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Topic: Legendre function

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	Legendre transformation - Wikipedia, the free encyclopedia
		Legendre transformations are used in thermodynamics to transform between the different thermodynamic potentials, and in classical mechanics to derive Hamiltonian mechanics from Lagrangian mechanics, as well as the other way around.
		For a strictly convex function the Legendre-transformation can be interpreted as the mapping between the graph of the function and the family of tangents of the graph.
		The convex conjugate of a polyhedral convex function (a convex function with polyhedral epigraph) is again a polyhedral convex function.
	en.wikipedia.org /wiki/Legendre_transformation (805 words)

	Legendre Polynomials
		The function 1/R is called the generating function of the Legendre polynomials, and can be used to investigate their properties.
		Generating functions are available for most orthogonal polynomials, but only in the Legendre case does the generating function have a clear and simple meaning.
		The functions are not orthogonal for different values of m; orthogonality of the spherical harmonics in this case depends on the φ functions.
	www.du.edu /~jcalvert/math/legendre.htm (1164 words)

	efg's Mathematics Projects -- Gauss-Legendre Quadrature
		This technique of integrand evaluation at n+1 functional values should yield the exact integral when the Integrand is a polynomial of degree 2n+1 or less.
		This is because of the use of orthogonal Legendre polynomials.
		The function "F" is passed to GaussLegendreQuadrature as the integrand of the definite integral evaluated from A to B. For a commerical package, take a look at Quadrature by Engineering Objects International.
	www.efg2.com /Lab/Mathematics/GaussLegendre.htm (185 words)

	Wolfram Research, Inc.
		The zeta function for complex arguments is central to number-theoretical studies of the distribution of primes.
		Legendre functions arise in studies of quantum-mechanical scattering processes.
		The Legendre functions, and the functions which give generalizations of other orthogonal polynomials, can be expressed in terms of the hypergeometric function.
	documents.wolfram.com /v3/MainBook/3.2.10.html (1325 words)

	dxlegf.f (Site not responding. Last check: 2007-11-05)
		Legendre function of the first kind of negative order, either C a.
		Legendre function of the second kind, either C a.
		In this case the C value of the Legendre function is contained entirely in PQA(I), C so it can be used in subsequent computations without further C consideration of extended-range arithmetic.
	www.cs.yorku.ca /~roumani/fortran/slatecAPI/dxlegf.f.html (651 words)

	Legendre Polynomials (Site not responding. Last check: 2007-11-05)
		Legendre ( biography) developed a set of orthogonal polynomials that can be used to approximate various functions.
		Given a continuous function f on a closed interval, scale the domain so that the interval becomes [-1,1].
		The error term is orthogonal to the first n+1 Legendre polynomials, hence it is orthogonal to all polynomials of degree n or less.
	www.mathreference.com /la,legpoly.html (361 words)

	IDL Help
		A user-supplied function or a built-in polynomial or ; legendre polynomial is fit to the data.
		For ; polynomials, M is equal to the degree of the polynomial + 1.
		; ; The function must be able to return R as a FLOAT vector or ; a DOUBLE vector depending on the input type of X. ; LEGENDRE: Set this keyword to use the IDL function SVDLEG in the lib ; directory to fit the data to an M element legendre polynomial.
	www.astro.washington.edu /deutsch-bin/getpro/library07.html?SVDFIT (690 words)

	Legendre Transformation (Site not responding. Last check: 2007-11-05)
		The Legendre Transformation allows to describe a function using a different set of variable.
		The Lagrangian is a function of the generalized coordinates
		Legendre Transformation are applied in many different areas of physics.
	mit.fnal.gov /~paus/8.21-IAP2001/notes/notes/node6.html (147 words)

	Untitled Document - Legendre Functions and Spherical Harmonics (Site not responding. Last check: 2007-11-05)
		The Legendre Functions and Legendre Polynomials are described in Abramowitz & Stegun, Chapter 8.
		This function computes an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ for @c{$m \ge 0$} m >= 0, @c{$l = m, \dots, lmax$} l = m,..., lmax, @c{$x
		This function computes an array of radial eigenfunctions L^{H3d}_l(\lambda, \eta) for @c{$0 \le l \le lmax$} 0 <= l <= lmax.
	www.ugcs.caltech.edu /info/gsl/specfunc_24.html (679 words)

	GNU Scientific Library -- Reference Manual - Special Functions (local)
		The Debye functions are defined by the integral D_n(x) = n/x^n \int_0^x dt (t^n/(e^t - 1)).
		The hazard function for the normal distrbution, also known as the inverse Mill's ratio, is defined as h(x) = Z(x)/Q(x) = \sqrt{2/\pi \exp(-x^2 / 2) / \erfc(x/\sqrt 2)}.
		The transport functions J(n,x) are defined by the integral representations J(n,x) := \int_0^x dt t^n e^t /(e^t - 1)^2.
	hektor.umcs.lublin.pl /~spectro/gsl-ref_7.html (6128 words)

	MATH2070: LAB 9: Legendre Polynomials and L2 Approximation (Site not responding. Last check: 2007-11-05)
		is the function to be approximated on the interval
		A discontinuous function exhibits the Gibb's phenomonon, but when convergence is measured using an integral norm it doesn't prevent convergence (although it slows it down).
		There are alternative approaches for using piecewise linears: piecewise linear functions on each interval with jumps at interval endpoints, as the piecewise constant functions have; and, piecewise linear functions that are continuous throughout whole interval.
	www.math.pitt.edu /~sussmanm/2070Fall04/lab_09 (3184 words)

	Octave Functions: L (Site not responding. Last check: 2007-11-05)
		For each element of X, compute the cumulative distribution function (CDF) at X of the Laplace distribution.
		Legendre Function of degree n and order m where all values for m = 0..N are returned.
		These routines compute the logarithm of the Gamma function, \\log(\\Gamma(x)), subject to x not a being negative integer.
	octave.sourceforge.net /index/L.html (923 words)

	Earliest Uses of Function Symbols
		The function symbol f(x) was first used by Leonhard Euler (1707-1783) in 1734 in Commentarii Academiae Scientiarum Petropolitanae (Cajori, vol.
		The use of the small letter zeta for this function was introduced by Bernhard Riemann (1826-1866) as early as 1857 (Cajori vol.
		The function is also referred to by Legendre who used the now obsolete notation E ( x)." The Gauss reference is to Theorematis arithmetici demonstratio nova.
	members.aol.com /jeff570/functions.html (1079 words)

	GNU Scientific Library -- Reference Manual - Special Functions
		The special functions are available in two calling conventions, a natural form which returns the numerical value of the function and an error-handling form which returns an error code.
		Since the arguments of the standard coupling coefficient functions are integer or half-integer, the arguments of the following functions are, by convention, integers equal to twice the actual spin value.
		The probability functions for the Normal or Gaussian distribution are described in Abramowitz & Stegun, Section 26.2.
	www.math.umn.edu /systems_guide/gsl-1.3/gsl-ref_7.html (6074 words)

	GNU Scientific Library -- Reference Manual - Associated Legendre Polynomials and Spherical Harmonics (Site not responding. Last check: 2007-11-05)
		This function computes an array of Legendre polynomials @math{P_l^m(x)} for @c{$m \ge 0$} @math{m >= 0}, @c{$l = m, \dots, lmax$} @math{l = m,..., lmax}, @c{$x
		These routines compute the normalized associated Legendre polynomial @math{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} suitable for use in spherical harmonics.
		This function computes an array of normalized associated Legendre functions @math{$\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$} for @c{$m \ge 0$} @math{m >= 0}, @c{$l = m, \dots, lmax$} @math{l = m,..., lmax}, @c{$x
	www.math.utah.edu /software/gsl/gsl-ref_121.html (259 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The Incomplete function is defined by the integral $\Gamma(a,z)=\int_{z}^{\infty} e^{-t}t^{a-1}dt$.
		The associated Legendre function $P_{\nu}^{\mu}(z)$ is defined by $P_{\nu}^{\mu}(z)=\frac{1}{\Gamma(1-\mu)} (\frac{z+1}{z-1})^{\frac{1}{2}\mu}F(-\nu, \nu+1; 1-\mu; \frac{1-z}{2})$ where $F(a, b; c;z)$ is a hypergeometric function and $1-z\lt2$.
		The hypergeometric function $F(a, b; c; z)$ is the solution of the hypergeometric differential equation $z(1-z)w^{\prime\prime}+[c-(a+b+1)z]w^{\prime}-abw=0$.
	www.csd.uwo.ca /faculty/watt/home/research/openmath/spfun1.ocd (3083 words)

	Michael Thomas Flanagan's Java Library: Class Integration
		See below ( Coding the function) for details on how to write the code for the function to be integrated.
		The mathematicl function to be integrated, f(x, a, b, c.
		The function is passed to the integration method, gaussQuad(...) as an instance of the class containing the method function(x) [func in the above example but this instance may be named at the user's choice].
	www.ee.ucl.ac.uk /~mflanaga/java/Integration.html (857 words)

	Citations: Legendre functions and the method of random Bregman projections - Bauschke, Borwein (ResearchIndex)
		Now, from the de nition of the Legendre Bregman conjugate we have (q; v) hv; L (q; v)i D (L (q; v) q) 2:22) for q 2 int( Properties A4 and A5 allow us to de ne : R m R as the continuous extension of : int( R m R, satisfying the same identity.
		....a convex function f on R n, nite at x; y 2 R n and di erentiable at y, the Bregman distance [2] between x and y, determined by f, is D f (x; y) f(x) f(y) hrf(y) x yi : Note that, by the convexity of f, the Bregman distance is always nonnegative.
		Moreover, the Legendre property is the most general condition known to date [5] that guarantees zone consistency.
	citeseer.ist.psu.edu /context/357566/357572 (2046 words)

	GNU Scientific Library -- Reference Manual: Associated Legendre Polynomials and Spherical Harmonics
		Note that this function grows combinatorially with l and can overflow for l larger than about 150.
		This function computes an array of Legendre polynomials P_l^m(x) for m >= 0, l = m,..., lmax, x
		This function computes an array of normalized associated Legendre functions $\sqrt{(2l+1)/(4\pi)} \sqrt{(l-m)!/(l+m)!} P_l^m(x)$ for m >= 0, l = m,..., lmax, x
	www.dulug.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_121.html (220 words)

	LEGENDRE (Site not responding. Last check: 2007-11-05)
		The LEGENDRE function returns the value of the associated Legendre polynomial
		The associated Legendre functions are solutions to the differential equation:
		If one argument is a scalar and the other arguments are arrays, the function uses the scalar value with each element of the arrays, and returns an array with the same dimensions as the smallest input array.
	star.pst.qub.ac.uk /idl/L30.html (298 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		) % % By dividing the function by the square root of this term, % the normalized associated Legendre functions have norm 1.
		% % Parameters: % % Input, integer N, the maximum first index of the Legendre % function, which must be at least 0.
		% % Input, integer M, the second index of the Legendre function, % which must be at least 0, and no greater than N. % Input, real X, the point at which the function is to be % evaluated.
	www.csit.fsu.edu /~burkardt/m_src/polpak/legendre_associated_normalized.m (315 words)

	[No title]
		types.nesl : definitions for data types and functions to access their ;; fields, and some constants.
		;; ;; The FMM algorithm is invoked using the function FMM(vect,level) where ;; vect is the vector of particles and level is an integer specifying the ;; number of levels that the uniform FMM tree should have.
		;; This function returns a vector of pairs (particle,force) where particle ;; is the particle that was given in the input of the type described above, ;; and force if (float,float,float) which is the force on the particle ;; in 3D.
	www.cs.cmu.edu /~scandal/nesl/algorithms/FMM.nesl (390 words)

	[No title]
		I don't know the implementation of maple's Legendre function, but if it is the same Legendre/Jacobi function that I know, it does _not_ require factorization, and should be of the same or slightly easier complexity than gcd(p,q).
		Maple's problem is in not clearly distinguishing between their function L(a,b) and Legendre's symbol (a/b), which is defined _only_ for prime b.
		Jacobi's symbol is the generalization of (a/b) to all odd b, but it loses some of the usefulness for non-prime b, since it no longer correctly identifies non-residues.
	www.math.niu.edu /~rusin/known-math/97/jacobi (1551 words)

	Special Functions
		Polylogarithm functions appear in Feynman diagram integrals in elementary particle physics, as well as in algebraic K-theory.
		is given in terms of the ordinary Struve function by
		This function appears for example in integrating cubic polynomials to arbitrary powers.
	documents.wolfram.com /v4/MainBook/3.2.10.html (1396 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The * function returns the value 0 unless the arguments are out of range, * in which case it returns 1.
		Because one of the recursion * coefficients depends on cot(t) the recursion fails when x = cos(t) * is close to 1 (sin(t) close to 0).
		* The function returns the value 0 unless the arguments nu and mu are out * of range, in which case it returns 1.
	soi.stanford.edu /technotes/91.028/source (915 words)

	GNU Scientific Library -- Reference Manual - Legendre Polynomials (Site not responding. Last check: 2007-11-05)
		These functions evaluate the Legendre polynomials @math{P_l(x)} using explicit representations for @math{l=1, 2, 3}.
		These routines compute the Legendre function @math{Q_0(x)} for @math{x > -1}, @c{$x \ne 1$} @math{x != 1}.
		These routines compute the Legendre function @math{Q_l(x)} for @math{x > -1}, @c{$x \ne 1$} @math{x != 1} and @c{$l \ge 0$} @math{l >= 0}.
	www.math.utah.edu /software/gsl/gsl-ref_120.html (141 words)

	DATAPLOT Update (Oct. 1995)
		In addition, the log-normal cdf, pdf, and ppf functions were upgraded to handle the standard deviation shape parameter (LGNCDF, LGNPDF, LGNPPF).
		These new functions are based on code from the SLATEC library.
		The maximum number of functions was increased from 100 to 200.
	stommel.tamu.edu /~baum/DATAPLOT/DATAPLOT-new.html (10423 words)

	AMS55 Online, p. 331
		Derivative of the Legendre Function-Second Kind Qk(z) ~=0(.01)1, n=0(1)3, 9.
		Derivative of the Legendre Function-Second Kind Q:(z) 2=1(.2)10, n=0(1)3.
		The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States.
	www.convertit.com /Go/AAAMA/Reference/AMS55.ASP?Res=150&Page=331 (119 words)

	[No title]
		Compute the discrete Legendre Transform of a function XLLTd.m --> Compute the discrete Legendre Transform of discrete data XLLTd2D.m --> Compute the discrete Legendre Transform of X discrete data corresponding to a bivariate X function.
		Two functions are proposed: Xfusionma.m --> use matlab syntax to speed up computations Xfusionca.m --> classical programming, affine parts are removed Xfusion.m --> switch between one of the above.
		However I do not know how Xthe sort function is coded, so I cannot say whether it has a Xlinear-time complexity.
	www.netlib.org /numeralgo/na13 (681 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.
		Generating Function : The generating function of a Legendre Polynomial is
	www.efunda.com /math/Legendre/index.cfm (213 words)

	Maxima Manual
		[specfun package] return the associated Legendre function of the first kind for integers n > -1 and m > -1.
		[specfun package] return the associated Legendre function of the second kind for integers n > -1 and m > -1.
		Many functions in specfun are computed as a special case of the Jacobi polynomials; they also enjoy the speed boost from the modedeclared version of jacobi.
	www.delorie.com /gnu/docs/maxima/maxima_55.html (520 words)

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