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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the associated Legendre polynomials are the canonical solutions of the general Legendre equation \left(1 - x^2\right) \frac P_\ell^m(x) - 2 x \frac P_\ell^m(x) + \left \ell (\ell + 1) - \frac \rightP_\ell^m(x) = 0, or equivalently \frac \left \left(1 - x^2\right) \frac P_\ell^m(x) \right+ \left \ell (\ell + 1) - \frac \rightP_\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 In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
. 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
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s when ''m'' is odd. The fully general class of functions with arbitrary real or complex values of ''ℓ'' and ''m'' are
Legendre function In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
s. In that case the parameters are usually labelled with Greek letters. The Legendre
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
is frequently encountered in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
and other technical fields. In particular, it occurs when solving
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \na ...
(and related
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
s) in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
. Associated Legendre polynomials play a vital role in the definition of
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
.


Definition for non-negative integer parameters and

These functions are denoted P_\ell^(x), where the superscript indicates the order and not a power of ''P''. Their most straightforward definition is in terms of derivatives of ordinary
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
(''m'' ≥ 0) P_\ell^(x) = (-1)^m (1-x^2)^ \frac \left( P_\ell(x) \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 : \left(1-x^2\right) \fracP_\ell(x) -2x\fracP_\ell(x)+ \ell(\ell+1)P_\ell(x) = 0. Moreover, since by
Rodrigues' formula In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by , and . The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out ...
, P_\ell(x) = \frac \ \frac\left x^2-1)^\ell\right the ''P'' can be expressed in the form P_\ell^(x) = \frac (1-x^2)^\ \frac(x^2-1)^\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 \frac (x^2-1)^ = c_ (1-x^2)^m \frac(x^2-1)^, then it follows that the proportionality constant is c_ = (-1)^m \frac , so that P^_\ell(x) = (-1)^m \frac P^_\ell(x).


Alternative notations

The following alternative notations are also used in literature: P_(x) = (-1)^m P_\ell^(x)


Closed Form

The Associated Legendre Polynomial can also be written as: P_l^m(x)=(-1)^ \cdot 2^ \cdot (1-x^2)^ \cdot \sum_^l \frac\cdot x^ \cdot \binom \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'': \int_^ P_k ^ P_\ell ^ dx = \frac\ \delta _ Where is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 & ...
. Also, they satisfy the orthogonality condition for fixed : \int_^ \fracdx = \begin 0 & \text m\neq n \\ \frac & \text m=n\neq0 \\ \infty & \text m=n=0 \end


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_\ell ^ = (-1)^m \frac P_\ell ^ (This followed from the Rodrigues' formula definition. This definition also makes the various recurrence formulas work for positive or negative .) \text\quad , m, > \ell\,\quad\text\quad P_\ell^ = 0.\, The differential equation is also invariant under a change from to , and the functions for negative are defined by P_ ^ = P_ ^,\ (\ell=1,\,2,\, \dots).


Parity

From their definition, one can verify that the Associated Legendre functions are either even or odd according to P_\ell ^ (-x) = (-1)^ P_\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 \begin P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=x \\ P_^(x)&=-(1-x^2)^ \end \begin P_^(x)&=\tfracP_^(x) \\ P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=\tfrac(3x^-1) \\ P_^(x)&=-3x(1-x^2)^ \\ P_^(x)&=3(1-x^2) \end \begin P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=\tfracP_^(x) \\ P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=\tfrac(5x^3-3x) \\ P_^(x)&=\tfrac(1-5x^)(1-x^2)^ \\ P_^(x)&=15x(1-x^2) \\ P_^(x)&=-15(1-x^2)^ \end \begin P_^(x)&=\tfracP_^(x) \\ P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=\tfracP_^(x) \\ P_^(x)&=-\tfracP_^(x) \\ P_^(x)&=\tfrac(35x^-30x^+3) \\ P_^(x)&=-\tfrac(7x^3-3x)(1-x^2)^ \\ P_^(x)&=\tfrac(7x^2-1)(1-x^2) \\ P_^(x)&= - 105x(1-x^2)^ \\ P_^(x)&=105(1-x^2)^ \end


Recurrence formula

These functions have a number of recurrence properties: (\ell-m-1)(\ell-m)P_^(x) = -P_^(x) + P_^(x) + (\ell+m)(\ell+m-1)P_^(x) (\ell-m+1)P_^(x) = (2\ell+1)xP_^(x) - (\ell+m)P_^(x) 2mxP_^(x)=-\sqrt\left _^(x)+(\ell+m)(\ell-m+1)P_^(x)\right/math> \fracP_\ell^m(x) = \frac \left P_^(x) + (\ell+m-1)(\ell+m)P_^(x) \right/math> \fracP_\ell^m(x) = \frac \left P_^(x) + (\ell-m+1)(\ell-m+2)P_^(x) \right/math> \sqrtP_\ell^m(x) = \frac1 \left (\ell-m+1)(\ell-m+2) P_^(x) - (\ell+m-1)(\ell+m) P_^(x) \right \sqrtP_\ell^m(x) = \frac \left P_^(x) - P_^(x) \right \sqrtP_\ell^(x) = (\ell-m)xP_^(x) - (\ell+m)P_^(x) \sqrtP_\ell^(x) = (\ell-m+1)P_^m(x) - (\ell+m+1)xP_\ell^m(x) \sqrt\frac(x) = \frac12 \left (\ell+m)(\ell-m+1)P_\ell^(x) - P_\ell^(x) \right (1-x^2)\frac(x) = \frac1 \left (\ell+1)(\ell+m)P_^m(x) - \ell(\ell-m+1)P_^m(x) \right (x^2-1)\frac(x) = xP_^(x) - (\ell+m)P_^(x) (x^2-1)\frac(x) = -(\ell+1)xP_^(x) + (\ell-m+1)P_^(x) (x^2-1)\frac(x) = \sqrtP_^(x) + mxP_^(x) (x^2-1)\frac(x) = -(\ell+m)(\ell-m+1)\sqrtP_^(x) - mxP_^(x) Helpful identities (initial values for the first recursion): P_^(x) = - (2\ell+1) \sqrt P_^(x) P_^(x) = (-1)^\ell (2\ell-1)!! (1- x^2)^ P_^(x) = x (2\ell+1) P_^(x) with the
double factorial In mathematics, the double factorial or semifactorial of a number , denoted by , is the product of all the integers from 1 up to that have the same parity (odd or even) as . That is, :n!! = \prod_^ (n-2k) = n (n-2) (n-4) \cdots. For even , the ...
.


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 \begin \frac \int_^1 P_l^u(x) P_m^v(x) P_n^w(x) dx =&(-1)^\frac \\ &\times \ \sum_^q (-1)^t \frac \end This formula is to be used under the following assumptions: # the degrees are non-negative integers l,m,n\ge0 # all three orders are non-negative integers u,v,w\ge 0 # u is the largest of the three orders # the orders sum up u=v+w # the degrees obey m\ge n Other quantities appearing in the formula are defined as 2s = l+m+n p = \max(0,\,n-m-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+n\ge l \ge m-n Dong and Lemus (2002) generalized the derivation of this formula to integrals over a product of an arbitrary number of associated Legendre polynomials.


Generalization via hypergeometric functions

These functions may actually be defined for general complex parameters and argument: P_^(z) = \frac \left frac\right \,_2F_1 (-\lambda, \lambda+1; 1-\mu; \frac) where \Gamma is the
gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers excep ...
and _2F_1 is the hypergeometric function \,_2F_1 (\alpha, \beta; \gamma; z) = \frac \sum_^\infty\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)\,y'' -2zy' + \left(\lambda
lambda+1 Lambda (}, ''lám(b)da'') is the 11th letter of the Greek alphabet, representing the voiced alveolar lateral approximant . In the system of Greek numerals, lambda has a value of 30. Lambda is derived from the Phoenician Lamed . Lambda gave ri ...
- \frac\right)\,y = 0.\, Since this is a second order differential equation, it has a second solution, Q_\lambda^(z), defined as: Q_^(z) = \frac\frac(1-z^2)^ \,_2F_1 \left(\frac, \frac; \lambda+\frac; \frac\right) P_\lambda^(z) and Q_\lambda^(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_\ell^(\cos\theta) = (-1)^m (\sin \theta)^m\ \frac\left(P_\ell(\cos\theta)\right) Using the relation (1 - x^2)^ = \sin\theta, the list given above yields the first few polynomials, parameterized this way, as: \begin P_0^0(\cos\theta) & = 1 \\ ptP_1^0(\cos\theta) & = \cos\theta \\ ptP_1^1(\cos\theta) & = -\sin\theta \\ ptP_2^0(\cos\theta) & = \tfrac (3\cos^2\theta-1) \\ ptP_2^1(\cos\theta) & = -3\cos\theta\sin\theta \\ ptP_2^2(\cos\theta) & = 3\sin^2\theta \\ ptP_3^0(\cos\theta) & = \tfrac (5\cos^3\theta-3\cos\theta) \\ ptP_3^1(\cos\theta) & = -\tfrac (5\cos^2\theta-1)\sin\theta \\ ptP_3^2(\cos\theta) & = 15\cos\theta\sin^2\theta \\ ptP_3^3(\cos\theta) & = -15\sin^3\theta \\ ptP_4^0(\cos\theta) & = \tfrac (35\cos^4\theta-30\cos^2\theta+3) \\ ptP_4^1(\cos\theta) & = - \tfrac (7\cos^3\theta-3\cos\theta)\sin\theta \\ ptP_4^2(\cos\theta) & = \tfrac (7\cos^2\theta-1)\sin^2\theta \\ ptP_4^3(\cos\theta) & = -105\cos\theta\sin^3\theta \\ ptP_4^4(\cos\theta) & = 105\sin^4\theta \end The orthogonality relations given above become in this formulation: for fixed ''m'', P_\ell^m(\cos\theta) are orthogonal, parameterized by θ over , \pi/math>, with weight \sin \theta: \int_0^\pi P_k^(\cos\theta) P_\ell^(\cos\theta)\,\sin\theta\,d\theta = \frac\ \delta _ Also, for fixed ''ℓ'': \int_0^\pi P_\ell^(\cos\theta) P_\ell^(\cos\theta) \csc\theta\,d\theta = \begin 0 & \text m\neq n \\ \frac & \text m=n\neq0 \\ \infty & \text m=n=0\end In terms of θ, P_\ell^(\cos \theta) are solutions of \frac + \cot \theta \frac + \left lambda - \frac\right,y = 0\, More precisely, given an integer ''m''\ge0, the above equation has nonsingular solutions only when \lambda = \ell(\ell+1)\, for ''ℓ'' an integer ≥ ''m'', and those solutions are proportional to P_\ell^(\cos \theta).


Applications in physics: spherical harmonics

In many occasions in
physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, associated Legendre polynomials in terms of angles occur where
spherical A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
is involved. The colatitude angle in
spherical coordinates In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' meas ...
is the angle \theta used above. The longitude angle, \phi, appears in a multiplying factor. Together, they make a set of functions called
spherical harmonic In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
s. These functions express the symmetry of the two-sphere under the action of the
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
SO(3). What makes these functions useful is that they are central to the solution of the equation \nabla^2\psi + \lambda\psi = 0 on the surface of a sphere. In spherical coordinates θ (colatitude) and φ (longitude), the
Laplacian In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
is \nabla^2\psi = \frac + \cot \theta \frac + \csc^2 \theta\frac. When the
partial differential equation In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
\frac + \cot \theta \frac + \csc^2 \theta\frac + \lambda \psi = 0 is solved by the method of
separation of variables In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs ...
, 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 \frac + \cot \theta \frac + \left lambda - \frac\right,y = 0\, for which the solutions are P_\ell^(\cos \theta) with \ellm and \lambda = \ell(\ell+1). Therefore, the equation \nabla^2\psi + \lambda\psi = 0 has nonsingular separated solutions only when \lambda = \ell(\ell+1), and those solutions are proportional to P_\ell^(\cos \theta)\ \cos (m\phi)\ \ \ \ 0 \le m \le \ell and P_\ell^(\cos \theta)\ \sin (m\phi)\ \ \ \ 0 < m \le \ell. For each choice of ''ℓ'', there are functions for the various values of ''m'' and choices of sine and cosine. They are all orthogonal in both ''ℓ'' and ''m'' when integrated over the surface of the sphere. The solutions are usually written in terms of
complex exponential The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, al ...
s: Y_(\theta, \phi) = \sqrt\ P_\ell^(\cos \theta)\ e^\qquad -\ell \le m \le \ell. The functions Y_(\theta, \phi) are the
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
, 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 identityThis 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_^*(\theta, \phi) = (-1)^m Y_(\theta, \phi). The spherical harmonic functions form a complete orthonormal set of functions in the sense of
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or '' ...
. Workers in the fields of geodesy, geomagnetism and spectral analysis use a different phase and normalization factor than given here (see
spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
). 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^2\psi(\theta, \phi) + \lambda\psi(\theta, \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 In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the addi ...
SO(3). There are many other Lie groups besides SO(3), and an analogous generalization 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 In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
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 In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
*
Gaussian quadrature In numerical analysis, a quadrature rule is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the domain of integration. (See numerical integration for mor ...
*
Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applica ...
*
Spherical harmonics In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form ...
*
Whipple's transformation of Legendre functions In the theory of special functions, Whipple's transformation for Legendre functions, named after Francis John Welsh Whipple, arise from a general expression, concerning associated Legendre functions. These formulae have been presented previously in ...
*
Laguerre polynomials In mathematics, the Laguerre polynomials, named after Edmond Laguerre (1834–1886), are solutions of Laguerre's equation: xy'' + (1 - x)y' + ny = 0 which is a second-order linear differential equation. This equation has nonsingular solutions on ...
*
Hermite polynomials In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence. The polynomials arise in: * signal processing as Hermitian wavelets for wavelet transform analysis * probability, such as the Edgeworth series, as well ...


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






Legendre and Related Functions in DLMF
{{Authority control Atomic physics Orthogonal polynomials