Equivariant functions and integrals of elliptic functions

In this paper, we introduce the theory of equivariant functions by studying their analytic, geometric and algebraic properties. We also determine the necessary and sufficient conditions under which an equivariant form arises from modular forms. This study was motivated by observing examples of functions for which the Schwarzian derivative is a modular form on a discrete group. We also investigate the Fourier expansions of normalized equivariant functions, and a strong emphasis is made on the connections to elliptic functions and their integrals.     

Generalised elliptic functions

We consider multiply periodic functions, sometimes called Abelian functions, defined with respect to the period matrices associated with classes of algebraic curves. We realise them as generalisations of the Weierstraß ?-function using two different approaches. These functions arise naturally as solutions to some of the important equations of mathematical physics and their differential equations, addition formulae, and applications have all been recent topics of study.The first approach discussed sees the functions defined as logarithmic derivatives of the σ-function, a modified Riemann θ-function. We can make use of known properties of the σ-function to derive power series expansions and in turn the properties mentioned above. This approach has been extended to a wide range of non hyperelliptic and higher genus curves and an overview of recent results is given.The second approach defines the functions algebraically, after first modifying the curve into its equivariant form. This approach allows the use of representation theory to derive a range of results at lower computational cost. We discuss the development of this theory for hyperelliptic curves and how it may be extended in the future. We consider how the two approaches may be combined, giving the explicit mappings for the genus 3 hyperelliptic theory. We consider the problem of generating bases of the functions and how these decompose when viewed in the equivariant form.     

Cubic elliptic functions

The function

On the Abelian Higgs model

In this paper we prove existence of a global non trivial solution for a nonlinear time-independent system arising in the Higgs field theory.     

Hypergeometric functions and elliptic curves

We provide uniform formulas for the real period and the trace of Frobenius associated to an elliptic curve in Legendre normal form. These are expressed in terms of classical and Gaussian hypergeometric functions, respectively. 2000 Mathematics Subject Classification Primary—11G05, 33C05 This research was supported by K. Ono’s NSF grant     

Fourier series and elliptic functions

Non-linear second-order differential equations whose solutions are the elliptic functions sn(t, k), cn(t, k) and dn(t, k) are investigated. Using Mathematica, high precision numerical solutions are generated. From these data, Fourier coefficients are determined yielding approximate formulas for these non-elementary functions that are correct to at least 11 decimal places. These formulas have the advantage over numerically generated data that they are computationally efficient over the entire real line. This approach is seen as further justification for the early introduction of Fourier series in the undergraduate curriculum, for by doing so, models previously considered hard or advanced, whose solution involves elliptic functions, can be solved and plotted as easily as those models whose solutions involve merely trigonometric or other elementary functions.     

Cubic analogues of the Jacobian theta functions and the Jacobian elliptic functions

In this paper, the Jacobian theta functions, the Jacobian elliptic functions, and their cubic analogues are introduced. Then, the properties satisfied the cubic analogues are discussed.     

Numerical calculation of elliptic integrals and elliptic functions. III

Summary This paper is a continuation of [2, 3]. It contains anALGOL program for the incomplete elliptic integral of the third kind based on a theory described in [4]. This program replaces the inadequate one based on the Gauß-transformation which was published in [2]. In addition, anAlgol program for a general complete elliptic integral is presented. Editor's note. In this fascicle, prepublication of algorithms from the Special Functions Series of the Handbook for Automatic Computation is continued. Algorithms are published inAlgol 60 reference language as approved by the IFIP. Contributions in this series should be styled after the most recently published ones.This research was supported by the Advanced Research Projects Agency of the Department of Defense and was monitored by the U. S. Army Research Office —Durham under Contract DA-31-124-ARO-D-257.     

Limits of functions and elliptic operators

We show that a subspaceS of the space of real analytical functions on a manifold that satisfies certain regularity properties is contained in the set of solutions of a linear elliptic differential equation. The regularity properties are thatS is closed inL ² (M) and that if a sequence of functions fn in ƒ_n converges inL ²(M), then so do the partial derivatives of the functions ƒ_n.     

Generalized elliptic genera and baker-Akhiezer functions

Translated from Matematicheskie Zametki, Vol. 47, No. 2, pp. 34–45, February, 1990.     

Scattering of vortices in the Abelian Higgs model

We study the scattering of vortices in the Abelian (2+1)-dimensional Higgs model. We show that in the case of the symmetric head-on collision of N vortices, their trajectories are rotated by the angle π/N after the collision. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 77–91, July, 2008.