Definitions

Legendre equation

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}right),y = 0,,

or

([1-x^2],y')' + left(ell[ell+1] - frac{m^2}{1-x^2}right),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 ≤ mell, 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.

Definition

These functions are denoted P_ell^{(m)}(x). We put the superscript in parentheses to avoid confusing it with an exponent. Their most straightforward definition is in terms of 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)right),

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]^ellright),

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: -lml. The definitions of Plm), 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 = begin{cases} 0 & mbox{if } mneq n frac{(ell+m)!}{m(ell-m)!} & mbox{if } m=nneq0 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.)

textrm{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_{0}^{0}(x)=1

P_{1}^{-1}(x)=-begin{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)=begin{matrix}frac{1}{24}end{matrix}P_{2}^{2}(x)
P_{2}^{-1}(x)=-begin{matrix}frac{1}{6}end{matrix}P_{2}^{1}(x)
P_{2}^{0}(x)=begin{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)=-begin{matrix}frac{1}{720}end{matrix}P_{3}^{3}(x)
P_{3}^{-2}(x)=begin{matrix}frac{1}{120}end{matrix}P_{3}^{2}(x)
P_{3}^{-1}(x)=-begin{matrix}frac{1}{12}end{matrix}P_{3}^{1}(x)
P_{3}^{0}(x)=begin{matrix}frac{1}{2}end{matrix}(5x^3-3x)
P_{3}^{1}(x)=-begin{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)=begin{matrix}frac{1}{40320}end{matrix}P_{4}^{4}(x)
P_{4}^{-3}(x)=-begin{matrix}frac{1}{5040}end{matrix}P_{4}^{3}(x)
P_{4}^{-2}(x)=begin{matrix}frac{1}{360}end{matrix}P_{4}^{2}(x)
P_{4}^{-1}(x)=-begin{matrix}frac{1}{20}end{matrix}P_{4}^{1}(x)
P_{4}^{0}(x)=begin{matrix}frac{1}{8}end{matrix}(35x^{4}-30x^{2}+3)
P_{4}^{1}(x)=-begin{matrix}frac{5}{2}end{matrix}(7x^3-3x)(1-x^2)^{1/2}
P_{4}^{2}(x)=begin{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

frac{1}{2} int_{-1}^1 dx P_l^u(x) P_m^v(x) P_n^w(x) = (-1)^{s-m-w}frac{(m+v)!(n+w)!(2s-2n)!s!}{(m-v)!(s-l)!(s-m)!(s-n)!(2s+1)!}
times sum_{t=p}^q (-1)^t frac{(l+u+t)!(m+n-u-t)!}{t!(l-u-t)!(m-n+u+t)!(n-w-t)!}
This formula is to be used under the following assumptions:

  1. the degrees are non-negative integers l,m,nge0
  2. all three orders are non-negative integers u,v,wge 0
  3. u is the largest of the three orders
  4. the orders sum up u=v+w
  5. 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

  1. the sum of degrees is even so that s is an integer
  2. 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}right]^{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, beta; gamma; z) = frac{Gamma(gamma)}{Gamma(alpha)Gamma(beta)} sum_{n=0}^inftyfrac{Gamma(n+alpha)Gamma(n+beta)}{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}right]^{mu/2} sum_{n=0}^inftyfrac{Gamma(n-lambda)Gamma(n+lambda+1)}{Gamma(n+1-mu) n!}left(frac{1-z}{2}right)^n.

They are called the Legendre functions when defined in this more general way. They satisfy the same differential equation as before:

(1-z^2),y -2zy' + left(lambda[lambda+1] - frac{mu^2}{1-z^2}right),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}right)

P_lambda^{(mu)}(z) and Q_lambda^{(mu)}(z) both obey the various recurrence formulas given previously.

Reparameterization in terms of angles

These functions are most useful when the argument is reparameterized in terms of angles, letting x = costheta:

P_ell^{(m)}(costheta) = (-1)^m (sin theta)^m frac{d^m}{d(costheta)^m}left(P_ell(costheta)right),

The first few polynomials, parameterized this way, are:

P_{0}^{0}(costheta)=1

P_{1}^{0}(costheta)=costheta
P_{1}^{1}(costheta)=-sintheta

P_{2}^{0}(costheta)=begin{matrix}frac{1}{2}end{matrix}(3cos^2theta-1)
P_{2}^{1}(costheta)=-3costhetasintheta
P_{2}^{2}(costheta)=3sin^2theta

P_{3}^{0}(costheta)=begin{matrix}frac{1}{2}end{matrix}(5cos^3theta-3costheta)
P_{3}^{1}(costheta)=-begin{matrix}frac{3}{2}end{matrix}(5cos^2theta-1)sintheta
P_{3}^{2}(costheta)=15costhetasin^2theta
P_{3}^{3}(costheta)=-15sin^3theta

P_{4}^{0}(costheta)=begin{matrix}frac{1}{8}end{matrix}(35cos^4theta-30costheta^{2}+3)
P_{4}^{1}(costheta)=-begin{matrix}frac{5}{2}end{matrix}(7cos^3theta-3costheta)sintheta
P_{4}^{2}(costheta)=begin{matrix}frac{15}{2}end{matrix}(7cos^2theta-1)sin^2theta
P_{4}^{3}(costheta)=-105costhetasin^3theta
P_{4}^{4}(costheta)=105sin^4theta

For fixed m, P_ell^{(m)}(costheta) are orthogonal, parameterized by θ over [0, pi], with weight sin theta:

int_{0}^{pi} P_k^{(m)}(costheta) P_ell^{(m)}(costheta),sintheta,dtheta = frac{2 (ell+m)!}{(2ell+1)(ell-m)!} delta _{k,ell}

Also, for fixed ell:

int_{0}^{pi}P_ell^{(m)}(costheta) P_ell^{(n)}(costheta) csctheta,dtheta = begin{cases} 0 & mbox{if } mneq n frac{(ell+m)!}{m(ell-m)!} & mbox{if } m=nneq0 infty & mbox{if } m=n=0end{cases}

In terms of θ, P_ell^{(m)}(cos theta) are solutions of

frac{d^{2}y}{dtheta^2} + cot theta frac{dy}{dtheta} + left[lambda - frac{m^2}{sin^2theta}right],y = 0,

More precisely, given an integer mge0, the above equation has nonsingular solutions only when lambda = ell(ell+1), for ell an integer{ge}m, and those solutions are proportional to P_ell^{(m)}(cos theta).

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 is the angle theta 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 nabla^2psi + lambdapsi = 0 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is

nabla^2psi = frac{partial^2psi}{partialtheta^2} + cot theta frac{partial psi}{partial theta} + csc^2 thetafrac{partial^2psi}{partialphi^2}.

When the partial differential equation

frac{partial^2psi}{partialtheta^2} + cot theta frac{partial psi}{partial theta} + csc^2 thetafrac{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}{dtheta^2} + cot theta frac{dy}{dtheta} + left[lambda - frac{m^2}{sin^2theta}right],y = 0,

for which the solutions are P_ell^{(m)}(cos theta) with ell{ge}m and lambda = ell(ell+1).

Therefore, the equation

nabla^2psi + lambdapsi = 0

has nonsingular separated solutions only when lambda = ell(ell+1), and those solutions are proportional to

P_ell^{(m)}(cos theta) cos (mphi) 0 le m le ell

and

P_ell^{(m)}(cos theta) sin (mphi) 0 < m le ell.

For each choice of ell, there are 2ell+1 functions for the various values of m and choices of sine and cosine. They are all orthogonal in both ell and m when integrated over the surface of the sphere.

The solutions are usually written in terms of complex exponentials:

Y_{ell, m}(theta, phi) = sqrt{frac{(2ell+1)(ell-m)!}{4pi(ell+m)!}} P_ell^{(m)}(cos theta) e^{imphi}qquad -ell le m le ell.
The functions Y_{ell, m}(theta, 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

Y_{ell, m}^*(theta, phi) = (-1)^m Y_{ell, -m}(theta, 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 typically of the form nabla^2psi(theta, phi) + lambdapsi(theta, phi) = 0, and hence the solutions are spherical harmonics.

See also

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. 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.

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