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Legendre polynomials In mathematics, Legendre functions are solutions to Legendre's differential equation: 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. 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 sequence of orthogonal polynomials called the Legendre polynomials. Each Legendre polynomial Pn(x) is an nth-degree polynomial. It may be expressed using Rodrigues' formula: That these polynomials satisfy the Legendre differential equation (1) follows by differentiating (n+1) times both sides of the identity and employing the general Leibniz rule for repeated differentiation.[1] The Pn can also be defined as the coefficients in a Taylor series expansion:[2] . In physics, this generating function is the basis for multipole expansions. 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 P0 and P1, allows the Legendre Polynomials to be generated recursively. The first few Legendre polynomials are: n 0 1 2 3 4 5 6 7 8 9 10 The graphs of these polynomials (up to n = 5) are shown below: 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: (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, where the Legendre polynomials are eigenfunctions of a Hermitian differential operator: 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 Legendre[3] as the coefficients in the expansion of the Newtonian potential where r and r' are the lengths of the vectors and respectively and γ is the angle between those two vectors. The series converges when r > r'. 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 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 zenith angle), the solution for the potential will be and are to be determined according to the boundary condition of each problem[4]. 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): which arise naturally in multipole expansions. The left-hand side of the equation is the generating function for the Legendre polynomials. 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 (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 expansion. 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 [2] 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 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 transformation) is chosen such that it bijectively maps 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' formula for the shifted Legendre polynomials is The first few shifted Legendre polynomials are: n 0 1 1 2x − 1 2 6x2 − 6x + 1 3 20x3 − 30x2 + 12x − 1 Legendre functions of fractional order Legendre functions 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 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 function P0n.

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posted: | 12/2/2011 |

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