In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
,
Legendre functions are solutions to
Legendre's differential equation:
They are named after
Adrien-Marie LegendreAdrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
. This
ordinary differential equationA differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders...
is frequently encountered in
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
and other technical fields. In particular, it occurs when solving
Laplace's equationIn mathematics, 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:where ∆ = ∇² is the Laplace operator and \varphi is a scalar function...
(and related
partial differential equationIn mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
s) in spherical coordinates.
The Legendre differential equation may be solved using the standard
power series method. The equation has
regular singular points at
x = ±1 so, in general, a series solution about the origin will only converge for |
x| < 1. When
n is an integer, the solution
Pn(
x) that is regular at
x = 1 is also regular at
x = −1, and the series for this solution terminates (i.e. is a polynomial).
These solutions for
n = 0, 1, 2, ... (with the normalization
Pn(1) = 1) form a
polynomial sequenceIn mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial...
of
orthogonal polynomialsIn 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 polynomials, the Chebyshev polynomials, and the...
called the
Legendre polynomials. Each Legendre polynomial
Pn(
x) is an
nth-degree polynomial. It may be expressed using
Rodrigues' formulaIn mathematics, Rodrigues's formula is a formula for Legendre polynomials independently introduced by , and...
:
That these polynomials satisfy the Legendre differential equation follows by differentiating (
n+1) times both sides of the identity
and employing the general Leibniz rule for repeated differentiation. The
Pn can also be defined as the coefficients in a
Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
expansion:
.
In physics, this
generating functionIn mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
is the basis for
multipole expansionA multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
s.
Recursive Definition
Expanding the Taylor series in equation (1) for the first two terms gives
for the first two Legendre Polynomials. To obtain further terms without resorting to direct expansion of the Taylor series, equation (1) is differentiated with respect to t on both sides and rearranged to obtain
Replacing the quotient of the square root with its definition in (1), and equating the coefficients of powers of t in the resulting expansion gives
Bonnet’s recursion formula
This relation, along with the first two polynomials
and
, allows the Legendre Polynomials to be generated recursively.
The first few Legendre polynomials are:
The graphs of these polynomials (up to
n = 5) are shown below:
Orthogonality
An important property of the Legendre polynomials is that they are orthogonal with respect to the
L2 inner productIn mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
on the interval −1 ≤
x ≤ 1:
(where δ
mn denotes the
Kronecker delta, equal to 1 if
m =
n and to 0 otherwise).
In fact, an alternative derivation of the Legendre polynomials is by carrying out the Gram-Schmidt process on the polynomials {1,
x,
x2, ...} with respect to this inner product. The reason for this orthogonality property is that the Legendre differential equation can be viewed as a Sturm–Liouville problem, where the Legendre polynomials are
eigenfunctionIn mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has...
s of a Hermitian
differential operatorIn mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation, accepting a function and returning another .This article considers only linear operators,...
:
where the eigenvalue λ corresponds to
n(
n + 1).
Applications of Legendre polynomials in physics
The Legendre polynomials were first introduced in 1782 by
Adrien-Marie LegendreAdrien-Marie Legendre was a French mathematician.The Moon crater Legendre is named after him.- Life :...
as the coefficients in the expansion of the
Newtonian potentialIn mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory...
where
and
are the lengths of the vectors
and
respectively and
is the angle between those two vectors. The series converges when
. The expression gives the
gravitational potential associated to a point mass or the Coulomb potential associated to a point charge. The expansion using Legendre polynomials might be useful, for instance, when integrating this expression over a continuous mass or charge distribution.
Legendre polynomials occur in the solution of Laplace equation of the
potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
,
, in a charge-free region of space, using the method of
separation of variablesIn mathematics, separation of variables 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 on a different side of the equation....
, where the boundary conditions have axial symmetry (no dependence on an
azimuthal angle). Where
is the axis of symmetry and
is the angle between the position of the observer and the
axis (the zenith angle), the solution for the potential will be
and
are to be determined according to the boundary condition of each problem.
They also appear when solving Schrödinger equation in three dimensions for a central force.
Legendre polynomials in multipole expansions
Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently):
which arise naturally in
multipole expansionA multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
s. The left-hand side of the equation is the
generating functionIn mathematics, a generating function is a formal power series in one indeterminate, whose coefficients encode information about a sequence of numbers an that is indexed by the natural numbers. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general...
for the Legendre polynomials.
As an example, the
electric potentialIn classical electromagnetism, the electric potential at a point within a defined space is equal to the electric potential energy at that location divided by the charge there...
(in spherical coordinates) due to a point charge located on the
z-axis at
(Figure 2) varies like
If the radius
r of the observation point
P is
greater than
a, the potential may be expanded in the Legendre polynomials
where we have defined
η =
a/
r < 1 and
x = cos
θ. This expansion is used to develop the normal
multipole expansionA multipole expansion is a mathematical series representing a function that depends on angles — usually the two angles on a sphere. These series are useful because they can often be truncated, meaning that only the first few terms need to be retained for a good approximation to the original...
.
Conversely, if the radius
r of the observation point
P is
smaller than
a, the potential may still be expanded in the
Legendre polynomials as above, but with
a and
r exchanged.
This expansion is the basis of interior multipole expansion.
Additional properties of Legendre polynomials
Legendre polynomials are symmetric or antisymmetric, that is
Since the differential equation and the orthogonality property are
independent of scaling, the Legendre polynomials' definitions are
"standardized" (sometimes called "normalization", but note that the
actual norm is not unity) by being scaled so that
The derivative at the end point is given by
As discussed above, the Legendre polynomials obey the three term recurrence relation known as Bonnet’s recursion formula
and
Useful for the integration of Legendre polynomials is
From the above one can see also that
or equivalently
where
is the norm over the interval −1 ≤ x ≤ 1
From Bonnet’s recursion formula one obtains by induction the explicit representation
Shifted Legendre polynomials
The
shifted Legendre polynomials are defined as
. Here the "shifting" function
(in fact, it is an
affine transformationIn geometry, an affine transformation or affine map or an affinity is a transformation which preserves straight lines. It is the most general class of transformations with this property...
) is chosen such that it
bijectively mapsA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
the interval [0, 1] to the interval [−1, 1], implying that the polynomials
are orthogonal on [0, 1]:
An explicit expression for the shifted Legendre polynomials is given by
The analogue of
Rodrigues' formulaIn mathematics, Rodrigues's formula is a formula for Legendre polynomials independently introduced by , and...
for the shifted Legendre polynomials is
The first few shifted Legendre polynomials are:
Legendre functions of fractional order
Legendre functions of fractional order exist and follow from insertion of fractional derivatives as defined by
fractional calculusFractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation operator.and the integration operator J...
and non-integer
factorialIn mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n...
s (defined by the
gamma functionIn mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
) into the
Rodrigues' formulaIn mathematics, Rodrigues's formula is a formula for Legendre polynomials independently introduced by , and...
. The resulting functions continue to satisfy the Legendre differential equation throughout (−1,1), but are no longer regular at the endpoints. The fractional order Legendre function
Pn agrees with the associated Legendre polynomial
P.
See also
- Associated Legendre functions
- 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....
- Gegenbauer polynomials
In mathematics, Gegenbauer polynomials or ultraspherical polynomials C are orthogonal polynomials on the interval [−1,1] with respect to the weight function α–1/2. They generalize Legendre polynomials and Chebyshev polynomials, and are special cases of Jacobi polynomials...
- Legendre rational functions
- Turán's inequalities
In mathematics, Turán's inequalities are some inequalities for Legendre polynomials found by . There are many generalizations to other polynomials, often called Turán's inequalities, given by and other authors....
- Legendre wavelet
Compactly supported wavelets derived from Legendre polynomials are termed spherical harmonic or Legendre wavelets. Legendre functions have widespread applications in which spherical coordinate system are appropriate. As with many wavelets there is no nice analytical formula for describing these...
- Jacobi polynomials
In mathematics, Jacobi polynomials are a class of classical orthogonal polynomials. They are orthogonal with respect to the weight ^\alpha ^\beta on the interval [-1, 1]...
- Spherical Harmonics
In mathematics, spherical harmonics are the angular portion of a set of solutions to Laplace's equation. Represented in a system of spherical coordinates, Laplace's spherical harmonics Y_\ell^m are a specific set of spherical harmonics that forms an orthogonal system, first introduced by Pierre...
External links