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# Topic: Elliptic function

###### In the News (Thu 23 May 19)

 Elliptic function - Wikipedia, the free encyclopedia Historically, elliptic functions were discovered as inverse functions of elliptic integrals; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. More generally, the study of elliptic functions is closely related to the study of modular functions and modular forms, examples of which include the j-invariant, the Eisenstein series and the Dedekind eta function. The derivative of an elliptic function is again an elliptic function, with the same periods. en.wikipedia.org /wiki/Elliptic_function   (718 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 Giulio 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. en.wikipedia.org /wiki/Elliptic_integral   (655 words)

 elliptic function The elliptic functions can be thought of as analogs of the trigonometric functions (which have only a single period). Leading eighteenth-century mathematicians, including Leonhard Euler and Joseph Lagrange, had studied elliptic integrals, such as the integral that gives the arc length of an ellipse; however, these cannot be expressed in terms of the elementary functions (polynomials, exponentials, and trigonometric functions). It was the insight of Karl Jacobi, and also of Karl Gauss and Niels Abel, that the inverse functions of elliptic integrals are much easier to study. www.daviddarling.info /encyclopedia/E/elliptic_function.html   (228 words)

 Weierstrass's elliptic functions - Wikipedia, the free encyclopedia One is as a function of a complex variable z and a lattice Λ in the complex plane. Further development of the theory of elliptic functions shows that the condition on Weierstrass's function (correctly called pe) is determined up to addition of a constant and multiplication by a non-zero constant by the condition on the poles alone, amongst all meromorphic functions with the given period lattice. are homogeneous functions of degree -4 and -6; that is, en.wikipedia.org /wiki/Weierstrass's_elliptic_functions   (1322 words)

 Elliptic function: Encyclopedia topic   (Site not responding. Last check: 2007-11-07) The elliptic functions can be seen as analogs of the trigonometric function (trigonometric function: Function of an angle expressed as a ratio of the sides of right-angled triangle containing the angle) s (which have a single period only). Formally, an elliptic function is a meromorphic function (meromorphic function: a meromorphic function is a function that is holomorphic on an open setopen subset... The Weierstrass elliptic function (Weierstrass elliptic function: more facts about this subject) is the prototypical elliptic function, and in fact, the field of elliptic functions with respect to a given lattice is generated by and its derivative. www.absoluteastronomy.com /reference/elliptic_function   (1039 words)

 PlanetMath: elliptic function   (Site not responding. Last check: 2007-11-07) Remark: An elliptic function which is holomorphic is constant. Theorem 1 The field of elliptic functions with respect to a lattice This is version 4 of elliptic function, born on 2003-07-22, modified 2003-08-04. planetmath.org /encyclopedia/EllipticFunction.html   (163 words)

 Elliptic function   (Site not responding. Last check: 2007-11-07) In complex analysis, an elliptic function is,roughly speaking, a function defined on the complex plane which is periodic in two directions. Unless the elliptic function is constant, anyfundamental parallelogram has at least one pole, a consequence of Liouville's theorem. The derivative of an elliptic function is again an elliptic function, withthe same periods. www.therfcc.org /elliptic-function-153226.html   (432 words)

 Elliptic function   (Site not responding. Last check: 2007-11-07) In developments of the theory of elliptic modern authors mostly follow Karl Weierstrass : the notations of Weierstrass's elliptic functions based on his $\wp-functionare convenient and any$elliptic function be expressed in terms of these. Elliptic functions are the inverse functions of elliptic integrals which is how they were introduced The derivative of an elliptic function is again elliptic function with the same periods. www.freeglossary.com /Elliptic_functions   (629 words)

 Elliptic function   (Site not responding. Last check: 2007-11-07) The elliptic functions be seen as analogs of the trigonometric functions (which have a single period only). Formally an elliptic function is a meromorphic function f defined on C for which there exist two non-zero numbers a and b such that The elliptic functions introduced by Carl Jacobi and the auxiliary theta functions (not doubly-periodic) are more complex; but both for the history and for general www.freeglossary.com /Elliptic_function   (629 words)

 Elliptic functions The theory of elliptic functions is one of the highlights of 19th century complex analysis, connected to names such as Gauss, Abel, Jacobi, and Weierstrass. This is the height of the rectangles in the plot, and the up-down period of the sn function is consequently 2iK′(k). The inverse of this function, the sinus lemniscaticus denoted w = sinlemn(z) or w = sl(z), is the original elliptic function; Gauss discovered around 1797 that it is doubly periodic as a function of a complex variable. www.mai.liu.se /~halun/complex/elliptic   (2365 words)

 ipedia.com: Elliptic curve Article   (Site not responding. Last check: 2007-11-07) Elliptic curves are non-singular, meaning they don't have cusps or self-intersections, and a binary operation can be defined for their points in a natural geometric fashion, thus turning the set of points into an abelian group. Elliptic curves can be defined over any field K; the formal definition of an elliptic curve is a non-singular projective algebraic curve over K with genus 1. Elliptic curves over finite fields are used in some cryptographic applications as well as for integer factorization. www.ipedia.com /elliptic_curve.html   (887 words)

 Elliptic Curves and Modular Functions Modular and elliptic functions are both special cases of the concept of an automorphic function, which is a meromorphic function of 1 or more complex variables defined on a particular complex manifold and invariant under a particular group of analytic transformations (symmetries) of the manifold. the modular functions, are essentially the meromorphic functions on D considered as a Riemann surface by its isomorphism with H/ We have stressed this idea of symmetry because of the way it relates the analytic and geometric properties of an object like a Riemann surface to the algebraic properties of a group. Thus the elliptic functions are essentially the automorphic functions on the extended complex plane corresponding to the group of translations by two non-collinear values. www.mbay.net /~cgd/flt/flt05.htm   (2994 words)

 A summary of the definitions and properties of the Jacobian elliptic functions The 12 Jacobian elliptic functions are denoted by two letters taken from the quartet s, c, d, n and may be classified into four groups, each with three members, according to the second letter of the function's name. The most popular Jacobian elliptic functions is a copolar trio of sine amplitude elliptic function - sn(x,k), cosine amplitude elliptic function - cn(x,k), and delta amplitude elliptic function - dn(x,k). The functions sn(x,k), cn(x,k), and dn(x,k) are real valued when their argument is real, and modulus k is either real or purely imaginary. www.cmmp.ucl.ac.uk /~jlg/Elliptic/appendix.html   (352 words)

 Elliptic Integrals and Elliptic Functions The elliptic integrals have a complicated structure in the complex plane. The inverse Jacobi elliptic functions are related to elliptic integrals. Modular elliptic functions are defined to be invariant under certain fractional linear transformations of their arguments. documents.wolfram.com /v5/TheMathematicaBook/AdvancedMathematicsInMathematica/MathematicalFunctions/3.2.11.html   (619 words)

 [No title] Two functions Ns and Ss are created to help in the study because of the complexity involved. To accomplish this task, we construct two functions Ns and Ss, which represent the normalized, and “smoothed” sums respectively of ap ‘s defined as p+1 minus the number of solutions modulo p, where p is a prime. Moreover, in the case of elliptic curves, deep properties of the curve can be revealed as a result of the properties of the Riemann zeta function and Dirichlet L-functions. www.math.princeton.edu /mathlab/projects/ellcurves/op/curvecubic.doc   (2824 words)

 Open Questions: Elliptic Curves and Modular Forms The subject of elliptic curves is a lot like a major city, in which many highways and railroad lines converge, the airport serves as a hub of major airlines, and a seaport connects with important inland waterways. It follows that all elliptic functions are of the form g(℘(z)) + ℘′(z)h(℘(z)), where g(t) and h(t) are quotients of polynomials in the indeterminate t. Now, we have already seen that an elliptic curve as a complex torus is essentially determined by the period lattice of the ℘ function that parameterizes the curve. www.openquestions.com /oq-ma017.htm   (18524 words)

 Elliptic Function An elliptic integral is an integral involving a rational function which contains square roots of cubic or quartic polynomials. Generally, the elliptic integrals CANNOT be expressed in terms of elementary functions. Elliptic Integral of the First Kind: See plots. www.efunda.com /math/elliptic/elliptic.cfm   (92 words)

 SERIES (a Latin word f... - Online Information article about SERIES (a Latin word f... The series whose general term is of the form Kan+..(n), where 0(n) is a rational integral algebraic function of degree r, is a recurring series whose scale of relation is (I—ax) (i —x)**1, but the general term of this series may be obtained by another method. simple function of n, or a sum of a finite number of simple functions, the number being independent of n, which shall be equal to this sum. If the limit, when x is infinite, of the first of the functions Po, 1?i, p2,..., whose limit is not unity, is greater than unity the series is convergent, if less than unity it is divergent. encyclopedia.jrank.org /SCY_SHA/SERIES_a_Latin_word_from_serere.html   (5488 words)

 [No title]   (Site not responding. Last check: 2007-11-07) I know that rho is used >as a coefficient of an elliptic curve over the complex plane, but I'd like >to know how to map or transform that into a curve over a specific field in >GF(2^n) (assuming all parameters are correct for doing that of course). An elliptic curve giving any specific j-invariant can be found defining the curve to be y^2=4x^3-cx-c and then solving for j in the corresponding equation. Weierstrass 'rho': the rho-function (in most literature denoted by a symbol that's generally known as 'Weierstrass p'..) or p-function is defined over a lattice generated by two complex numbers omega_1 and omega_2, or possibly normalized to but one complex number tau as an infinite sum of certain reciprocals. www.maths.tcd.ie /~tim/EllipticCurves/nn   (406 words)

 Bibliography Allasia and R. Besenghi, Numerical calculation of the Riemann zeta function and generalizations by means of the trapezoidal rule, Numerical and Applied Mathematics, Part 2 (Paris 1988) (C. Brezinski, ed.), IMACS Ann. A portable Fortran subroutine for derivatives of the psi function, ACM Trans. Coleman, A Fortran subroutine for the Bessel function gams.cam.nist.gov /mcsd/Reports/2001/nesf/node38.html   (8398 words)

 elliptic function, rf. - Numerical Recipes Forum I changed it such that I don't have to use "nr.h", and in "MINE/utility.h" has the functions for max, min. Now I also wrote the function for "complete elliptic integral of the first kind in Legendre form" which is It seems everything is in order, by the way, the function "sqr(double) is a function that is also defined in "MINE/utility.h", and it complies fine. www.nr.com /forum/showthread.php?p=227   (241 words)

 Mathematics : Publications since January 2005 Beale and A. Layton, On the accuracy of finite difference methods for elliptic problems with interfaces, Commun. J.M. Hawkins and L Koss, Connectivity properties of Julia sets of Weierstrass elliptic functions, Topology and its Applications, vol. Takemura and R. Yoshida, Fundamental holes and saturation points of a commutative semigroup and their application to contingency tables, STATISTICA SINICA on Algebraic Statistics and Computational Biology, submitted 2006 [math.ST/0603108] [abs]. fds.duke.edu /db/aas/math/publications.html   (2413 words)

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