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Topic: Weierstrass elliptic functions

	Weierstrass's elliptic functions - Wikipedia, the free encyclopedia
		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,
		is a meromorphic function in the complex plane with poles at the lattice points.
	en.wikipedia.org /wiki/Weierstrass's_elliptic_functions (1115 words)

	List of mathematical functions - Wikipedia, the free encyclopedia
		A modern, abstract point of view contrasts large function spaces, which are infinite-dimensional and within which most functions are 'anonymous', with special functions picked out by properties such as symmetry, or relationship to harmonic analysis and group representations.
		Superadditive function : The value of a sum is greater than or equal to the sum of the values of the summands.
		Elliptic functions : The inverses of elliptic integrals; used to model double-periodic phenomena.
	en.wikipedia.org /wiki/List_of_mathematical_functions (861 words)

	Elliptic function - free-definition (Site not responding. Last check: 2007-10-14)
		In complex analysis, an elliptic function is, roughly speaking, a function defined on the complex plane which is periodic in two directions.
		The elliptic functions introduced by Carl Jacobi, and the auxiliary theta functions (not doubly-periodic), are more complex; but important both for the history and for general theory.
		Elliptic functions are the inverse functions of elliptic integrals, which is how they were introduced historically.
	www.free-definition.com /Elliptic-function.html (478 words)

	Articles - Elliptic function (Site not responding. Last check: 2007-10-14)
		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.
		The primary difference between these two theories is that the Weierstrass functions have high-order poles located at the corners of the periodic lattice, whereas the Jacobi functions have simple poles.
		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.
	www.kamero.net /articles/Elliptic_function (760 words)

	Elliptic curve - Enpsychlopedia (Site not responding. Last check: 2007-10-14)
		Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in the proof of Fermat's last theorem.
		The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions.
		The Weierstrass functions are doubly-periodic; that is, they are period with respect to a lattice Λ; in essence, the Weierstrass functions are naturally defined on a torus $T=\mathbb{C}/\Lambda .$
	www.grohol.com /psypsych/Elliptic_curve (1362 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 2 iK ′( 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 (2603 words)

	Weierstrass (Site not responding. Last check: 2007-10-14)
		Weierstrass had made a decision to become a mathematician but he was still supposed to be on a course studying public finance and administration.
		Weierstrass attended Gudermann 's lectures on elliptic functions, some of the first lectures on this topic to be given, and Gudermann strongly encouraged Weierstrass in his mathematical studies.
		The concepts on which Weierstrass based his theory of functions of a complex variable in later years after 1857 are found explicitly in his unpublished works written in Münster from 1841 through 1842, while still under the influence of Gudermann.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Weierstrass.html (2487 words)

	Talk:Weierstrass's elliptic functions - Wikipedia, the free encyclopedia
		Given a period lattice Î, what is true is that all meromorphic functions periodic under Î form a field that is actually C ( P, P '), i.e.
		rational functions in the Weierstrass P and its derivative.
		In a sense I think this page should be about Pe-related formulae, and the general elliptic function and elliptic curve theory should live somewhere else.
	en.wikipedia.org /wiki/Talk:Weierstrass's_elliptic_functions (425 words)

	Elliptic function (Site not responding. Last check: 2007-10-14)
		The elliptic functions can be seen as analogs of the trigonometric function s (which have a single period only).
		The elliptic functions introduced by Carl Jacobi, and the auxiliary theta function s (not doubly-periodic), are more complex; but important both for the history and for general theory.
		Elliptic functions are the inverse function s of elliptic integral s, which is how they were introduced historically.
	www.worldhistory.com /wiki/E/Elliptic-function.htm (572 words)

	Karl Weierstrass [Definition] (Site not responding. Last check: 2007-10-14)
		Karl Weierstrass was the son of Wilhem Weierstrass, a government official, and Theodora Vonderforst.
		During this period of study, Weierstrass attended the lectures of Christoph Gudermann Christoph Gudermann (March 25, 1798 - September 25, 1852) was born in Vienenburg, Germany.
		Weierstrass was greatly influenced by this course, which marked...
	www.wikimirror.com /Karl_Weierstrass (937 words)

	Elliptic Curves and Elliptic Functions
		For every algebraic function, it is possible to construct a specific surface such that the function is "single-valued" on the surface as a domain of definition.
		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.
	cgd.best.vwh.net /home/flt/flt03.htm (3548 words)

	PlanetMath: examples of elliptic functions
		are of special relevance in the theory of elliptic curves.
		"examples of elliptic functions" is owned by alozano.
		This is version 3 of examples of elliptic functions, born on 2003-08-25, modified 2003-08-27.
	planetmath.org /encyclopedia/ExamplesOfEllipticFunctions.html (62 words)

	Weierstrass
		Weierstrass M-test In infinite series, and applies to a series whose terms are themselves functions of a real variable.
		Weierstrass preparation theorem In coefficients are analytic functions in the remaining variables and zero at P. There a...
		Weierstrass's elliptic functions In elliptic functions that have become the basis for the most standard notations used.
	www.brainyencyclopedia.com /topics/weierstrass.html (124 words)

	PlanetMath: elliptic function
		Remark : An elliptic function which is holomorphic is constant.
		-function (see elliptic curve) is an elliptic function, probably the most important.
		This is version 4 of elliptic function, born on 2003-07-22, modified 2003-08-04.
	planetmath.org /encyclopedia/EllipticFunction.html (162 words)

	Jacobi's elliptic functions
		In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that have historical importance with also many features that show up important structure, and have direct relevance to some applications (e.g.
		The three Jacobi elliptic functions are doubly periodic, meromorphic functions of z, whose periods are expressible in terms of τ and θ.
		If we call the periods of cn the lattice Λ, then both sn and dn are periodic with respect to Λ, but their full lattices of periods are larger (in each case, Λ is a subgroup of index 2).
	www.sciencedaily.com /encyclopedia/jacobi_s_elliptic_functions (407 words)

	Weierstraß Elliptic Function (Site not responding. Last check: 2007-10-14)
		The Weierstraß elliptic function is analytic at the origin and therefore at all points congruent to the origin.
		By Liouville's Elliptic Function Theorem, it is therefore a constant.
		The periods of the Weierstraß elliptic function are given as follows.
	www.itu.dk /edu/documentation/mathworks/math/math/w/w060.htm (263 words)

	Elliptic Functions: Introductory
		Elliptic functions — as developed by Jacobi, Weierstrass, Eisenstein, Dedekind, and others — are one of the crowning achievements of 19th century mathematics and are widely applied in physics and engineering.
		Their study is the natural continuation of the analysis of polynomial, exponential, and trigonometric functions of a complex variable.
		In the 20th century, the analysis of the beautiful transformation properties of elliptic functions developed into the theory of elliptic curves and modular forms, a central topic of algebraic geometry and number theory.
	www.e.kth.se /~tkatchev/teaching/Elliptic-main.htm (196 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		Circular Functions: Starting with (1), the cotangent function, pi and finally the other circular functions and their properties are developed.
		Elliptic Integrals: Elliptic integrals as inverses of elliptic functions are discussed now.
		Applications: The use of elliptic functions in connection with waves, number theory and elliptic curves are discussed.
	www.math.yorku.ca /siamopsf/books/elliptic.html (313 words)

	Weierstrass P-Function
		A singularly periodic function p(x) on R which satisfies f(x + nw) = f(x) is determined by the values on the interval [0,w] - the values at 0 and w are the same, so it can be pictured as two endpoints that have been glued together.
		Elliptic functions, due to their doubly periodic nature, wrap around both ways (forming a torus) - with the values on opposite sides of the boundary equal to each other.
		Proof: Looking at an elliptic function, p(z), the logarithmic derivative (as we did previously in the presentation) would be equal to p?(z)/p(z).
	www.willamette.edu /~zizza/Courses/SeniorSeminar/G2.3/weierstrass.html (1245 words)

	Elliptic and Modular Functions (Site not responding. Last check: 2007-10-14)
		More information on elliptic functions can be found for example in Chandrasekharan [Cha85], and for modular functions and their use see Koblitz [Kob84].
		Given a lattice L = [a, b] in the complex plane, this function returns the value of the elliptic j-invariant of L. This is the j-invariant of tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
		Given a pair L = [a,b] of complex numbers generating a lattice in C, return the normalized q-series expansion of the discriminant Delta(q) evaluated at tau where tau = a/b or tau = b / a, whichever is in the upper half complex plane.
	www.msri.org /info/computing/docs/magma/text570.htm (1242 words)

	Wolfram Research, Inc.
		The inverse Jacobi elliptic functions are related to elliptic integrals.
		The Weierstrass zeta and sigma functions are not strictly elliptic functions since they are not periodic.
		Modular elliptic functions are defined to be invariant under certain fractional linear transformations of their arguments.
	documents.wolfram.com /v3/MainBook/3.2.11.html (620 words)

	Elliptic Functions - Cambridge University Press (Site not responding. Last check: 2007-10-14)
		The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and then the reader is introduced to the richly varied applications of the elliptic and related functions.
		Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.
		An application of elliptic functions in algebra solution of the general quintic equation; 11.
	www.cambridge.org /uk/catalogue/catalogue.asp?isbn=0521785634 (201 words)

	Weierstrass's elliptic functions
		The sum extends over the lattice { n + m τ : n and m in Z } with the origin omitted.
		The Weierstrass theory also includes the Weierstrass zeta-function, which is an indefinite integral of
		and not doubly-periodic, and a theta function called the Weierstrass sigma-function, of which his zeta-function is the log-derivative.
	www.sciencedaily.com /encyclopedia/weierstrass_s_elliptic_functions (485 words)

	Jacobi's elliptic functions - Art History Online Reference and Guide
		The twelve Jacobian elliptic functions are then pq, where p and q are one of the letters s,c,d,n.
		There is a simpler, but completely equivalent definition, giving the elliptic functions as inverses of the incomplete elliptic integral of the first kind.
		Equivalently, Jacobi's elliptic functions can be defined in terms of his theta functions.
	www.arthistoryclub.com /art_history/Jacobi%27s_elliptic_functions (1038 words)

	Numerical Evaluation Of Special Functions - Lozier, Olver (ResearchIndex)
		This document is an excerpt from the current hypertext version of an article that appeared in Walter Gautschi (ed.), Mathematics of Computation 1943--1993: A Half-Century of Computational Mathematics, Proceedings of Symposia in Applied Mathematics 48, American Mathematical Society, Providence, RI 02940, 1994.
		2 Rational approximations to the incomplete elliptic integrals..
		1 Transformations of the Jacobian amplitude function and its c..
	citeseer.ist.psu.edu /101146.html (654 words)

	Creation Functions (Site not responding. Last check: 2007-10-14)
		An elliptic curve E can currently only be created by specifying Weierstrass coordinates for the curve over a field K (integer coordinates are regarded as rational elements).
		Points on an elliptic curve over a field are given in terms of projective coordinates: a point (a, b, c) is equivalent to (x, y, z) if and only if there exists an element u (in the field of definition) such that ua=x, ub=y, and uc=z.
		Given an elliptic curve E over R and coefficients x, y, z in R satisfying the equation for E, return the normalized point P=(x:y:z) on E. If z is not specified it is assumed to be 1.
	www.math.ufl.edu /help/magma/text473.html (444 words)

	INI : Abstracts : MAA : Model theory of elliptic functions (Site not responding. Last check: 2007-10-14)
		The lectures will consider the structure of definitions in the theory of individual Weierstrass elliptic functions, paying as much attention as possible to uniformities as the function, or its associated lattice, varies.
		The setting will be that of an o-minimal expansion of the real field, and one will interpret therein the elliptic function on a semi-algebraic fundamental parallelogram.
		Peterzil and Starchenko began the model theory of families of elliptic functions, and showed in particular that this is interpretable in an o-minimal theory, so is certainly not undecidable in any Godelian way.
	www.newton.cam.ac.uk /programmes/MAA/angus.html (332 words)

	Conformal Geometry and Dynamics
		Abstract: We study parametrized dynamics of the Weierstrass elliptic
		function by looking at the underlying lattices; that is, we study parametrized families
		We show that within each shape space there is a wide variety of dynamical behavior, and we conduct a deeper study into certain lattice shapes such as triangular and square.
	www.ams.org /ecgd/2004-08-01/S1088-4173-04-00103-1/home.html (432 words)

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