Associated Legendre polynomials explained

In mathematics, the associated Legendre polynomials are the canonical solutions of the general Legendre equation

$$\backslash left(1\; -\; x^2\backslash right)\; \backslash frac\; P\_\backslash ell^m(x)\; -\; 2\; x\; \backslash frac\; P\_\backslash ell^m(x)\; +\; \backslash left[\backslash ell\; (\backslash ell\; +\; 1)\; -\; \backslash frac\{m^2\}\{1\; -\; x^2\}\; \backslash right]\; P\_\backslash ell^m(x)\; =\; 0,$$

or equivalently

$$\backslash frac\; \backslash left[\backslash left(1\; -\; x^2\backslash right)\; \backslash frac\{d\}\{d\; x\}\; P\_\backslash ell^m(x)\; \backslash right]\; +\; \backslash left[\backslash ell\; (\backslash ell\; +\; 1)\; -\; \backslash frac\{m^2\}\{1\; -\; x^2\}\; \backslash right]\; P\_\backslash ell^m(x)\; =\; 0,$$

where the indices ℓ and m (which are integers) are referred to as the degree and order 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 and

These functions are denoted

P ^{m} _{\ell} (x) , where the superscript indicates the order and not a power of P. Their most straightforward definition is in termsof derivatives of ordinary Legendre polynomials (m ≥ 0)

$$P\_\backslash ell^(x)\; =\; (-1)^m\; (1-x^2)^\; \backslash frac\; \backslash left(P\_\backslash ell(x)\; \backslash right),$$

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 :^[1] $$\backslash left(1-x^2\backslash right)\; \backslash fracP\_\backslash ell(x)\; -2x\backslash fracP\_\backslash ell(x)+\; \backslash ell(\backslash ell+1)P\_\backslash ell(x)\; =\; 0.$$

Moreover, since by Rodrigues' formula,$$P\_\backslash ell(x)\; =\; \backslash frac\; \backslash \; \backslash frac\backslash left[(x^2-1)^\backslash ell\backslash right],$$the P can be expressed in the form$$P\_\backslash ell^(x)\; =\; \backslash frac\; (1-x^2)^\backslash \; \backslash frac(x^2-1)^\backslash ell.$$

This equation allows extension of the range of m to: . The definitions of, resulting from this expression by substitution of, are proportional. Indeed, equate the coefficients of equal powers on the left and right hand side of$$\backslash frac\; (x^2-1)^\; =\; c\_\; (1-x^2)^m\; \backslash frac(x^2-1)^,$$then it follows that the proportionality constant is$$c\_\; =\; (-1)^m\; \backslash frac,$$so that$$P^\_\backslash ell(x)\; =\; (-1)^m\; \backslash frac\; P^\_\backslash ell(x).$$

Alternative notations

The following alternative notations are also used in literature:$$P\_(x)\; =\; (-1)^m\; P\_\backslash ell^(x)$$

Closed Form

The Associated Legendre Polynomial can also be written as:$$P\_l^m(x)=(-1)^\; \backslash cdot\; 2^\; \backslash cdot\; (1-x^2)^\; \backslash cdot\; \backslash sum\_^l\; \backslash frac\backslash cdot\; x^\; \backslash cdot\; \backslash binom\; \backslash binom$$with simple monomials and the generalized form of the binomial coefficient.

Orthogonality

The associated Legendre polynomials are not mutually orthogonal in general. For example,

P ^{1} _{1} is not orthogonal to P ^{2} _{2} . However, some subsets are orthogonal. Assuming 0 ≤ m ≤ ℓ, they satisfy the orthogonality condition for fixed m:

$$\backslash int\_^\; P\_k\; ^\; P\_\backslash ell\; ^\; dx\; =\; \backslash frac\backslash \; \backslash delta\; \_$$

Where is the Kronecker delta.

Also, they satisfy the orthogonality condition for fixed :

Negative and/or negative

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:$$P\_\backslash ell\; ^\; =\; (-1)^m\; \backslash frac\; P\_\backslash ell\; ^$$

(This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative .)$$\backslash text\backslash quad\; |m|\; >\; \backslash ell\backslash ,\backslash quad\backslash text\backslash quad\; P\_\backslash ell^\; =\; 0.\backslash ,$$

The differential equation is also invariant under a change from to, and the functions for negative are defined by

$$P\_\; ^\; =\; P\_\; ^,\backslash \; (\backslash ell=1,\backslash ,2,\backslash ,\; \backslash dots).$$

Parity

From their definition, one can verify that the Associated Legendre functions are either even or odd according to

$$P\_\backslash ell\; ^\; (-x)\; =\; (-1)^\; P\_\backslash ell\; ^(x)$$

The first few associated Legendre functions

The first few associated Legendre functions, including those for negative values of m, are:

$$P\_^(x)=1$$

Recurrence formula

These functions have a number of recurrence properties:

$$(\backslash ell-m-1)(\backslash ell-m)P\_^(x)\; =\; -P\_^(x)\; +\; P\_^(x)\; +\; (\backslash ell+m)(\backslash ell+m-1)P\_^(x)$$

$$(\backslash ell-m+1)P\_^(x)\; =\; (2\backslash ell+1)xP\_^(x)\; -\; (\backslash ell+m)P\_^(x)$$

$$2mxP\_^(x)=-\backslash sqrt\backslash left[P\_\{\backslash ell\}^\{m+1\}(x)+(\backslash ell+m)(\backslash ell-m+1)P\_\{\backslash ell\}^\{m-1\}(x)\backslash right]$$

$$\backslash fracP\_\backslash ell^m(x)\; =\; \backslash frac\; \backslash left[P\_\{\backslash ell-1\}^\{m+1\}(x)\; +\; (\backslash ell+m-1)(\backslash ell+m)P\_\{\backslash ell-1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash fracP\_\backslash ell^m(x)\; =\; \backslash frac\; \backslash left[P\_\{\backslash ell+1\}^\{m+1\}(x)\; +\; (\backslash ell-m+1)(\backslash ell-m+2)P\_\{\backslash ell+1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash sqrtP\_\backslash ell^m(x)\; =\; \backslash frac1\; \backslash left[(\backslash ell-m+1)(\backslash ell-m+2)\; P\_\{\backslash ell+1\}^\{m-1\}(x)\; -\; (\backslash ell+m-1)(\backslash ell+m)\; P\_\{\backslash ell-1\}^\{m-1\}(x)\; \backslash right]$$

$$\backslash sqrtP\_\backslash ell^m(x)\; =\; \backslash frac\; \backslash left[P\_\{\backslash ell+1\}^\{m+1\}(x)\; -\; P\_\{\backslash ell-1\}^\{m+1\}(x)\; \backslash right]$$

$$\backslash sqrtP\_\backslash ell^(x)\; =\; (\backslash ell-m)xP\_^(x)\; -\; (\backslash ell+m)P\_^(x)$$

$$\backslash sqrtP\_\backslash ell^(x)\; =\; (\backslash ell-m+1)P\_^m(x)\; -\; (\backslash ell+m+1)xP\_\backslash ell^m(x)$$

$$\backslash sqrt\backslash frac(x)\; =\; \backslash frac12\; \backslash left[(\backslash ell+m)(\backslash ell-m+1)P\_\backslash ell^\{m-1\}(x)\; -\; P\_\backslash ell^\{m+1\}(x)\; \backslash right]$$

$$(1-x^2)\backslash frac(x)\; =\; \backslash frac1\; \backslash left[(\backslash ell+1)(\backslash ell+m)P\_\{\backslash ell-1\}^m(x)\; -\; \backslash ell(\backslash ell-m+1)P\_\{\backslash ell+1\}^m(x)\; \backslash right]$$

$$(x^2-1)\backslash frac(x)\; =\; xP\_^(x)\; -\; (\backslash ell+m)P\_^(x)$$

$$(x^2-1)\backslash frac(x)\; =\; -(\backslash ell+1)xP\_^(x)\; +\; (\backslash ell-m+1)P\_^(x)$$

$$(x^2-1)\backslash frac(x)\; =\; \backslash sqrtP\_^(x)\; +\; mxP\_^(x)$$

$$(x^2-1)\backslash frac(x)\; =\; -(\backslash ell+m)(\backslash ell-m+1)\backslash sqrtP\_^(x)\; -\; mxP\_^(x)$$

Helpful identities (initial values for the first recursion):

$$P\_^(x)\; =\; -\; (2\backslash ell+1)\; \backslash sqrt\; P\_^(x)$$$$P\_^(x)\; =\; (-1)^\backslash ell\; (2\backslash ell-1)!!\; (1-\; x^2)^$$$$P\_^(x)\; =\; x\; (2\backslash ell+1)\; P\_^(x)$$

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 ^[2] This formula is to be used under the following assumptions:

1. the degrees are non-negative integers

l,m,n\ge 0

1. all three orders are non-negative integers

u,v,w\ge 0 u is the largest of the three orders

1. the orders sum up

u = v + w

1. the degrees obey

m\ge n

Other quantities appearing in the formula are defined as$$2s\; =\; l+m+n$$$$p\; =\; \backslash max(0,\backslash ,n-m-u)$$$$q\; =\; \backslash min(m+n-u,\backslash ,l-u,\backslash ,n-w)$$

The integral is zero unless

1. the sum of degrees is even so that

s is an integer

1. the triangular condition is satisfied

m + n\ge l \ge m - n

Dong and Lemus (2002)^[3] generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.

Generalization via hypergeometric functions

See main article: Legendre function. These functions may actually be defined for general complex parameters and argument:

$$P\_^(z)\; =\; \backslash frac\; \backslash left[\backslash frac\{1+z\}\{1-z\}\backslash right]^\; \backslash ,\_2F\_1\; (-\backslash lambda,\; \backslash lambda+1;\; 1-\backslash mu;\; \backslash frac)$$

where

\Gamma is the gamma function and _{2F} _{1} is the hypergeometric function

$$\backslash ,\_2F\_1\; (\backslash alpha,\; \backslash beta;\; \backslash gamma;\; z)\; =\; \backslash frac\; \backslash sum\_^\backslash infty\backslash fracz^n,$$

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

$$(1-z^2)\backslash ,y$$ -2zy' + \left(\lambda[\lambda+1] - \frac\right)\,y = 0.\,

Since this is a second order differential equation, it has a second solution,

Q ^{\mu} _{λ} (z) , defined as:

$$Q\_^(z)\; =\; \backslash frac\backslash frac(1-z^2)^\; \backslash ,\_2F\_1\; \backslash left(\backslash frac,\; \backslash frac;\; \backslash lambda+\backslash frac;\; \backslash frac\backslash right)$$

P ^{\mu} _{λ} (z) and Q ^{\mu} _{λ} (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 = \cos\theta :

$$P\_\backslash ell^(\backslash cos\backslash theta)\; =\; (-1)^m\; (\backslash sin\; \backslash theta)^m\backslash \; \backslash frac\backslash left(P\_\backslash ell(\backslash cos\backslash theta)\backslash right)$$

Using the relation

( 1 - x ^{2)} ^{1} = \sin\theta , 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,

P ^{m(\cos\theta)} _{\ell} are orthogonal, parameterized by θ over [ 0 , \pi] , with weight \sin \theta :

$$\backslash int\_0^\backslash pi\; P\_k^(\backslash cos\backslash theta)\; P\_\backslash ell^(\backslash cos\backslash theta)\backslash ,\backslash sin\backslash theta\backslash ,d\backslash theta\; =\; \backslash frac\backslash \; \backslash delta\; \_$$

Also, for fixed ℓ:

In terms of θ,

P ^{m} _{\ell} (\cos \theta) are solutions of

$$\backslash frac\; +\; \backslash cot\; \backslash theta\; \backslash frac\; +\; \backslash left[\backslash lambda\; -\; \backslash frac\{m^2\}\{\backslash sin^2\backslash theta\}\backslash right]\backslash ,y\; =\; 0\backslash ,$$

More precisely, given an integer m

\ge 0, the above equation hasnonsingular solutions only when λ = \ell(\ell +1 ) for ℓan integer ≥ m, and those solutions are proportional toP ^{m} _{\ell} (\cos \theta) .

Applications in physics: spherical harmonics

See main article: 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

\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).

What makes these functions useful is that they are central to the solution of the equation

\nabla ^{2\psi} + λ \psi = 0 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the Laplacian is

$$\backslash nabla^2\backslash psi\; =\; \backslash frac\; +\; \backslash cot\; \backslash theta\; \backslash frac\; +\; \backslash csc^2\; \backslash theta\backslash frac.$$

When the partial differential equation

$$\backslash frac\; +\; \backslash cot\; \backslash theta\; \backslash frac\; +\; \backslash csc^2\; \backslash theta\backslash frac\; +\; \backslash lambda\; \backslash psi\; =\; 0$$

is solved by the method of separation of variables, one gets a φ-dependent part

\sin(m\phi) or \cos(m\phi) for integer m≥0, and an equation for the θ-dependent part

$$\backslash frac\; +\; \backslash cot\; \backslash theta\; \backslash frac\; +\; \backslash left[\backslash lambda\; -\; \backslash frac\{m^2\}\{\backslash sin^2\backslash theta\}\backslash right]\backslash ,y\; =\; 0\backslash ,$$

for which the solutions are

P ^{m} _{\ell} (\cos \theta) with \ell{\ge}m and λ = \ell(\ell +1 ) .

Therefore, the equation

$$\backslash nabla^2\backslash psi\; +\; \backslash lambda\backslash psi\; =\; 0$$

has nonsingular separated solutions only when

λ = \ell(\ell +1 ) ,and those solutions are proportional to

$$P\_\backslash ell^(\backslash cos\; \backslash theta)\backslash \; \backslash cos\; (m\backslash phi)\backslash \; \backslash \; \backslash \; \backslash \; 0\; \backslash le\; m\; \backslash le\; \backslash ell$$

and

For each choice of ℓ, there are functionsfor the various values of m and choices of sine and cosine.They are all orthogonal in both ℓ and m when integrated over thesurface of the sphere.

The solutions are usually written in terms of complex exponentials:

$$Y\_(\backslash theta,\; \backslash phi)\; =\; \backslash sqrt\backslash \; P\_\backslash ell^(\backslash cos\; \backslash theta)\backslash \; e^\backslash qquad\; -\backslash ell\; \backslash le\; m\; \backslash le\; \backslash ell.$$The functions

Y _{\ell,} (\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^[4]

$$Y\_^*(\backslash theta,\; \backslash phi)\; =\; (-1)^m\; Y\_(\backslash theta,\; \backslash phi).$$

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 typicallyof the form

$$\backslash nabla^2\backslash psi(\backslash theta,\; \backslash phi)\; +\; \backslash lambda\backslash psi(\backslash theta,\; \backslash phi)\; =\; 0,$$

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.

See also

• Angular momentum
• Gaussian quadrature
• Legendre polynomials
• Spherical harmonics
• Whipple's transformation of Legendre functions
• Laguerre polynomials
• Hermite polynomials

Notes and references

• ; Section 12.5. (Uses a different sign convention.)
• .
• ; Chapter 3.
• .
  • ; Chapter 2.
• .
  • Schach, S. R. (1973) New Identities for Legendre Associated Functions of Integral Order and Degree , Society for Industrial and Applied Mathematics Journal on Mathematical Analysis, 1976, Vol. 7, No. 1 : pp. 59–69

External links

• Associated Legendre polynomials in MathWorld
• Legendre polynomials in MathWorld
• Legendre and Related Functions in DLMF

Notes and References

1. .
2. 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)
3. Dong S.H., Lemus R., (2002), "The overlap integral of three associated Legendre polynomials", Appl. Math. Lett. 15, 541-546.
4. 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.