- Associated Legendre function
:"Note: This article describes a very general class of functions. An important subclass of these functions—those with integer and "m"—are commonly called "associated Legendre polynomials", even though they are not
polynomials when "m" is odd. The fully general class of functions described here, with arbitrary real or complex values of and "m", are sometimes called "generalized Legendre functions", or just "Legendre functions". In that case the parameters are usually renamed with Greek letters."
mathematics, the associated Legendre functions are the canonical solutions of the general Legendre equation
where the indices and "m" (which in general are complex quantities) are referred to as the degree and order of the associated Legendre function respectively. This equation has solutions that are nonsingular on [−1, 1] only if and "m" are integers with 0 ≤ "m" ≤ , or with trivially equivalent negative values. When in addition "m" is even, the function is a
polynomial. When "m" is zero and integer, these functions are identical to the Legendre polynomials.
This ordinary differential equation is frequently encountered in
physicsand other technical fields. In particular, it occurs when solving Laplace's equation(and related partial differential equations) in spherical coordinates.
These functions are denoted . We put the superscript in parenthesesto avoid confusing it with an exponent. Their most straightforward definition is in termsof derivatives of ordinary
Legendre polynomials("m" ≥ 0)
The factor in this formula is known as the Condon-Shortley phase. Some authors omit it.
Since, by Rodrigues' formula,
This equation allows extension of the range of "m" to: -"l" ≤ "m" ≤ "l". The definitions of "P""l"(±"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 following notations are used in literature:::
Assuming , they satisfy the orthogonality condition for fixed "m":
Where is the
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 "m".)
The differential equation is also invariant under a change from to, and the functions for negative are defined by
The first few associated Legendre polynomials
The first few associated Legendre polynomials, including those for negative values of "m", are:
These functions have a number of recurrence properties:
Helpful identities (initial values for the first recursion):
with !! the double factorial.
The integral over the product of three associated Legendre polynomials (with orders matching as shown below)turns out to be necessary when doing atomic calculations of the
Hartree-Fockvariety where matrix elements of the Coulomb operator are needed. For this we have Gaunt's formula [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 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
The Legendre functions, and the hypergeometric function
These functions may be defined for general complex parameters and argument:
where is the
gamma functionand is the hypergeometric function
They are called the Legendre functions when defined in this more general way. They satisfythe same differential equation as before:
Since this is a second order differential equation, it has a second solution, , defined as:
and both obey the variousrecurrence formulas given previously.
Reparameterization in terms of angles
These functions are most useful when the argument is reparameterized in terms of angles,letting :
The first few polynomials, parameterized this way, are:
For fixed "m", are orthogonal, parameterized by θ over , with weight :
Also, for fixed :
In terms of θ, are solutions of
More precisely, given an integer "m"0, the above equation hasnonsingular solutions only when for an integer, and those solutions are proportional to.
Applications in physics: Spherical harmonics
In many occasions in
physics, associated Legendre polynomials in terms of angles occur where spherical symmetryis involved. The colatitude angle in spherical coordinatesisthe 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 groupSO(3). As such, Legendre polynomials can be generalized to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces.
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
partial differential equation
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
For each choice of , there are functionsfor the various values of "m" and choices of sine and cosine.They are all orthogonal in both and "m" when integrated over thesurface of the sphere.
The solutions are usually written in terms of
: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 [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.]
The spherical harmonic functions form a complete orthonormal set of functions in the sense of
Fourier series. It should be noted that 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 typicallyof the form , and hence the solutions are spherical harmonics.
Whipple's transformation of Legendre functions
* Arfken G.B., Weber H.J., "Mathematical methods for physicists", (2001) Academic Press, ISBN 0-12-059825-6 "See Section 12.5". (Uses a different sign convention.)
* A.R. Edmonds, "Angular Momentum in Quantum Mechanics", (1957) Princeton University Press, ISBN 0-691-07912-9 "See chapter 2".
* E. U. Condon and G. H. Shortley, "The Theory of Atomic Spectra", (1970) Cambridge, England: The University Press. Oclc number|5388084 "See chapter 3"
* F. B. Hildebrand, "Advanced Calculus for Applications", (1976) Prentice Hall, ISBN 0-13-011189-9
* Belousov, S. L. (1962), "Tables of normalized associated Legendre polynomials", Mathematical tables series Vol. 18, Pergamon Press, 379p.
* [http://mathworld.wolfram.com/LegendrePolynomial.html Legendre polynomials in MathWorld]
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