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Topic: Jacobian elliptic functions

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	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 3 of elliptic integrals and Jacobi elliptic functions, born on 2003-09-30, modified 2005-02-08.
	planetmath.org /encyclopedia/JacobianEllipticFunction.html (232 words)

	Jacobian Elliptic Functions
		Elliptic functions have provided a lot of entertainment for mathematicians, however, and are as fascinating as any useless knowledge can be.
		Elliptic integrals came first, invented by the Bernoullis, and were studied by Maclaurin, Euler and Lagrange in the 18th century, and later by Legendre, when there was great interest in evaluating the integrals that appeared in scientific applications, after it was realized that most integrals could not be evaluated in terms of the elementary functions.
		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.
	www.du.edu /~jcalvert/math/jacobi.htm (2457 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.
	www.maumae.net /yorick/doc/html_i/elliptic_i.php (613 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)

	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 second argument of the functions k - is a modulus of the elliptic function and
	www.cmmp.ucl.ac.uk /~jlg/Elliptic/appendix.html (352 words)

	About Jacobi
		Karl Gustav Jacob Jacobi was born on the 10th of December 1804 in Potsdam, and died on the 18th of February, 1851, in Berlin.
		His results in elliptic functions were published in Fundamenta Nova Theoriae Functionum Ellipticarum (1829; "New Foundations of the Theory of Elliptic Functions").
		In 1832 he demonstrated that just as elliptic functions can be obtained by inverting elliptic integrals, hyperelliptic functions can be obtained by inverting hyperelliptic integrals.
	beige.ucs.indiana.edu /B673/node28.html (306 words)

	Maxima Manual: 17. Elliptic Functions
		In particular, all elliptic functions and integrals use the parameter m instead of the modulus k or the modular angle \alpha.
		This is one area where we differ from Abramowitz and Stegun who use the modular angle for the elliptic functions.
		The elliptic functions and integrals are primarily intended to support symbolic computation.
	maxima.sourceforge.net /docs/manual/en/maxima_17.html (413 words)

	A printable page of our project
		The quotients of the theta functions yield the three Jacobian elliptic functions: sn z, cn z, and dn z.
		The Jacobi elliptic functions are defined in terms of the integral:
		Some definitions of the elliptic functions use the modulus k instead of the parameter m.
	www.andrews.edu /~calkins/math/biograph/199900/biojacob.htm (760 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.
		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.
		Elliptic integrals are usually defined to be integrals of the form
	www.getnet.com /~cherry/mathml/jsc.html (1932 words)

	Talk Abstracts: IMA 2002 Summer Program: Special Functions in the Digital Age, July 22 - August 2, 2002
		In order to analyze the wave functions of electrons in molecules with a crystal structure it is necessary to use the structure of spherical harmonics invariant under the symmetry point group of the crystal.
		In particular, those special functions that arise as explicit solutions of the partial differential equations of mathematical physics, such as via separation of variables, can be characterized in terms of their transformation properties under the Lie symmetry groups and algebras of the differential equations.
		Khare and Sukhatme have recently introduced scores of identities for sums of Jacobi functions with arguments, z, augmneted by addition of 2(i-1)K/p, K being the "quarter period" which is a complete elliptic integral of the first kind, and i =3D 1,2,3,...p.
	www.ima.umn.edu /digital-age/abstracts (7679 words)

	The Miracle of Theta Functions
		Modular functions are functions which are meromorphic in H, the upper half of the complex plane, and which are invariant under a group of linear fractional transformations, G, in the sense that
		Thus the behaviour of a modular function is uniquely determined by its behaviour on a fundamental region.
		Modular functions are, in a sense, an extension of elliptic (or doubly periodic) functions --- functions such as sn which are invariant under linear transformations and which arise naturally in the inversion of elliptic integrals.
	www.cecm.sfu.ca /organics/papers/borwein/paper/html/node12.html (740 words)

	Elliptic Functions - Cambridge University Press (Site not responding. Last check: 2007-10-22)
		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.
		Jacobian elliptic functions of a complex variable; 3.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521780780 (198 words)

	Module scipy.special.info (Site not responding. Last check: 2007-10-22)
		Struve Functions struve -- Struve function --- Hv(x) modstruve -- Modified struve function --- Lv(x) itstruve0 -- Integral of H0(t) from 0 to x it2struve0 -- Integral of H0(t)/t from x to Inf.
		erfc -- Complemented error function (1- erf(x)) erfinv -- Inverse of error function erfcinv -- Inverse of erfc erf_zeros -- **Complex zeros of erf(z) fresnel -- Fresnel sine and cosine integrals.
		in the description indicates a function which is not a universal function** and does not follow broadcasting and automatic array-looping rules.
	www.scipy.org /doc/api_docs/scipy.special.info.html (1219 words)

	Elliptic integral - Article from FactBug.org - the fast Wikipedia mirror site
		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.
		with u as defined above: thus, the jacobian elliptic functions are inverses to the elliptic integrals.
	www.factbug.org /cgi-bin/a.cgi?a=9960 (703 words)

	MUG: JacobiSN is wrong in Maple 6.0 (15.12.00)
		Remember that there are two conventions in the literature for the Jacobian elliptic functions, regarding the use of k versus the use in Abramowitz and Stegun of m=k^2 (this is less useful---see the book by D. Lawden).
		The error is in the JacobiAM function (all the other Jacobi functions are simple trig combination of it and therefore the test of Robert Corless: sn^2+dn^2=1 is just a test on the trig functions not on the Jacobi).
		We pay a lot of money for upgrades that have nice GUI and other functions but the stability of Maple for the evaluation of elementary and special functions is a cornerstone of its power and use.
	www.math.rwth-aachen.de /mapleAnswers/html/1206.html (614 words)

	FORTRAN Routines for Computation of Special Functions
		For example, the computation of of special functions based on polynomial approximations does not have to use double precision.
		To compute functions with a higher order or degree, all you need to do is simply set the dimension of proper arrays higher.
		mlgama.for (LGAMA) Evaluate the gamma function or the logarithm of the gamma function.
	jin.ece.uiuc.edu /routines/routines.html (2278 words)

	ellipke (Site not responding. Last check: 2007-10-22)
		Complete elliptic integrals of the first and second kind
		The complete elliptic integral of the first kind [1] is:
		returns the complete elliptic integral of the first and second kinds.
	grove.ufl.edu /matlab_help/techdoc/ref/ellipke.html (133 words)

	M. Abramowitz and I. A. Stegun. Handbook of mathematical functions
		A greater variety of functions and higher accuracy of tabulation are now required as a result of scientific advances and, especially, of the increasing use of automatic computers.
		The enthusiastic reception accorded the "Handbook of Mathematical Functions" is little short of unprecedented in the long history of mathematical tables that began when John Napier published his tables of logarithms in 1614.
		The classification of functions and organization of the chapters in this Handbook is similar to that of An Index of Mathematical Tables by A. Fletcher, J. Miller, and L. Rosenhead.
	mintaka.sdsu.edu /faculty/wfw/ABRAMOWITZ-STEGUN/intro.htm (2412 words)

	Maxima Manual - Numerical
		It expects that fn is a Lisp function (or translated Macsyma function) which accepts a floating point argument and that it always returns a floating point value.
		This may be due to too much noise in function (relative to the given error requirements) or due to an ill-behaved integrand.
		When INTERPOLATE is called, it determines whether or not the function to be interpolated satisfies the condition that the values of the function at the endpoints of the interpolation interval are opposite in sign.
	www.ma.utexas.edu /maxima/maxima_22.html (2270 words)

	Elliptic integral - Wikipedia, the free encyclopedia
		The incomplete elliptic integral of the first kind F is defined as
		Main article: complete elliptic integral of the first kind.
		Main article: complete elliptic integral of the second kind.
	en.wikipedia.org /wiki/Elliptic_integral (791 words)

	135-139
		Consequently, the higher mathematical functions, such as Bessel functions, hypergeometric functions, and elliptic functions, would form the core of the work.
		New functions have emerged in impor-tance, and new properties of well-known functions have been discovered.
		In spite of the fact that sophisticated numerical methods have been embodied in well- designed commercial software for many functions, there continues to be a need for a compendium of information on the properties of mathematical functions.
	nvl.nist.gov /pub/nistpubs/sp958-lide/html/135-139.html (2156 words)

	Matlab Routines for Computation of Special Functions (Site not responding. Last check: 2007-10-22)
		mlgama.m (LGAMA) Evaluate the gamma function or the logarithm of the gamma function.
		mitjyb.m (ITJYB) Evaluate the integral of Bessel functions J0(t) and Y0(t) from 0 to x using polynomial approximations.
		mairya.m (AIRYA) Evaluate the Airy functions and their derivatives by means of Bessel functions.
	ceta.mit.edu /comp_spec_func (2288 words)

	GNU Scientific Library -- Reference Manual: Elliptic Functions (Jacobi) (Site not responding. Last check: 2007-10-22)
		The Jacobian Elliptic functions are defined in Abramowitz & Stegun, Chapter 16.
		The functions are declared in the header file
		This function computes the Jacobian elliptic functions sn(um), cn(um), dn(um) by descending Landen transformations.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.0/gsl-ref_94.html (55 words)

	Cephes double precision special functions suite
		<= 34 are reduced by recurrence and the function * approximated by a rational function of degree 6/7 in the * interval (2,3).
		/* lmdif.c * * The purpose of lmdif is to minimize the sum of the squares of * M nonlinear functions in N variables by a modification of * the Levenberg-Marquardt algorithm.
		* * The function is approximated by a Chebyshev expansion in * the interval [0,1].
	www.netlib.org /cephes/doubldoc.html (16786 words)

	Jacobian Elliptic Functions and Elliptic Integrals for the HP-41
		and the 3 Jacobian elliptic functions sn ; cn ; dn are defined by:
		From these, all the 9 other elliptic functions can be determined by the relations mentioned above.
		Then, the function could be evaluated by a Taylor series.
	www.hpmuseum.org /software/41/41jacob.htm (1232 words)

	Extended precision special functions library
		The value of * MAXPOL is set by calling the function * * polini(maxpol); * * where maxpol is the desired maximum degree.
		* * This is accomplished using the inverse beta integral * function and the relations * * z = incbi(df2/2, df1/2, p) * x = df2 (1-z) / (df1 z).
		* Since the function is symmetric about t=0, the area under the * right tail of the density is found by calling the function * with -t instead of t.
	www.moshier.net /qlibdoc.html (5926 words)

	[No title]
		Bessel's Functions: the most useful special functions of all.
		Jacobian Elliptic Functions: What is the mysterious function sn(u)?.
		Complex Variables: a synopsis of the theory of analytic functions.
	www.du.edu /~jcalvert/math/mathom.htm (616 words)

	Cephes long double precision special functions suite
		<= 13 are reduced by recurrence and the function * approximated by a rational function of degree 7/8 in the * interval (2,3).
		Computation is via the functions * erf and erfc with care to avoid error amplification in computing exp(-x^2).
		* N 0 * * The function p1evll() assumes that coef[N] = 1.0 and is * omitted from the array.
	www.moshier.net /ldoubdoc.html (10123 words)

	Abramowitz and Stegun. Subject Index
		Cn, Dn, Sn integrals of the squares of Jacobian elliptic functions.....
		E($\phi$ \a) elliptic integral of the second kind.....
		n characteristic of the elliptic integral of the third kind.....
	mintaka.sdsu.edu /faculty/wfw/ABRAMOWITZ-STEGUN/subj.htm (463 words)

	DIGITAL LIBRARY OF MATHEMATICAL FUNCTIONS
		These functions (often known as special functions) are used extensively in mathematical analysis in many fields, such as physics and chemistry, and they are essential tools in modern computational modeling of phenomena in the physical sciences and engineering.
		In July 1997, NIST hosted an Invitational Workshop on Mathematical Functions, which was attended by well-known experts in the theory and application of special functions.
		The concept of the DLMF project, to develop a successor to AMS 55 by conducting a thorough survey of the pertinent scientific literature and to take advantage of the latest relevant advances in information technology, was developed in broad outline at this meeting.
	www.itl.nist.gov /lab/pub/newsaug00.htm (1661 words)

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