The different languages of q-calculus

An addition formula for the Jacobian theta function and its applications

It's a reality that there is a relationship between the sigma function of Weierstrass and theta functions. An elliptic function can be set up using the theta functions just as it can be astablished with the help of sigma function of Weierstrass and two relations between the Dedekind's h-function and - theta function were established by the using characteristic values (mod2) for θ-function according to the (<i>u</i>,τ) pair and <i>u</i>,τ complex numbers, satisfying Im τ>0. In this study, the transformations among the theta functions according to the quarter periods have been given and a Jacobian style elliptic functions has been set up the theta function by the help of a defined function.

General remarks. Any integral of the type ∫ R \(\left( {z,{Z^{\frac{1}{2}}}} \right)\) is a rational function of x and y and Z is a polynomial of the third or fourth degree in z with real coefficients and no repeated factors is called an elliptic integral.

Circular summation of theta functions in Ramanujan's Lost Notebook

Wilker- and Huygens-type inequalities for Jacobian elliptic functions and classical theta functions are established. For the limiting values of the modulus parameter of the elliptic functions obtained results simplify to known ones which have been established earlier for circular and hyperbolic functions. The main results in this paper are derived with the aid of two inequalities proven in Neuman [Inequalities for weighted sums of powers and their applications. Math Inequal Appl. 2012;15(4):995–1005].

Report of the Committee on Mathematical Tables. The objects for which the Committee were appointed at Edinburgh were twofold, viz., the preparation of a list of tables scattered about in books and mathematical journals and transactions, and the calculation of new tables. With regard to the first object, the tables were roughly divided into three classes, viz. (1) ordinary tables (such as trigonometrical and logarithmic) usually published in books; (2) tables of continuously varying quantities, generally definite integrals; and (3) theory-of-numberf tables. On the first class Mr. J. W. L. Glaisher had already written a report, to which it was intended, after the lapse of several years, to add a supplement; with the second some progress had been made; while Prof. Cayley proposed to undertake the third. The Committee had to acknowledge the assistance of several foreigners, and chiefly of Prof. Bierens de Haan, who had forwarded to them an account of 128 logarithmic and 105 non-logarithmic tables; to Dr. Carl Ohrtmann, of Berlin; and Profs. W. W. Johnson and J. M. Rice, of Annapolis, Maryland. The principal achievement, however, which the Committee had to report related to the second object, and was the completion of the tables of the Elliptic Functions, the commencement of which was noticed in NATURE nearly two years ago, and on which six or seven computers, under the superintendence of Mr. J. Glaisher, F.R.S., and Mr. J. W. L. Glaisher, have since been constantly engaged. These tables (which are of double entry) give the four theta functions, which form the numerators and denominators of the three elliptic functions, and their logarithms for 8,100 arguments; so that they contain nearly 65,000 tabular results. The calculation has been carried to ten figures, but only eight will be printed, the tabular portion of the work occupying 360 pages. Parts of the introduction will be written by Prof. Cayley, Sir William Thomson, and Prof. H. J. S. Smith, and. it is hoped that before the next meeting of the Association the whole work, which will form one of the largest tables that have appeared as the result of an original calculation, will be in print. It is perhaps desirable to state that the elliptic functions which have thus been tabulated are, as it were, generalised sines and cosines. Sines and cosines may be combined so as to represent any singly periodic function, as is well known; and in the same way elliptic functions represent every possible doubly periodic function; and no quantities can be of a higher degree of periodicity. The elliptic functions (which are in a sense inverse to Legendre's Elliptic Integrals) are thus quantities of the highest importance and generality in mathematics, and they are daily becoming of more importance in physics. They appear conspicuously in the investigation of the motion of a rigid body and in electrostatics, and have also numerous applications in the theory of numbers. The calculations were just completed before the meeting, and the printing will commence immediately: it is intended that the tables shall be stereotyped to ensure freedom from typographical errors.

Transcendental numbers form a fascinating subject: so little is known about the nature of analytic constants that more research is needed in this area. Even when one is interested only in numbers like $\pi$ and $e^\pi$ which are related with the classical exponential function, it turns out that elliptic functions are required (so far -- this should not last forever!) to prove transcendence results and get a better understanding of the situation. First we briefly recall some of the basic transcendence results related with the exponential function. Next, we survey the main properties of elliptic functions that are involved in transcendence theory. We survey transcendence theory of values of elliptic functions, linear independence and algebraic independence. This splitting is somewhat artificial but convenient. Moreover, we restrict ourselves to elliptic functions, even when many results are only special cases of statements valid for abelian functions. A number of related topics are not considered here (e.g. heights, $p$-adic theory, theta functions, diophantine geometry on elliptic curves...).

Ramanujan studied the analytic properties of many q-hypergeometric series. Of those, mock theta functions have been particularly intriguing, and by work of Zwegers, we now know how these curious q-series fit into the theory of automorphic forms. The analytic theory of partial theta functions however, which have q-expansions resembling modular theta functions, is not well understood. Here we consider families of q-hypergeometric series which converge in two disjoint domains. In one domain, we show that these series are often equal to one another, and define mock theta functions, including the classical mock theta functions of Ramanujan, as well as certain combinatorial generating functions, as special cases. In the other domain, we prove that these series are typically not equal to one another, but instead are related by partial theta functions.

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.

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.

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.

In previous papers [4, 6], B.-Y. Chen introduced a Riemannian invariant δM for a Riemannian n-manifold Mn, namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold Mn in a Riemannian space form Rm(ε) satisfies: δM[les ][n2(n−2)]/ 2(n−1)H2+[half](n+1)(n−2)ε. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.

This book presents several results on elliptic functions and Pi, using Jacobi’s triple product identity as a tool to show suprising connections between different topics within number theory such as theta functions, Eisenstein series, the Dedekind delta function, and Ramanujan’s work on Pi. The included exercises make it ideal for both classroom use and self-study.

Elliptic Functions and Modular Forms

The main object of study in this paper is the double holomorphic Eisenstein series $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ having two complex variables $$\mathbf{s}=(s_1,s_2)$$ and two parameters $$\mathbf{z}= (z_1,z_2)$$ which satisfies either $$\mathbf{z}\in (\mathfrak {H}^+)^2$$ or $$\mathbf{z}\in (\mathfrak {H}^-)^2$$ , where $$\mathfrak {H}^{\pm }$$ denotes the complex upper and lower half-planes, respectively. For $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ , its transformation properties and asymptotic aspects are studied when the distance $$|z_2-z_1|$$ becomes both small and large under certain natural settings on the movement of $$\mathbf{z}\in (\mathfrak {H}^{\pm })^2$$ . Prior to the proofs our main results, a new parameter $$\eta $$ , which plays a pivotal role in describing our results, is introduced in connection with the difference $$z_2-z_1$$ . We then establish complete asymptotic expansions for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ when $$\mathbf{z}$$ moves within the poly-sector either $$(\mathfrak {H}^+)^2$$ or $$(\mathfrak {H}^-)^2$$ , so as to $$\eta \rightarrow 0$$ through $$|\arg \eta |<\pi /2$$ in the ascending order of $$\eta $$ (Theorem 1). This further leads us to show that counterpart expansions exist for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ in the descending order of $$\eta $$ as $$\eta \rightarrow \infty $$ through $$|\arg \eta |<\pi /2$$ (Theorem 2). Our second main formula in Theorem 2 yields a functional equation for $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ (Corollaries 2.1, 2.2), and also reduces naturally to various expressions of $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ in closed forms for integer lattice point $$\mathbf{s}\in \mathbb {Z}^2$$ (Corollaries 2.3–2.17). Most of these results reveal that the particular values of $$\widetilde{\zeta _{\mathbb {Z}^2}}(\mathbf{s};\mathbf{z})$$ at $$\mathbf{s}\in \mathbb {Z}^2$$ are closely linked to Weierstras’ elliptic function, the classical Eisenstein series reformulated by Ramanujan, and the Jordan–Kronecker type functions, each associated with the bases $$2\pi (1, z_j)$$ , $$j=1,2$$ . The latter two functions were extensively utilized by Ramanujan in the course of developing his theories of Eisenstein series, elliptic functions, and theta functions. As for the methods used, crucial roles in the proofs are played by the Mellin–Barnes type integrals, manipulated with several properties of hypergeometric functions; the transference from Theorem 1 to Theorem 2 is, for instance, achieved by a connection formula for Kummer’s confluent hypergeometric functions.