Elliptic functions

In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15.

On complex soliton solutions, complex elliptic solutions and complex rational function solutions for the Sasa-Satsuma model equation with variable coefficients

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

A modified elliptic low-pass filter function with progressively diminishing ripples in both the passband and the stopband is proposed and analyzed. The modified elliptic function is realizable by the passive doubly-terminated ladder network for the order n even or odd, thus lending themselves amenable to high-quality low-sensitivity active RC or switched capacitor filters through the simulation techniques. Besides the passive ladder network realization, the modified elliptic function has improved characteristics both in the frequency domain and the time domain as compared with the classical elliptic function. In addition, the modified function exhibits better magnitude sensitivity properties than the classical function.

These lecture notes discuss some of the basics of elliptic hypergeometric functions. These are fairly recent generalizations of ordinary hypergeometric functions. In this chapter we first discuss both ordinary hypergeometric functions and elliptic functions, as you need to know both to define elliptic hypergeometric series. We subsequently discuss some of the important properties these series satisfy, in particular we consider the biorthogonal functions found by Spiridonov and Zhedanov, both with respect to discrete and continuous measure. In doing so we naturally encounter the most important evaluation and transformation formulas for elliptic hypergeometric series, and for the associated elliptic beta integral.

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.

In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. 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.

Based on the computerized symbolic computation, a new rational expansion method using the Jacobian elliptic function was presented by means of a new general ansätz and the relations among the Jacobian elliptic functions. The results demonstrated an effective direction in terms of a uniformed construction of the new exact periodic solutions for nonlinear differential–difference equations, where two representative examples were chosen to illustrate the applications. Various periodic wave solutions, including Jacobian elliptic sine function, Jacobian elliptic cosine function and the third elliptic function solutions, were obtained. Furthermore, the solitonic solutions and trigonometric function solutions were also obtained within the limit conditions in this paper.

Recently, we established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. The same theorem was subsequently applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this article, we discuss what are, in a sense, inverse applications of this theorem. We first prove a Lemma that if two meromorphic on the whole complex plane functions f(z) and g(z) have the same zeroes and poles, taking into account their orders, and have appropriate asymptotic for large |z|, then for some integer n, dnln(f(z))dzn=dnln(g(z))dzn. The use of this Lemma enables proofs of many identities between elliptic functions, their transformation and n-tuple product rules. In particular, we show how exactly for any complex number a, ℘(z)-a, where ℘(z) is the Weierstrass ℘ function, can be presented as a product and ratio of three elliptic θ1 functions of certain arguments. We also establish n-tuple rules for some elliptic theta functions.

5 - Indefinite Integrals of Special Functions

Introduction. In spite of its wide applicability in various branches of the theory of functions, the elliptic modular function is often used with a certain hesitation. This is mainly due to the fact that its application presupposes familiarity with a comparatively intricate formalism, in particular when the determination of numerical constants is involved. In fact, the endeavor to avoid the elliptic modular function has given rise to an extensive mathematical literature aiming at proving certain theorems in an way, the word being used here as a synonym for without making use of the elliptic modular function. As an impressive example, Picard's theorem on integral functions might be quoted. The difficulties which beset the numerical treatment of the elliptic modular function go essentially back to the fact that, on the one hand, the formalism of this function can only be developed with the help of the Jacobian elliptic functions while, on the other hand, what is needed in the applications are the conformal mapping properties of the modular function, and the connection between these two different aspects of the modular function has to be established through the medium of the theory of Schwarz' differential parameter or by a very detailed study of the periodic properties of the Jacobian elliptic functions. The object of the first part of this paper is to show how those properties of the elliptic modular function which are required for the applications may be derived in a simple way by the exclusive use of elementary principles of the theory of conformal representation. It will be shown that once the modular surface' is defined, the functional equation

CHAPTER 1 - ELLIPTIC FUNCTIONS

The purpose of this chapter is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We are primarily focused on constructing examples of various classes of compactly nonrecurrent elliptic functions. All these examples are either Weierstrass elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice. The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass elliptic functions generated by square and triangular lattices. We then provide simple constructions of many classes of elliptic functions discerned in the previous chapter. We essentially cover all of them. All these examples stem from Weierstrass $\wp$ functions. Finally, we also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass $\wp$ elliptic functions and their modifications coming from a series of papers by Hawkins and her collaborators.

Circular summation of theta functions in Ramanujan's Lost Notebook

We give a survey of the different schools in q-analysis and introduce difference calculus and Bernoulli numbers to make a preparation for the important fourth chapter. We summarize the different attempts at elliptic and Theta functions, both of which are intimately related to q-calculus. We present the history of trigonometry, prosthaphaeresis, logarithms and calculus, because we claim that Fermat introduced the precursor of the q-integral long before calculus was invented. The Hindenburg combinatoric School gives a background to the discovery of the Schweins q-binomial theorem. The so-called Fakultaten was a forerunner to the Γ function and q-factorial. In the year 1844, Gudermann published his book on elliptic functions and two years later, in 1846, Heine published his important article on q-hypergeometric series, referring to Gauss’s Disquisitiones, pointing out the two q-analogues of the exponential function.