Associated Legendre function

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 polynomials 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,,


:( [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 the Legendre polynomials.

This 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.


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)


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 to-ell-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_{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 n

Other 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-n

The 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 the hypergeometric 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 integer{ge}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 where spherical symmetry is involved. The colatitude angle in spherical coordinates isthe angle heta used above. The longitude angle, phi, 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). 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 abla^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


: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 exponentials:

: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 &plusmn;"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 (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 abla^2psi( heta, phi) + lambdapsi( heta, phi) = 0, and hence the solutions are spherical harmonics.

ee also

* Angular momentum
* Gaussian quadrature
* Legendre polynomials
* 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.

External links

* [ Legendre polynomials in MathWorld]

Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Legendre wavelet — Legendre wavelets: spherical harmonic wavelets = Compactly supported wavelets derived from Legendre polynomials are termed spherical harmonic or Legendre wavelets [1] . Legendre functions have widespread applications in which spherical coordinate …   Wikipedia

  • Legendre polynomials — Note: People sometimes refer to the more general associated Legendre polynomials as simply Legendre polynomials . In mathematics, Legendre functions are solutions to Legendre s differential equation::{d over dx} left [ (1 x^2) {d over dx} P n(x)… …   Wikipedia

  • Hough function — In applied mathematics, the Hough functions are the eigenfunctions of Laplace s tidal equations which govern fluid motion on a rotating sphere. As such, they are relevant in geophysics and meteorology where they form part of the solutions for… …   Wikipedia

  • special function — ▪ mathematics       any of a class of mathematical functions (function) that arise in the solution of various classical problems of physics. These problems generally involve the flow of electromagnetic, acoustic, or thermal energy. Different… …   Universalium

  • Spherical harmonics — In mathematics, the spherical harmonics are the angular portion of an orthogonal set of solutions to Laplace s equation represented in a system of spherical coordinates. Spherical harmonics are important in many theoretical and practical… …   Wikipedia

  • List of mathematics articles (A) — NOTOC A A Beautiful Mind A Beautiful Mind (book) A Beautiful Mind (film) A Brief History of Time (film) A Course of Pure Mathematics A curious identity involving binomial coefficients A derivation of the discrete Fourier transform A equivalence A …   Wikipedia

  • Classical orthogonal polynomials — In mathematics, the classical orthogonal polynomials are the most widely used orthogonal polynomials, and consist of the Hermite polynomials, the Laguerre polynomials, the Jacobi polynomials together with their special cases the ultraspherical… …   Wikipedia

  • Spectral theory of ordinary differential equations — In mathematics, the spectral theory of ordinary differential equations is concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his dissertation Hermann Weyl… …   Wikipedia

  • Pierre-Simon Laplace — Laplace redirects here. For the city in Louisiana, see LaPlace, Louisiana. For the joint NASA ESA space mission, see Europa Jupiter System Mission. Pierre Simon, marquis de Laplace Pierre Simon Laplace (1749–1827). Posthumous portrait …   Wikipedia

  • Calculus of variations — is a field of mathematics that deals with extremizing functionals, as opposed to ordinary calculus which deals with functions. A functional is usually a mapping from a set of functions to the real numbers. Functionals are often formed as definite …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.