- Associated Legendre function
:"Note: This article describes a very general class of functions. An important subclass of these functions—those with integer ell and "m"—are commonly called "associated Legendre polynomials", even though they are not
polynomial s when "m" is odd. The fully general class of functions described here, with arbitrary real or complex values of ell, and "m", are sometimes called "generalized Legendre functions", or just "Legendre functions". In that case the parameters are usually renamed with Greek letters."In
mathematics , the associated Legendre functions are the canonical solutions of the general Legendre equation:1-x^2),y" -2xy' + left(ell [ell+1] - frac{m^2}{1-x^2} ight),y = 0,,
or
:1-x^2] ,y')' + left(ell [ell+1] - frac{m^2}{1-x^2} ight),y = 0,,
where the indices ell 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 ell, and "m" are integers with 0 ≤ "m" ≤ ell, or with trivially equivalent negative values. When in addition "m" is even, the function is a
polynomial . When "m" is zero and ell, integer, these functions are identical to theLegendre polynomial s.This ordinary differential equation is frequently encountered in
physics and other technical fields. In particular, it occurs when solvingLaplace's equation (and relatedpartial differential equation s) inspherical coordinates .Definition
These functions are denoted P_ell^{(m)}(x). 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):P_ell^{(m)}(x) = (-1)^m (1-x^2)^{m/2} frac{d^m}{dx^m}left(P_ell(x) ight),
The 1)^m factor in this formula is known as the Condon-Shortley phase. Some authors omit it.
Since, by Rodrigues' formula,
:P_ell(x) = frac{1}{2^ell,ell!} frac{d^ell}{dx^ell}left( [x^2-1] ^ell ight),
one obtains
:P_ell^{(m)}(x) = frac{(-1)^m}{2^ell ell!} (1-x^2)^{m/2} frac{d^{ell+m{dx^{ell+m(x^2-1)^ell.
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 :frac{d^{ell-m{dx^{ell-m (x^2-1)^{ell} = c_{lm} (1-x^2)^m frac{d^{ell+m{dx^{ell+m(x^2-1)^{ell},then it follows that the proportionality constant is:c_{lm} = (-1)^m frac{(ell-m)!}{(ell+m)!} ,so that :P^{(-m)}_ell(x) = (-1)^m frac{(ell-m)!}{(ell+m)!} P^{(m)}_ell(x).
Alternative notations
The following notations are used in literature::P_{ell} ^{m}(x) = P_ell^{(m)}(x) :P_{ell m}(x) = (-1)^m P_ell^{(m)}(x)
Orthogonality
Assuming 0 le m le ell, they satisfy the orthogonality condition for fixed "m":
:int_{-1}^{1} P_k ^{(m)} P_ell ^{(m)} dx = frac{2 (ell+m)!}{(2ell+1)(ell-m)!} delta _{k,ell}
Where delta _{k,ell} is the
Kronecker delta .Also, they satisfy the orthogonality condition for fixed ell:
:int_{-1}^{1} frac{P_ell ^{(m)} P_ell ^{(n){1-x^2}dx = egin{cases} 0 & mbox{if } m eq n \ frac{(ell+m)!}{m(ell-m)!} & mbox{if } m=n eq0 \ infty & mbox{if } m=n=0end{cases}
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":
:P_ell ^{(-m)} = (-1)^m frac{(ell-m)!}{(ell+m)!} P_ell ^{(m)}
(This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative "m".)
extrm{If}quad {mid}m{mid} > ell,quadmathrm{then}quad P_ell^{(m)} = 0.,
The differential equation is also invariant under a change from ell toell-1, and the functions for negative ell are defined by
:P_{-ell} ^{(m)} = P_{ell-1} ^{(m)}.,
The first few associated Legendre polynomials
The first few associated Legendre polynomials, including those for negative values of "m", are:
:P_{0}^{0}(x)=1
:P_{1}^{-1}(x)=-egin{matrix}frac{1}{2}end{matrix}P_{1}^{1}(x):P_{1}^{0}(x)=x:P_{1}^{1}(x)=-(1-x^2)^{1/2}
:P_{2}^{-2}(x)=egin{matrix}frac{1}{24}end{matrix}P_{2}^{2}(x):P_{2}^{-1}(x)=-egin{matrix}frac{1}{6}end{matrix}P_{2}^{1}(x):P_{2}^{0}(x)=egin{matrix}frac{1}{2}end{matrix}(3x^{2}-1):P_{2}^{1}(x)=-3x(1-x^2)^{1/2}:P_{2}^{2}(x)=3(1-x^2)
:P_{3}^{-3}(x)=-egin{matrix}frac{1}{720}end{matrix}P_{3}^{3}(x):P_{3}^{-2}(x)=egin{matrix}frac{1}{120}end{matrix}P_{3}^{2}(x):P_{3}^{-1}(x)=-egin{matrix}frac{1}{12}end{matrix}P_{3}^{1}(x):P_{3}^{0}(x)=egin{matrix}frac{1}{2}end{matrix}(5x^3-3x):P_{3}^{1}(x)=-egin{matrix}frac{3}{2}end{matrix}(5x^{2}-1)(1-x^2)^{1/2}:P_{3}^{2}(x)=15x(1-x^2):P_{3}^{3}(x)=-15(1-x^2)^{3/2}
:P_{4}^{-4}(x)=egin{matrix}frac{1}{40320}end{matrix}P_{4}^{4}(x):P_{4}^{-3}(x)=-egin{matrix}frac{1}{5040}end{matrix}P_{4}^{3}(x):P_{4}^{-2}(x)=egin{matrix}frac{1}{360}end{matrix}P_{4}^{2}(x):P_{4}^{-1}(x)=-egin{matrix}frac{1}{20}end{matrix}P_{4}^{1}(x):P_{4}^{0}(x)=egin{matrix}frac{1}{8}end{matrix}(35x^{4}-30x^{2}+3):P_{4}^{1}(x)=-egin{matrix}frac{5}{2}end{matrix}(7x^3-3x)(1-x^2)^{1/2}:P_{4}^{2}(x)=egin{matrix}frac{15}{2}end{matrix}(7x^2-1)(1-x^2):P_{4}^{3}(x)= - 105x(1-x^2)^{3/2}:P_{4}^{4}(x)=105(1-x^2)^{2}
Recurrence formula
These functions have a number of recurrence properties:
:ell-m+1)P_{ell+1}^{(m)}(x) = (2ell+1)xP_{ell}^{(m)}(x) - (ell+m)P_{ell-1}^{(m)}(x)
:P_{ell+1}^{(m)}(x) = P_{ell-1}^{(m)}(x) - (2ell+1)sqrt{1-x^2}P_{ell}^{(m-1)}(x)
:sqrt{1-x^2}P_{ell}^{(m+1)}(x) = (ell-m)xP_{ell}^{(m)}(x) - (ell+m)P_{ell-1}^{(m)}(x)
:x^2-1){P_{ell}^{(m)'(x) = {ell}xP_{ell}^{(m)}(x) - (ell+m)P_{ell-1}^{(m)}(x)
:x^2-1){P_{ell}^{(m)'(x) = -(ell+m)(ell-m+1)sqrt{1-x^2}P_{ell}^{(m-1)}(x) - mxP_{ell}^{(m)}(x)
Helpful identities (initial values for the first recursion):
:P_{ell}^{(ell)}(x) = (-1)^l (2ell-1)!! (1- x^2)^{(l/2)}:P_{ell +1}^{(ell)}(x) = x (2ell+1) P_{ell}^{(ell)}(x)
with !! the double factorial.
Gaunt's formula
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-Fock variety 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 l,m,nge0
# all three orders are non-negative integers u,v,wge 0
# u is the largest of the three orders
# the orders sum up u=v+w
# the degrees obey mge nOther quantities appearing in the formula are defined as: 2s = l+m+n : p = max(0,n-w-u) : q = min(m+n-u,l-u,n-w)
The integral is zero unless
# the sum of degrees is even so that s is an integer
# the triangular condition is satisfied m+nge l ge m-nThe Legendre functions, and the hypergeometric function
These functions may be defined for general complex parameters and argument:
:P_{lambda}^{(mu)}(z) = frac{1}{Gamma(1-mu)} left [frac{1+z}{1-z} ight] ^{mu/2} ,_2F_1 (-lambda, lambda+1; 1-mu; frac{1-z}{2})
where Gamma is the
gamma function and 2F_1 is thehypergeometric function :2F_1 (alpha, eta; gamma; z) = frac{Gamma(gamma)}{Gamma(alpha)Gamma(eta)} sum_{n=0}^inftyfrac{Gamma(n+alpha)Gamma(n+eta)}{Gamma(n+gamma) n!}z^n,
so that
:P_{lambda}^{(mu)}(z) = frac{1}{Gamma(-lambda)Gamma(lambda+1)} left [frac{1+z}{1-z} ight] ^{mu/2} sum_{n=0}^inftyfrac{Gamma(n-lambda)Gamma(n+lambda+1)}{Gamma(n+1-mu) n!}left(frac{1-z}{2} ight)^n.
They are called the Legendre functions when defined in this more general way. They satisfythe same differential equation as before:
:1-z^2),y" -2zy' + left(lambda [lambda+1] - frac{mu^2}{1-z^2} ight),y = 0.,
Since this is a second order differential equation, it has a second solution, Q_lambda^{(mu)}(z), defined as:
:Q_{lambda}^{(mu)}(z) = frac{sqrt{pi} Gamma(lambda+mu+1)}{2^{lambda+1}Gamma(lambda+3/2)}frac{1}{z^{lambda+mu+1(1-z^2)^{mu/2} ,_2F_1 left(frac{lambda+mu+1}{2}, frac{lambda+mu+2}{2}; lambda+frac{3}{2}; frac{1}{z^2} ight)
P_lambda^{(mu)}(z) and Q_lambda^{(mu)}(z) 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 x = cos heta:
:P_ell^{(m)}(cos heta) = (-1)^m (sin heta)^m frac{d^m}{d(cos heta)^m}left(P_ell(cos heta) ight),
The first few polynomials, parameterized this way, are:
:P_{0}^{0}(cos heta)=1
:P_{1}^{0}(cos heta)=cos heta:P_{1}^{1}(cos heta)=-sin heta
:P_{2}^{0}(cos heta)=egin{matrix}frac{1}{2}end{matrix}(3cos^2 heta-1):P_{2}^{1}(cos heta)=-3cos hetasin heta:P_{2}^{2}(cos heta)=3sin^2 heta
:P_{3}^{0}(cos heta)=egin{matrix}frac{1}{2}end{matrix}(5cos^3 heta-3cos heta):P_{3}^{1}(cos heta)=-egin{matrix}frac{3}{2}end{matrix}(5cos^2 heta-1)sin heta:P_{3}^{2}(cos heta)=15cos hetasin^2 heta:P_{3}^{3}(cos heta)=-15sin^3 heta
:P_{4}^{0}(cos heta)=egin{matrix}frac{1}{8}end{matrix}(35cos^4 heta-30cos heta^{2}+3):P_{4}^{1}(cos heta)=-egin{matrix}frac{5}{2}end{matrix}(7cos^3 heta-3cos heta)sin heta:P_{4}^{2}(cos heta)=egin{matrix}frac{15}{2}end{matrix}(7cos^2 heta-1)sin^2 heta:P_{4}^{3}(cos heta)=-105cos hetasin^3 heta:P_{4}^{4}(cos heta)=105sin^4 heta
For fixed "m", P_ell^{(m)}(cos heta) are orthogonal, parameterized by θ over 0, pi] , with weight sin heta:
:int_{0}^{pi} P_k^{(m)}(cos heta) P_ell^{(m)}(cos heta),sin heta,d heta = frac{2 (ell+m)!}{(2ell+1)(ell-m)!} delta _{k,ell}
Also, for fixed ell:
:int_{0}^{pi}P_ell^{(m)}(cos heta) P_ell^{(n)}(cos heta) csc heta,d heta = egin{cases} 0 & mbox{if } m eq n \ frac{(ell+m)!}{m(ell-m)!} & mbox{if } m=n eq0 \ infty & mbox{if } m=n=0end{cases}
In terms of θ, P_ell^{(m)}(cos heta) are solutions of
:frac{d^{2}y}{d heta^2} + cot heta frac{dy}{d heta} + left [lambda - frac{m^2}{sin^2 heta} ight] ,y = 0,
More precisely, given an integer "m"ge0, the above equation hasnonsingular solutions only when lambda = ell(ell+1), for ellan integerge}m, and those solutions are proportional toP_ell^{(m)}(cos heta).
Applications in physics: Spherical harmonics
In many occasions in
physics , associated Legendre polynomials in terms of angles occur wherespherical symmetry is involved. The colatitude angle inspherical coordinates isthe angle heta used above. The longitude angle, phi, appears in a multiplying factor. Together, they make a set of functions calledspherical harmonic s.These functions express the symmetry of the two-sphere under the action of the
Lie group SO(3). As such, Legendre polynomials can be generalized to express the symmetries of semi-simple Lie groups andRiemannian symmetric space s.What makes these functions useful is that they are central to the solution of the equationabla^2psi + lambdapsi = 0 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the
Laplacian is:abla^2psi = frac{partial^2psi}{partial heta^2} + cot heta frac{partial psi}{partial heta} + csc^2 hetafrac{partial^2psi}{partialphi^2}.
When the
partial differential equation :frac{partial^2psi}{partial heta^2} + cot heta frac{partial psi}{partial heta} + csc^2 hetafrac{partial^2psi}{partialphi^2} + lambda psi = 0
is solved by the method of
separation of variables , one gets a φ-dependent part sin(mphi) or cos(mphi) for integer m≥0, and an equation for the θ-dependent part:frac{d^{2}y}{d heta^2} + cot heta frac{dy}{d heta} + left [lambda - frac{m^2}{sin^2 heta} ight] ,y = 0,
for which the solutions are P_ell^{(m)}(cos heta) with ell{ge}mand lambda = ell(ell+1).
Therefore, the equation
:abla^2psi + lambdapsi = 0
has nonsingular separated solutions only when lambda = ell(ell+1),and those solutions are proportional to
:P_ell^{(m)}(cos heta) cos (mphi) 0 le m le ell
and
:P_ell^{(m)}(cos heta) sin (mphi) 0 < m le ell.
For each choice of ell, there are 2ell+1 functionsfor the various values of "m" and choices of sine and cosine.They are all orthogonal in both ell and "m" when integrated over thesurface of the sphere.
The solutions are usually written in terms of
complex exponential s::Y_{ell, m}( heta, phi) = sqrt{frac{(2ell+1)(ell-m)!}{4pi(ell+m)! P_ell^{(m)}(cos heta) e^{imphi}qquad -ell le m le ell.The functions Y_{ell, m}( heta, phi) 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.]:Y_{ell, m}^*( heta, phi) = (-1)^m Y_{ell, -m}( heta, phi).
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 (seespherical 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 abla^2psi( heta, phi) + lambdapsi( heta, phi) = 0, and hence the solutions are spherical harmonics.
ee also
*
Angular momentum
*Gaussian quadrature
*Legendre polynomials
*Spherical harmonic s
*Whipple's transformation of Legendre functions Notes
References
* 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.External links
* [http://mathworld.wolfram.com/LegendrePolynomial.html Legendre polynomials in MathWorld]
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