Related papers: Elliptic functions revisited
We study functions of an elliptic parameter, which are defined as iterated integrals of elliptic functions. We establish their relation with the "elliptic associators" of our previous work, by means of a functional realization of Lie…
An algorithm is given to compute a normal form for hyperelliptic curves. The elliptic case has been treated in a previous paper. In this paper the hyperelliptic case is treated.
Summability has been a central object of study in difference algebra over the past half-century. It serves as a cornerstone of algebraic methods to study linear recurrences over various fields of coefficients and with respect to various…
In this work we derive results concerning Elliptic Functions using as tools general formulas from previus work.
We construct harmonic weak Maass forms that map to cusp forms of weight $k\geq 2$ with rational coefficients under the $\xi$-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished…
The elliptic gamma function is a generalization of the Euler gamma function. Its trigonometric and rational degenerations are the Jackson q-gamma function and the Euler gamma function. We prove multiplication formulas for the elliptic gamma…
In this article we give evaluations of certain series of hyperbolic functions using Jacobi elliptic functions theory. We also define some new functions that enable us to give characterization of not solvable class of series.
Recently, Hong, Mertens, Ono and Zhang proved a conjecture of C\u{a}ld\u{a}raru, He, and Huang that expresses the Taylor series of the modular $j$-function around the elliptic points $i$ and $\rho=e^{\pi i/3}$ as rational functions arising…
A class of log-trigonometric integrals are evaluated in terms of elliptic functions. From this, by using the elliptic integral singular values, one can obtain closed form evaluations of integrals such as \[…
In this paper we are interested in developments of elliptic functions of Jacobi. In particular a trigonometric expansion of the classical theta functions introduced by the author (Algebraic methods and q-special functions, Editors: C.R.M.…
The problem of the motion of a particle in an asymmetric double well is solved explicitly in terms of the Weierstrass and Jacobi elliptic functions. While the solution of the orbital motion is expressed simply in terms of the Weierstrass…
A short review will be made of elliptic integrals, widely applied in GPS (Global Positioning System) communications (accounting for General Relativity Theory-effects), cosmology, Black hole physics and celestial mechanics. Then a novel…
Based on the distinction between the covariant and contravariant metric tensor components in the framework of the affine geometry approach and the s.c. "gravitational theories with covariant and contravariant connection and metrics", it is…
Summation formulae are classical tools in analysis: Taylor-MacLaurin, Euler-MacLaurin, Poisson, Vorono\"i, Circle formulae\ldots We will show how, from a single equation - referred to as the mother-equation - it is possible to unify these…
We present new ideas for computing elliptic Gau{\ss} sums, which constitute an analogue of the classical cyclotomic Gau{\ss} sums and whose use has been proposed in the context of counting points on elliptic curves and primality tests. By…
By using the theory of the elliptic integrals a new method of summation is proposed for a certain class of series and their derivatives involving hyperbolic functions. It is based on the termwise differentiation of the series with respect…
A cubic algebraic equation for the effective parametrizations of the standard gravitational Lagrangian has been obtained without applying any variational principle.It was suggested that such an equation may find application in gravity…
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
The Weierstrassian $\wp, \zeta$ and $\sigma $ functions are generalized to ${\bf R}^{n}$. The $n=3$ and $n=4$ cases have already been used in gravitational and Yang-Mills instanton solutions which may be interpreted as explicit realizations…
Exact solutions are found for Euler's equations of rigid body motion for general asymmetrical bodies under the influence of torque by using Jacobi elliptic functions. Differential equations are determined for the amplitudes and the…