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In mathematics, Legendre functions are solutions to Legendre's differential equation: Euclid, Greek mathematician, 3rd century BC, known today as the father of geometry; shown here in a detail of The School of Athens by Raphael. ...
![{d over dx} left[ (1-x^2) {d over dx} P(x) right] + n(n+1)P(x) = 0.](http://upload.wikimedia.org/math/9/7/4/97426c1b358960a3da8430c9d4b3e931.png) They are named after Adrien-Marie Legendre. 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. Adrien-Marie Legendre (September 18, 1752 - January 10, 1833) was a French mathematician. ...
An illustration of a differential equation. ...
Physics (from the Greek, (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and understanding of the fundamental laws which govern matter, energy, space and time. ...
In mathematics, Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...
In mathematics, a partial differential equation (PDE) is a relation involving an unknown function of several independent variables and its partial derivatives with respect to those variables. ...
This article describes some of the common coordinate systems that appear in elementary mathematics. ...
The Legendre differential equation may be solved using the standard power series method. The solution is finite (i.e. the series converges) provided |x| < 1. Furthermore, it is finite at x = ± 1 provided n is a non-negative integer, i.e. n = 0, 1, 2,... . In this case, the solutions form a polynomial sequence of orthogonal polynomials called the Legendre polynomials. In mathematics, a power series (in one variable) is an infinite series of the form where the coefficients an, the center c, and the argument x are usually real or complex numbers. ...
The integers are commonly denoted by the above symbol. ...
In 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. ...
In mathematics, an orthogonal polynomial sequence is an infinite sequence of polynomials p0(x), p1(x), p2(x) ... , in which each pn(x) has degree n, and such that any two different polynomials in the sequence are orthogonal to each other in the following sense: One can define an inner...
Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula: In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...
![P_n(x) = {1 over 2^n n!} {d^n over dx^n } left[ (x^2 -1)^n right].](http://upload.wikimedia.org/math/7/0/a/70a1790c4fb6b05d9ccea331b0dabe51.png)
The orthogonality property An important property of the Legendre polynomials is that they are orthogonal with respect to the L2 inner product on the interval −1 ≤ x ≤ 1: In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ...
In mathematics, the Lp and lp spaces are spaces of p-power integrable functions, and corresponding sequence spaces. ...
 (where δmn 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 In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ...
In mathematics and numerical analysis, the Gram-Schmidt process of linear algebra is a method of orthogonalizing a set of vectors in an inner product space, most commonly the Euclidean space Rn. ...
In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form where the functions p(x), q(x), and w(x) are specified at the outset...
![{d over dx} left[ (1-x^2) {d over dx} P(x) right] = -lambda P(x),](http://upload.wikimedia.org/math/b/9/3/b93b0519eba36d1149a5ad46e123f14d.png) where the eigenvalue λ corresponds to n(n+1).
Examples of Legendre polynomials These are the first few Legendre polynomials: The graphs of these polynomials (up to n = 5) are shown below: Image File history File links Legendre_poly. ...
Applications of Legendre polynomials in physics Legendre polynomials are useful in expanding functions like  where r and r' are the lengths of the vectors  and  respectively and γ is the angle between those two vectors. This expansion hold where r > r'. This expression is used, for example, to obtain the potential of a point charge, felt at point while the charge is located at point  . The expansion using Legendre polynomials might be useful when integrating this expression over a continuous charge distribution. This page is a candidate for speedy deletion. ...
Legendre polynomials occur in the solution of Laplace equation of the potential,  , in a charge-free region of space, using the method of separation of variables, where the boundary conditions have axial symmetry (no dependence on an azimuthal angle). Where  is the axis of symmetry and θ is the angle between the position of the observer and the axis, the solution for the potential will be Laplaces equation is a partial differential equation named after its discoverer Pierre-Simon Laplace. ...
Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ...
In mathematics, separation of variables is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to re-write an equation so that each of two variables occurs on a different side of the equation. ...
Azimuth is the horizontal component of a direction (compass direction), measured around the horizon toward the East, i. ...
![Phi(r,theta)=sum_{ell=0}^{infty} left[ A_ell r^ell + B_ell r^{-(ell+1)} right] P_ell(costheta).](http://upload.wikimedia.org/math/d/3/a/d3a683c19b5e5e533d574488ec388893.png)  and  are to be determined according to the boundary condition of each problem[1].
Legendre polynomials in multipole expansions
 Figure 2 Legendre polynomials are also useful in expanding functions of the form (this is the same as before, written a little differently): Image File history File links Point_axial_multipole. ...
 which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials. This article is in need of attention from an expert on the subject. ...
In mathematics a generating function is a formal power series whose coefficients encode information about a sequence an that is indexed by the natural numbers. ...
As an example, the electric potential Φ(r,θ) (in spherical coordinates) due to a point charge located on the z-axis at z = a (Fig. 2) varies like Electric potential is the potential energy per unit of charge associated with a static (time-invariant) electric field, also called the electrostatic potential, typically measured in volts. ...
This article describes some of the common coordinate systems that appear in elementary mathematics. ...
This page is a candidate for speedy deletion. ...
 If the radius r of the observation point P is much greater than a, the potential may be expanded in the Legendre polynomials  where we have defined η = a / r < 1 and x = cosθ. This expansion is used to develop the normal multipole expansion. This article is in need of attention from an expert on the subject. ...
Conversely, if the radius r of the observation point P is much 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  Legendre polynomials can be constructed using the three term recurrence relations  and  Useful for the integration of Legendre polynomials is ![(2n+1) P_n = {d over dx} left[ P_{n+1} - P_{n-1} right].](http://upload.wikimedia.org/math/f/9/5/f952fd5ea27624b5b8f54f68d7caca68.png)
Shifted Legendre polynomials The shifted Legendre polynomials  are defined as being orthogonal on the unit interval [0,1]  An explicit expression for these polynomials is given by  The analogue of Rodrigues' formula for the shifted Legendre polynomials is: In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...
![tilde{P_n}(x) = ( n!)^{-1} {d^n over dx^n } left[ (x^2 -x)^n right].,](http://upload.wikimedia.org/math/b/7/4/b74d56e4dd062b9b3fb0267aa33f4aa4.png) The first few shifted Legendre polynomials are: | n | | | 0 | 1 | | 1 | 2x − 1 | | 2 | 6x2 − 6x + 1 | | 3 | 20x3 − 30x2 + 12x − 1 |
Legendre polynomials of fractional order Legendre polynomials of fractional order exist and follow from insertion of fractional derivatives as defined by fractional calculus and non-integer factorials (defined by the gamma function) into the Rodrigues' formula. The exponents of course become fractional exponents which represent roots. In mathematics, fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers of the differential operator and the integration operator J. (Usually not I, to avoid confusion with other I-like glyphs, or identities; but J must not be confused with Bessel functions...
The beginning of the sequence of factorials (sequence A000142 in OEIS) In mathematics, the factorial of a natural number n is the product of all positive integers less than or equal to n. ...
The Gamma function along part of the real axis In mathematics, the Gamma function extends the factorial function to complex and non integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). ...
In mathematics, two polynomials f and g are orthogonal to each other with respect to a nonnegative weight function w precisely if In other words, if polynomials are treated as vectors and the inner product of two polynomials f(x) and g(x) is defined as then the orthogonal polynomials...
In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ...
See also 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. ...
In mathematics, the associated Legendre polynomials, named after Adrien-Marie Legendre, are defined by: These differ from the Legendre polynomials. ...
Plot of the Legendre rational functions for n=0,1,2 and 3 for x between 0. ...
External links
References - ^ Jackson, J.D. Classical Electrodynamics, 3rd edition, Wiley & Sons, 1999. page 103
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