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Topic: Legendre form

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Elliptic integral

Carlson symmetric form

Elliptic function

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	Adrien-Marie Legendre
		Legendre's researches connected with the gamma function are of importance, and are well known; the subject was also treated by Carl Friedrich Gauss in his memoir Disquisitiones Generales Circa Series Infinitas (1816), but in a very different manner.
		Legendre's theorem is a fundamental one in geodesy, and his contributions to the subject are of the greatest importance.
		Legendre's name is most widely known on account of his Eléments de Géométrie, the most successful of the numerous attempts that have been made to supersede Euclid as a text-book on geometry.
	www.nndb.com /people/891/000093612 (1750 words)

	Form
		Carlson symmetric form In mathematics, the Carlson symmetric forms of elliptic integrals, RC(x,y), RD(x,y,z), RF(x,y,z)...
		Form (biology) In biology a form is a trinomial nomenclature notation is: Genusname speciesname forma formname.
		Legendre form In mathematics, the Legendre forms of elliptic integrals, F(φ,k), E(φ,k) and P(φ,k,n) are def...
	www.brainyencyclopedia.com /topics/form.html (1298 words)

	Adrien-Marie Legendre Summary
		Legendre's interest in celestial mechanics eventually led to two further papers, one on the attraction of certain ellipsoids, and the other on the form and density of fluid planets.
		Legendre succeeded Laplace as the examiner in mathematics of students assigned to the artillery in 1799, a position he held until 1815.
		Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely.
	www.bookrags.com /Adrien-Marie_Legendre (2649 words)

	Adrian Marie Legendre (1752 - 1833)
		Legendre's analysis is of a high order of excellence, and is second only to that produced by Lagrange and Laplace, though it is not so original.
		The second memoir was communicated in 1784, and is on the form of equilibrium of a mass of rotating liquid which is approximately spherical.
		Legendre's investigations had commenced with a paper written in 1786 on elliptic arcs, but here and in his other papers he treated the subject merely as a problem in the integral calculus, and did not see that it might be considered as a higher trigonometry, and so constitute a distinct branch of analysis.
	www.maths.tcd.ie /pub/HistMath/People/Legendre/RouseBall/RB_Legendre.html (1138 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-gap.dcs.st-and.ac.uk /~history/Biographies/Legendre.html (1850 words)

	Legendre.htm
		Legendre's major work on elliptic functions in Exercises du Calcul Intégral appeared in three volumes in 1811, 1817, and 1819.
		In the first volume Legendre introduced basic properties of elliptic integrals and also of beta and gamma functions.
		Legendre polynomials form an orthonormal basis for the vector space of polynomials.
	www.cse.ohio-state.edu /~brinkmei/math/Legendre.htm (136 words)

	Legendre polynomials - Wikipedia, the free encyclopedia
		Note: The term Legendre polynomials is sometimes used (wrongly) to indicate the associated Legendre polynomials.
		The Legendre differential equation may be solved using the standard power series method.
		An important property of the Legendre polynomials is that they are orthogonal with respect to the L
	en.wikipedia.org /wiki/Legendre_polynomials (249 words)

	PlanetMath: elliptic integrals and Jacobi elliptic functions
		The first three functions are known as Legendre's form of the incomplete elliptic integrals of the first, second, and third kinds respectively.
		The latter three are known as Jacobi's form of those integrals.
		has a “closed form” in terms of the above incomplete elliptic integrals, together with elementary functions and their inverses.
	planetmath.org /encyclopedia/JacobianEllipticFunction.html (236 words)

	Reciprocity Laws. Rule of Quadratic Reciprocity - Numericana
		Modulo a prime number, the residues form a field (a finite ring without divisors of zero is necessarily a field).
		Although special cases had been noted by Euler and Lagrange, the fully general theorem is credited to Legendre, who devised a special notation to express it.
		The Legendre symbol was introduced specifically to stress the nice symmetrical relationship between (mn) and (nm) when m and n are both odd primes.
	home.att.net /~numericana/answer/reciprocity.htm (1523 words)

	Legendre
		Legendre polynomials In mathematics, Legendre functions are solutions to Legendre's differential equation : References M...
		Legendre symbol The Legendre symbol is used by quadratic residues.
		Legendre transformation Legendre transformations are used in Lagrangian mechanics.
	www.brainyencyclopedia.com /topics/legendre.html (115 words)

	Gaussian Quadrature
		The simplest form of Gaussian Integration is based on the use of an optimally chosen polynomial to approximate the integrand f(t) over the interval [-1,+1].
		The simplest form uses a uniform weighting over the interval, and the particular points at which to evaluate f(t) are the roots of a particular class of polynomials, the Legendre polynomials, over the interval.
		Thus the carefully designed choice of function evaluation points in the Gauss-Legendre form results in the same accuracy for about half the number of function evaluations, and thus at about half the computing effort.
	pathfinder.scar.utoronto.ca /~dyer/csca57/book_P/node44.html (634 words)

	SLATEC
		DSLUI4-D Routine to solve a system of the form (LDU)' X = B, where L is a unit lower triangular matrix, D is a diagonal matrix, and U is a unit upper triangular matrix and ' denotes transpose.
		REBAK-S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC or REDUC2.
		REBAKB-S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC2.
	www.glue.umd.edu /~NSW/ench250/slatec.htm (13660 words)

	PSTSWM Platform Comparision I
		A: We use one-dimensional parallelizations of the form 8x1 or 1x8, 16x1 or 1x16, and 32x1 or 1x32, where the first decomposition in each pair is for examining parallel Fourier transform algorithms, and the second is for examining parallel Lengendre transform algorithms.
		The "larger" dimension is used to examine the parallel Fourier or Legendre transform options, while a fixed parallel algorithm is used for the "smaller" dimension.
		The parallel Legendre transforms are paired with either dfft using (1,6) or swtrans using a nonoverlap implementation.
	www.csm.ornl.gov /~worley/studies/platformsI.html (1785 words)

	SLATEC Table of Contents
		Form eigenvectors from eigenvalues BANDV -S Form the eigenvectors of a real symmetric band matrix associated with a set of ordered approximate eigenvalues by inverse iteration.
		REBAK -S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC or REDUC2.
		REBAKB -S Form the eigenvectors of a generalized symmetric eigensystem from the eigenvectors of derived matrix output from REDUC2.
	sdphca.ucsd.edu /slatec_source/toc.htm (11918 words)

	Boundary(1)
		(b) Using the solutions to Laplace's equation in spherical coordinates for the case of azimuthal symmetry, write the general form of the potential inside and outside of the sphere.
		The sphere is coated with an insulating material of relative dielectric constant, K, and the combined radius of the sphere and dielectric layer is b.
		Use an expansion in Legendre polynomials to find the potential for a point P located within the dielectric layer.
	electron6.phys.utk.edu /phys594/archives/e&m/Boundary/boundary1.htm (386 words)

	Elliptic integral - Wikipedia, the free encyclopedia
		In integral calculus, elliptic integrals originally arose in connection with the problem of giving the arc length of an ellipse and were first studied by Fagnano and Leonhard Euler.
		Besides the forms given below, the elliptic integrals may also be expressed in Legendre form and Carlson symmetric form.
		Additional insight into the theory of the indefinite integral may be gained through the study of the Schwarz-Christoffel mapping.
	en.wikipedia.org /wiki/Elliptic_integral (637 words)

	Arithmetic Geometry Report
		Other areas, such as modular forms and p-adic methods, were given lesser prominence, because of recent programmes at the IAS in Princeton and Centre Emile Borel in Paris that were devoted to them.
		Q form a finitely generated abelian group: this is a famous theorem of Mordell.
		Legendre's result was generalized to quadratic equations in any number of variables over Q and over arbitrary number fields by Minkowski and Hasse, respectively.
	www.newton.cam.ac.uk /reports/9798/amg.html (5177 words)

	Elliptic integral (Site not responding. Last check: 2007-10-19)
		In integral calculus, an elliptic integral is any function f which can be expressed in the form
		Or in form of integral of sine, when 0 ≤ k ≤ 1
		The incomplete elliptic integral of the first kind F is defined, in Jacobi 's form, as
	www.encyclopedia-1.com /e/el/elliptic_integral.html (194 words)

	Mathematical Methods Special Functions Legendre’s Equation and Legendre Polynomials
		The particular form given here is somewhat arbitrary, but it is consistent with most of the literature on this subject.
		Note that the choice of the specific orthogonal polynomial for a given application is often dictated by the domain of interest.
		In particular, Legendre polynomials are used extensively where the directional dependence of some quantity is treated explicitly - such as particle transport problems.
	gershwin.ens.fr /vdaniel/Doc-Locale/Cours-Mirrored/Methodes-Maths/white/math/s8/s8legd/s8legd.html (1033 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.
		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 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)

	Background: Spherical functions
		This is also the normalisation that is used in the model of Jensen and Cain (1962).
		This form is used most in geomagnetic data, as it is the form which is used in the International Geomagnetic Reference Field (see Peddie, 1982 and Langel, 1987).
		The Schmidt quasi-normalized associated Legendre functions are not completely normalized harmonics, in the sense that the average square value of P
	www.spenvis.oma.be /spenvis/help/background/magfield/legendre.html (416 words)

	Toward Symbolic Integration of Elliptic Integrals
		However, it is assumed that when all polynomials in the integrand have been factored symbolically into linear factors, the exponents of all distinct linear factors are known.
		An integral with neither limit of integration at a branch point would have to be split into two parts, doubling the number of canonical forms, and even this remedy was not always available because of divergence at both neighboring branch points.
		If the basic integrals are to be expressed in terms of Legendre's canonical forms, each of 136 formulas has to be accompanied by inequalities relating the branch points of the integrand and the interval of integration.
	www.getnet.net /~cherry/tth/jsc.html (2763 words)

	[No title] (Site not responding. Last check: 2007-10-19)
		\begin{center} \textbf{ADRIEN-MARIE LEGENDRE (1752--1833)} \end{center} \noindent Adrien-Marie Legendre (or Le Gendre) was born in Paris from a well-to-do family and got a good education in science, especially mathematics, at the Coll\`ege Mazarin (also called the Coll\`ege des Quatre Nations).
		\begin{center} \textbf{THE WORK OF LEGENDRE AND GAUSS} \end{center} \noindent The technique of combining observations on a single quantity by forming their arithmetic mean had appeared by the end of the seventeenth century (see Plackett's article in the Pearson and Kendall volume).
		It should be added that Legendre did not distinguish between errors and residuals, nor did he find the sum of squares of his calculated residuals (which of course added to zero before squaring).
	www.york.ac.uk /depts/maths/teaching/pml/hos/legendre.tex (2635 words)

	legendre_chi_function (Site not responding. Last check: 2007-10-19)
		legendre symbol legendre polynomials legendre form legendre s constant legendre crater legendre chi function transformation transformational grammar transformation problem transformation geometry...
		In mathematics, the Legendre chi function is defined as The discrete fourier transform of the Legendre chi function with respect to the...
		Â Â In mathematics, the Legendre chi function is defined as The discrete fourier transform of the Legendre chi function with respect to the...
	legendre_chi_function.networklive.org (319 words)

	GNU Scientific Library: 7.13 Elliptic Integrals
		The Legendre forms of elliptic integrals F(\phi,k), E(\phi,k) and P(\phi,k,n) are defined by,
		Further information on the Legendre forms of elliptic integrals can be found in Abramowitz and Stegun, Chapter 17.
		The Carlson symmetric forms of elliptical integrals RC(x,y), RD(x,y,z), RF(x,y,z) and RJ(x,y,z,p) are defined by,
	www.math.unizh.ch /sepp/gsl-1.4-mo/gsl-ref_68.html (278 words)

	Rational Points on Conics
		It uses an essentially factorization-free reduction algorithm where the only factorization that occurs is that of the coefficients of the reduced Legendre polynomial of the conic.
		We define three conics in Legendre form, having the same prime divisors in their discriminants, which differ only by a sign change (twisting by Sqrt(- 1)) of one of the coefficients.
		If the curve is not a reduced Legendre model, then the test is done by passing to this model for the test.
	www.umich.edu /~gpcc/scs/magma/text1049.htm (1126 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.
	people.scs.fsu.edu /~burkardt/math2070/lab_09.html (1352 words)

	Legendre Polynomials (Site not responding. Last check: 2007-10-19)
		One of the varieties of special functions which are encountered in the solution of physical problems is the class of functions called Legendre polynomials.
		From the Legendre polynomials can be generated another important class of functions for physical problems, the associated Legendre functions.
		The associated Legendre functions can be used to construct another important set of functions, the spherical harmonics.
	hyperphysics.phy-astr.gsu.edu /hbase/math/legend.html (255 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 (1384 words)

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