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Topic: Elliptic integral

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	NationMaster - Encyclopedia: Elliptic integral
		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 Giulio Fagnano and Leonhard Euler.
		The complete elliptic integral of the second kind E may be defined as or It is a special case of the incomplete elliptic integral of the second kind: Category:...
		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.
	www.nationmaster.com /encyclopedia/Elliptic_integral (0 words)

	elliptic integral standard forms@Everything2.com
		The actual method for transforming a general elliptic integral into these standard forms is actually quite involved, and requires a lot of fairly tedious algebra, but before automatic computers were widely available for performing numerical quadrature this was the best way to proceed.
		Once the integral has been thus reduced into elementary integrals and the Legendre standard forms, algorithms involving Landen's transformation and the arithmetic geometric mean are available to calculate a definite integral for the entire expression.
		Tables giving the values of the integral for various values of the amplitude (the upper limit of the integrand), the modulus, and in the case of integrals of the third kind, the parameter are also available.
	www.everything2.com /index.pl?node_id=1621468 (0 words)

	math lessons - Elliptic integral
		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.
		Elliptic integrals are often expressed as functions of a variety of different arguments.
		Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular; we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.
	www.mathdaily.com /lessons/Elliptic_integral (589 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.
		When the Jacobian elliptic functions are extended to complex arguments, they are doubly periodic and have two poles in any parallelogram of periods; both poles are simple.
		This is version 4 of elliptic integrals and Jacobi elliptic functions, born on 2003-09-30, modified 2006-12-10.
	www.planetmath.org /encyclopedia/EllipticIntegral.html (235 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.
		When the Jacobian elliptic functions are extended to complex arguments, they are doubly periodic and have two poles in any parallelogram of periods; both poles are simple.
		This is version 4 of elliptic integrals and Jacobi elliptic functions, born on 2003-09-30, modified 2006-12-10.
	planetmath.org /encyclopedia/JacobianEllipticFunction.html (235 words)

	Elliptic integral
		The incomplete elliptic integral of the first kind F is defined, in Jacobi's form, as
		Historically, elliptic functions were discovered as inverse functions of elliptic integrals, and this one in particular; we have F(sn(z;k);k) = z where sn is one of Jacobi's elliptic functions.
		Historically properties of these integrals were studied in connection with the problem of the arc length of an ellipse, by Fagnano and Leonhard Euler.
	www.xasa.com /wiki/en/wikipedia/e/el/elliptic_integral.html (198 words)

	Elliptic Functions
		Elliptic integrals were studied in detail by Euler and Legendre, but they did not considered the inverse functions.
		He was led to elliptic functions through his preceding research on algebraic equations, in particular, the ``lemniscate equations'' that Gauss considered in his book on number theory (but without any explicit comment on the relation to elliptic functions).
		This is a generalization of Euler's results on elliptic integrals, and became a prototype of the subsequent studies on algebraic curves and Riemann surfaces.
	www.math.h.kyoto-u.ac.jp /~takasaki/soliton-lab/chron/elliptic.html (737 words)

	News \| TimesDaily.com \| TimesDaily \| Florence, Alabama (AL)
		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 Giulio Fagnano and Leonhard Euler.
		Additional insight into the theory of the elliptic integral may be gained through the study of the Schwarz-Christoffel mapping.
		Elliptic integrals are often expressed as functions of a variety of different arguments.
	www.timesdaily.com /apps/pbcs.dll/section?category=NEWS&template=wiki&text=elliptic_integral (706 words)

	14H52: Elliptic Curves
		This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography).
		Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.
		Two (unstructured) equations equations in three unknowns lead to an elliptic curve (although integer points are not fully known).
	www.math.niu.edu /~rusin/known-math/index/14H52.html (750 words)

	Math Forum Discussions - Re: Elliptic integral of the seconnd kind with complex amplitude
		There is an article on elliptic integrals and elliptic functions.
		Elliptic integral of the seconnd kind with complex amplitude
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.org /kb/thread.jspa?forumID=226&threadID=1405596&messageID=4868152 (0 words)

	Elliptic integral
		An elliptic integral is any function f which can be expressed in the form
		The complete elliptic integral of the first kind K is defined as
		The complete elliptic integral of the second kind E is defined as
	www.ebroadcast.com.au /lookup/encyclopedia/el/Elliptic_integral.html (102 words)

	Toward Symbolic Integration of Elliptic Integrals
		A method is proposed by which elliptic integrals can be integrated symbolically without the kind of information about limits of integration and branch points of the integrand that is required in integral tables using Legendre's integrals.
		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)

	Elliptic Integral
		I thought elliptic integrals arose in giving the arc length of elliptic curves, which as far as I know are a lot different than ellipses.
		Elliptic integrals did, indeed, arise from trying to find the arc length of an ellipse which is NOT as simple as he seems to think.
		Elliptic functions are the inverse functions to elliptic integrals.
	www.physicsforums.com /showthread.php?t=92098 (321 words)

	Elliptic curves (Site not responding. Last check: )
		On the rank of an elliptic surface, J. Silverman.
		Beppo Levi and the arithmetic of elliptic curves, by Norbert Schappacher.
		On an elliptic analogue of Zagier's conjecture, Jörg Wildeshaus.
	www.fermigier.com /fermigier/elliptic.html.en (746 words)

	Elliptic Curves and Elliptic Functions
		This mapping is, in effect, a parameterization of the elliptic curve by points in a "fundamental parallelogram" in the complex plane.
		Classically, such doubly periodic functions were called elliptic functions, since they occurred in the elliptic integrals which represent the arc length of an ellipse.
		Further, the definition of an elliptic curve requires that there are no repeated roots of the polynomial in x, and this may fail to be true when reducing mod p for some primes.
	cgd.best.vwh.net /home/flt/flt03.htm (3513 words)

	PlanetMath: computation of surface area of portion of paraboloid
		Since the quartic has distinct roots, it is not possible to evaluate this integral in tems of elementary functions.
		In many ways, the complexities of the present calculation are typical of what happens when one attempts to compute even the simplest of integrals with respect to area explicitly.
		Because of the square root which appears in the fomulae for reducing integrals with respect to surface area, one is faced with the evaluation of a double integral involving a square root.
	planetmath.org /encyclopedia/Example8OfIntegrationWithRespectToSurfaceArea.html (874 words)

	i/elliptic
		The yorick elliptic functions in terms of M may need to be written ell_am(u,k^2) or ell_am(u,sin(alpha)^2) in order to agree with the definitions in other references.
		The exceptions are the complete elliptic integrals ellip_k and ellip_e which accept an array of M values.
		Note that the function ellip_k is infinite for M=1 and for large negative M. The "natural" range for M is 0<=M<=1; all other real values can be "reduced" to this range by various transformations; the logarithmic singularity of ellip_k is actually very mild, and other functions such as ell_am are perfectly well-defined there.
	yorick.sourceforge.net /html_i/elliptic_i.php (613 words)

	WWU Math Department - Colloquium
		Variants of this integral are called elliptic integrals and occur in a variety of physical and mathematical contexts.
		From one of these contexts --- the period of a simple pendulum --- we can derive an elliptic integral whose inverse can be regarded as a doubly periodic function on the complex plane.
		In this talk, I will explore the relationship between elliptic integrals and elliptic functions, show direct methods for constructing elliptic functions as well as deriving some of their simple properties, and discuss the relationship between elliptic functions and results in number theory.
	www.wwu.edu /depts/math/colloquium/c_042706.html (136 words)

	TU Delft - Research
		Current work focusses on extending the stochastic integral to stochastic integrands and on exploring connections between stochastic integration in Banach spaces and the geometry of the underlying space.
		For second order elliptic equations a so-called maximum principle exists which implies that a positive source term yields that the solution is positive.
		We want to describe the irreducible representations, study direct sum of direct integral decompositions of natural representations or tensor products, determine explicitly the matrix elements in terms of special functions, etc. Usually for compact (dynamical) quantum groups and finite dimensional representations the problems can be dealt with in a more or less algebraic way.
	www.tudelft.nl /live/pagina.jsp?id=915f3a54-cf96-40cc-ab77-bc982abb914d&lang=en (660 words)

	Jacobian Elliptic Functions
		Later, the brilliant and ingenious Gauss conceived of inverting the functions defined by incomplete elliptic integrals, unlocking a treasure chest of analytical investigations using the new methods of complex variables.
		The invention of elliptic functions is shared with C. Jacobi and Abel, who published their investigations around 1827, though Gauss knew many of the results as early as 1809.
		In the definite integral, the variable of integration is a dummy variable, in that which letter is used to denote it is immaterial.
	www.du.edu /%7Ejcalvert/math/jacobi.htm (2457 words)

	Elliptic functions
		The study of elliptical integrals can be said to start in 1655 when Wallis began to study the arc length of an ellipse.
		This integral, which is clearly satisfies the above definition so is an elliptic integral, became known as the lemniscate integral.
		The other good features of the lemniscate integral are the fact that it is general enough for many of its properties to be generalised to more general elliptic functions, yet the geometric intuition from the arc length of the lemniscate curve aids understanding.
	www-groups.dcs.st-and.ac.uk /~history/HistTopics/Elliptic_functions.html (0 words)

	Wolfram Research, Inc.
		for a discussion of argument conventions for elliptic integrals.
		This is the definition of the elliptic integral of the second kind.
		elliptic integral of the second kind, a particular case of the hypergeometric function.
	documents.wolfram.com /v3/RefGuide/EllipticE.html (91 words)

	L25.html
		An elliptic curve is the solution set to the bivariate polynomial equation f(x,y)=0, where f(x,y) is of total degree 3 or 4, and f(x,y) is irreducible.
		An elliptic curve is in standard form if its equation may be written as y^2=h(x), where h(x) is a cubic polynomial in x.
		This tangent line would also intersect our elliptic curve in one other point (x3,y3), and once once again we reflect this point in the x-axis and define the elliptic curve sum of q1 and q1 to be the point q3=(x3,-y3).
	www.math.sfu.ca /~gfee/Math342/L251.html (758 words)

	Elliptic Arcs
		It must be considered as defining a new function, the elliptic integral of the second kind.
		Tables of the complete elliptic integral are easy to find, for example in Dwight (see References).
		They are tools for the numerical evaluation of integrals, of practical importance at least.
	www.du.edu /%7Ejcalvert/math/ellarc.htm (941 words)

	info: Elliptic_integral (Site not responding. Last check: )
		Elliptic Integrals: An elliptic integral is an integral involving a rational function which...
		Elliptic Integral of the First KindElliptic Integral of the First Kind...
		Elliptic Integrals and Elliptic FunctionsAny elliptic integral can be expressed in terms of the three standard kinds of Legendre-Jacobi elliptic integrals.
	www.napoli-pizza.net /Elliptic_integral.html (946 words)

	Elliptic Functions (Site not responding. Last check: )
		There are three basic forms of elliptic integrals, only two of which are implemented in this research at this time.
		Elliptic functions are defined as the inverses of functions developed by elliptic integrals.
		The amplitude of Jacobi elliptic functions is defined as the inverse of the elliptic integral of the first kind.
	www.phy.davidson.edu /StuHome/timv/SUSY/Jacobi/Jacobi.htm (145 words)

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