Wikipedia / Associated Legendre polynomials
Wikipedia / Associated Legendre polynomials
In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre differential equation
or equivalently
where the indices ℓ and m (which are integers) are referred to as the order and degree of the associated Legendre polynomial respectively. This equation has nonzero solutions that are nonsingular on 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.
The Legendre ordinary differential equation is frequently encountered in physics and other technical fields. In particular, it occurs when solving Laplace's equation (and related partial differential equations) in spherical coordinates. Associated Legendre polynomials play a vital role in the definition of spherical harmonics.
Definition for non-negative integer parameters ℓ and m
, 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)
The 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 :
Moreover, since by Rodrigues' formula the P
This equation allows extension of the range of m to: . The definitions of , resulting from this expression by substitution of
Alternative notations
The following alternative notations are also used in literature:
Closed Form
Starting from the explicit form provided in the article of Legendre Polynomials
-fold derivatives for powers
with simple monomials and the generalized form of the binomial coefficient is zero.
Summarizing results of Doha
where
and where
is the Neumann factor.
Orthogonality
. However, some subsets are orthogonal. Assuming 0≤m≤ℓ, they satisfy the orthogonality condition for fixed m:
Where is the Kronecker delta.
Also, they satisfy the orthogonality condition for fixed :
Negative m and/or negative ℓ
The differential equation is clearly invariant under a change in sign of m.
The functions for negative m were shown above to be proportional to those of positive m
(This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative
The differential equation is also invariant under a change from to , and the functions for negative are defined by
Parity
From their definition, one can verify that the Associated Legendre functions are either even or odd according to
The first few associated Legendre functions
The first few associated Legendre functions, including those for negative values of m, are:
Recurrence formula
These functions have a number of recurrence properties:
Helpful identities (initial values for the first recursion):
with the double factorial.
Gaunt's formula
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 This formula is to be used under the following assumptions:
is the largest of the three orders
The integral is zero unless
is an integer
Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.
Generalization via hypergeometric functions
These functions may actually be defined for general complex parameters and argument:
is the gamma function is the hypergeometric function
They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:
, defined as:
both obey the various recurrence formulas given previously.
Reparameterization in terms of angles
):
, 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:
Also, for fixed ℓ:
are solutions of
More precisely, given an integer m for ℓ an integer ≥m.
Applications in physics: spherical harmonics
In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved. The colatitude angle in spherical coordinates, 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).
on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is
When the partial differential equation
is solved by the method of separation of variables for integer m≥0, and an equation for the θ-dependent part
.
Therefore, the equation
, and those solutions are proportional to
and
For each choice of ℓ, there are 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 solutions are usually written in terms of complex exponentials:
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
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
and hence the solutions are spherical harmonics.
Generalizations
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.
) Legendre Polynomials in higher than 3D can be defined.
See also
Notes and references
; Section 12.5. (Uses a different sign convention.)
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; Chapter 3.
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; Chapter 2.
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External links