latter. The relation between associated Legendre functions of ±m can then be proved from the complex conjugation identity of the spherical harmonics.
Arfken, G.B.; Weber, H.J. (2001), Mathematical methods for physicists, Academic Press, ISBN978-0-12-059825-0; Section 12.5. (Uses a different sign convention.)
Belousov, S. L. (1962), Tables of normalized associated Legendre polynomials, Mathematical tables, vol. 18, Pergamon Press.
Condon, E. U.; Shortley, G. H. (1970), The Theory of Atomic Spectra, Cambridge, England: Cambridge University Press, OCLC5388084; Chapter 3.
Courant, Richard; Hilbert, David (1953), Methods of Mathematical Physics, Volume 1, New York: Interscience Publischer, Inc.