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Topic: Legendre polynomials

	Legendre polynomials - Wikipedia, the free encyclopedia
		The left-hand side of the equation is the generating function for the Legendre polynomials.
		Legendre polynomials are symmetric or antisymmetric, that is
		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.
	en.wikipedia.org /wiki/Legendre_polynomials (708 words)

	PlanetMath: Legendre polynomial
		The Legendre polynomials are also known as Legendre functions of the first kind.
		The Legendre functions of the second kind also satisfy the Legendre ODE but are not regular at the origin.
		This is version 10 of Legendre polynomial, born on 2005-04-22, modified 2006-10-01.
	planetmath.org /encyclopedia/LegendrePolynomials.html (193 words)

	Legendre biography
		In 1770, at the age of 18, Legendre defended his thesis in mathematics and physics at the Collège Mazarin but this was not quite as grand an achievement as it sounds to us today, for this consisted more of a plan of research rather than a completed thesis.
		Legendre's work replaced Euclid's "Elements" as a textbook in most of Europe and, in succeeding translations, in the United States and became the prototype of later geometry texts.
		Gauss was correct, but one could understand how hurtful Legendre must have found an attack on the rigour of his results by such a young man. Of course Gauss did not state that he was improving Legendre's result but rather claimed the result for himself since his was the first completely rigorous proof.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Legendre.html (1850 words)

	Legendre Polynomials (Site not responding. Last check: 2007-10-12)
		Legendre (biography) developed a set of orthogonal polynomials that can be used to approximate various functions.
		The error term is orthogonal to the first n+1 Legendre polynomials, hence it is orthogonal to all polynomials of degree n or less.
		Expand (h+e).(h+e), remembering that h is orthogonal to every polynomial of degree n or less, hence h is orthogonal to e.
	www.mathreference.com /la,legpoly.html (360 words)

	LAB #9: Legendre Polynomials
		The Legendre polyonomials are a basis for the set of polynomials, appropriate for use on the interval [-1,1].
		The Legendre polynomials form a basis for the linear space of polynomials.
		It will also be most convenient to have a "vector" version of the Legendre polynomial routine, that is, something that we can give a vector x of arguments to, and which will return the corresponding vector of values.
	www.csit.fsu.edu /~burkardt/math2070/lab_09.html (1352 words)

	Legendre Polynomials
		The weight function w(x) of the Legendre polynomials is unity, and this is what distinguishes them from the others and determines them.
		The polynomials can also be found by solving the differential equation by determining the coefficients of a power series substituted in the equation.
		The more general problem requires the introduction of related functions called the associated Legendre functions that are actually built up from Jacobi polynomials, and can also be expressed in terms of derivatives of the Legendre polynomials.
	www.du.edu /~jcalvert/math/legendre.htm (1164 words)

	Springer Online Reference Works
		The Legendre polynomials can be defined as the coefficients in the expansion of the generating function
		Legendre polynomials belong to the families of Gegenbauer polynomials; Jacobi polynomials and classical orthogonal polynomials.
		A noteworthy consequence of this interpretation is the addition formula for Legendre polynomials.
	eom.springer.de /l/l058050.htm (235 words)

	Associated Legendre polynomials - Wikipedia, the free encyclopedia
		They are a generalization of the "Legendre polynomials", but are not as general as the "Legendre functions".
		In many occasions in physics, associated Legendre polynomials in terms of angles occur where spherical symmetry is involved.
		As such, Legendre polynomials can be generalized to express the symmetries of semi-simple Lie groups and Riemannian symmetric spaces.
	en.wikipedia.org /wiki/Associated_Legendre_polynomials (856 words)

	Lulu.com - Self Publishing - Free
		The material presented here can be covered in eight to ten 2-hour classroom lectures; however, it is also written in a way that it can be used as a self study text.
		The book is intended to help students in engineering, physics and applied sciences understand various aspects of Special Functions and Orthogonal Polynomials that very often occur in engineering, physics, mathematics and applied sciences.
		Legendre Polynomials and Functions are studied in Chapter 3.
	www.lulu.com /browse/search.php?fKeywords=Legendre (419 words)

	PlanetPhysics: Legendre polynomials
		This ordinary differential equation with variable coefficients is named in honor of Adrien-Marie Legendre (1752-1833).
		While quite literally following in the footsteps of Laplace, he developed the Legendre polynomials in a paper on celestial mechanics.
		This is version 4 of Legendre polynomials, born on 2006-05-22, modified 2006-06-12.
	planetphysics.org /encyclopedia/LegendrePolynomials.html (126 words)

	Orthogonal Polynomials (Site not responding. Last check: 2007-10-12)
		are obtained from derivatives of the Legendre polynomials according to
		Section 3.2.10 discusses the generalization of Legendre polynomials to Legendre functions, which can have non-integer degrees.
		Legendre, Gegenbauer and Chebyshev polynomials can all be viewed as special cases of Jacobi polynomials.
	documents.wolfram.com /v4/MainBook/3.2.9.html (240 words)

	PlanetMath: orthogonal polynomials
		of the polynomial is not normally defined by criteria other than the quality of the approximation.
		are equal to one, the polynomials are called orthonormal.
		This is version 6 of orthogonal polynomials, born on 2002-01-04, modified 2005-04-24.
	planetmath.org /encyclopedia/OrthogonalPolynomials.html (278 words)

	LAB #9: Legendre Polynomials (Site not responding. Last check: 2007-10-12)
		In the same way, in our generalized vector space, the "length" of a vector is its L2 norm, which is the square root of its integral dot product with itself.
		Exercise 3: Using the normalized Legendre polynomials, estimate the L2-norms of:
		Plot both y and its Legendre polynomial approximation on the same graph.
	www.math.pitt.edu /~troy/math2070/lab_09.html (1475 words)

	Legendre polynomials
		Associated Legendre polynomials are the colatitudinal part of the spherical harmonics which are common to all separations of Laplace's equation in spherical polar coordinates.
		Associated polynomials have to be used when the solutions have an azimuthal component
		and m, and even then it may be only one of the two linearly independent solutions which is a polynomial, but the full implications of this situation will only become fully apparent when the spectral density is discussed.
	delta.cs.cinvestav.mx /~mcintosh/comun/complex/node58.html (287 words)

	Legendre Polynomial
		In the Sturm-Liouville Boundary Value Problem, there is an important special case called Legendre's Differential Equation which arises in numerous problems, especially in those exhibiting spherical symmetry.
		Since Legendre's differential equation is a second order ordinary differential equation, two sets of functions are needed to form the general solution.
		Recurrence Relation: A Legendre Polynomial at one point can be expressed by neighboring Legendre Polynomials at the same point.
	www.efunda.com /math/legendre/index.cfm (210 words)

	legendre.html (Site not responding. Last check: 2007-10-12)
		To approximate a function f(x) on the interval [-1,1] by a polynomial of degree n, we compute the orthogonal projection on the space spanned by Legendre polynomials L
		Since we can compute the inner product of our delta function with the Legendre polynomials, we can go ahead and compute its best approximation in each degree n, using the usual formulas.
		Here are graphs of the approximating polynomials in degrees 10 through 18, computed using Mathematica, a computer algebra system.
	math.berkeley.edu /~mhaiman/mathH54/legendre.html (230 words)

	Legendre Polynomials as a Basis for Expansion (Site not responding. Last check: 2007-10-12)
		The general prescription for expansion in a basis is illustrated by the Fourier series method.
		Due to the orthogonality and norms of the Legendre polynomials, we obtain
		Create a plot showing the step function, together with the first 6 terms of the Legendre series found in the exercise.
	www.sci.wsu.edu /idea/quantum/legendre_series.htm (167 words)

	legendre.html (Site not responding. Last check: 2007-10-12)
		This worksheet explores some of the properties of Legendre polynomials.
		The Legendre polynomials of the first kind are denoted as P(n,x), where n is the order and x is the argument; the Legendre polynomials of the second kind are LegendreQ(n,x) (these are the solutions of Legendre's equation which diverge logarithmically at x=+/-1).
		Now when we separate variables in spherical coordinates we find that the argument of the Legendre polynomials is cos(theta), where theta is the polar angle (measured from the z axis).
	www.phys.ufl.edu /~dorsey/phys6346/worksheets/legendre1.html (188 words)

	Legendre Polynomials
		Here are the first seven Legendre Polynomials, along with a plot.
		and the corresponding eigenfunctions are the Legendre polynomials LegendreP[n,x] for n=0, 1, 2,.
		By one of the major Sturm-Liouville theorems, the Legendre polynomials are orthogonal with respect to the weight p(x)=1.
	www.ma.iup.edu /projects/CalcDEMma/sturm/sl1.html (393 words)

	Fitting Data (Curve Fitting Toolbox)
		It is sometimes useful to describe a variable expressed as a function of angle in terms of Legendre polynomials
		Because the Legendre polynomials depend only on the predictor variable and constants, you use the Linear Equations pane on the Create Custom Equation GUI.
		The numerical results indicate that the odd Legendre terms do not contribute significantly to the fit, and the even Legendre terms are essentially unchanged from the previous fit.
	www.weizmann.ac.il /matlab/toolbox/curvefit/ch_fit11.html (1281 words)

	Bioline International Official Site (site up-dated regularly)
		Numerical examples are given which show the superiority of Legendre polynomials to Chebyshev polynomials.
		The so-called Canonical polynomials introduced by Lanczos(A) have hitherto been used in application to the Tau method for the solution of ordinary differential equation via Legendre polynomials and Chebyshev polynomials.
		From a computational point of view, the canonical polynomials are attractive, easily generated, uing a simple recursive relation and its associated conditional of the given problem via Legendre and Chebyshev polynomials is of great importance.
	www.bioline.org.br /request?ja05071 (756 words)

	Legendre Polynomials
		The constructions were based on a discrete set of interpolation points in the interval.
		Our final construction will use Legendre polynomials that were first studied by the French mathematician
		Compare the "discrete interpolation polynomial" and "discrete least squares approximation" using equally spaced nodes.
	math.fullerton.edu /mathews/n2003/LegendrePolyMod.html (486 words)

	Linear Methods of Applied Mathematics Legendre polynomials (Site not responding. Last check: 2007-10-12)
		We ought to do a better job of approximating Cos[Pi x/2] with a polynomial of 3rd or 4th degree.
		You may initially think of the Taylor series at x = 0, and wish regard Sin[x] as close to 0 + x + 0 - x^3/6.
		Perhaps it is surprising to learn that there are other polynomial expressions which do a much better job of approximating a function such as the sine than does the Taylor series about any point.
	www.mathphysics.com /pde/Legendrenb.html (342 words)

	GNU Scientific Library -- Reference Manual: Associated Legendre Polynomials and Spherical Harmonics
		These routines compute the associated Legendre polynomial P_l^m(x) for m >= 0, l >= m, x
		This function computes an array of Legendre polynomials P_l^m(x) for m >= 0, l = m,..., lmax, x
		These routines compute the normalized associated Legendre polynomial $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ suitable for use in spherical harmonics.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_121.html (220 words)

	Properties of Legendre Polynomials (Site not responding. Last check: 2007-10-12)
		You should NOT learn the equations in properties 3, 4, and 5 by heart, but when given them you should be able to use them.
		The arbitrary constant multiplying the polynomial solution of the equation is chosen to make
		This is particularly useful for numerical calculation of Legendre polynomials using a computer.
	www.maths.tcd.ie /~saram/2S2/P_L/node1.html (52 words)

	Appendix A: List of associated Legendre functions. Three-dimensional mathematical analysis of particle shape using ...
		The associated Legendre functions with m = -M < 0 are simply given in terms of the equivalent functions with M > 0 according to
		Table 3: List of associated Legendre polynomials from n = 0 to n = 5.
		Table 4: List of associated Legendre polynomials from n = 6 to n = 8.
	ciks.cbt.nist.gov /~garbocz/paper134/node10.html (97 words)

	40
		We then did examples and applications of Legendre polynomials in laplace equation BVPs in spherical coordinates.
		Because Legendre polynomials terminate, simplifications arise and allow an alternative algebraic approach to the eigenfunction expansion.
		This leaves only a small number of coefficients in a function's expansion in terms of Legendre polynomials.
	www.physics.emory.edu /faculty/benson/320/notes/40/40.html (621 words)

	SFU MATH 252, Vector Calculus: Spring 2005, Week 13
		In particular, section 8.2 (and 8.3) of the text by Burden and Faires (on reserve as the textbook for MACM 316) have information on orthogonal polynomials, the Gram-Schmidt process and Legendre polynomials, as well as on trigonometric polynomials and Fourier series.
		Legendre polynomials come up in many contexts in differential equations (especially partial differential equations), and it is difficult to find references that do not proceed from the DE context.
		Approximation of functions by Legendre polynomials, comparison with Taylor polynomials; trigonometric approximation, orthogonality of sines/cosines, Fourier coefficients
	www.math.sfu.ca /~ralfw/math252/week13.html (654 words)

	LEGENDRE
		The LEGENDRE function returns the value of the associated Legendre polynomial
		The associated Legendre functions are solutions to the differential equation:
		Compute the value of the associated Legendre polynomial at the same X values:
	www.astro.princeton.edu /~esirko/idl_html_help/L30.html (290 words)

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