Inverse Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytic 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 have 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. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function, ℘z(z,τ), leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points m+nτ and “demi-lattice” points m+1/2+nτ, m+(n+1/2)τ, m+1/2+(n+1/2)τ. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative θ1′(z|τ) of the elliptic theta-function and the Weierstrass ζ-function, and that far from the origin of coordinates such zeroes of the ζ-function tend to the positions of the simple poles of this function.

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.

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.

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].

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

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.

Circular summation of theta functions in Ramanujan's Lost Notebook

Newton flows are dynamical systems generated by a continuous, desingularized Newton method for mappings from a Euclidean space to itself. We focus on the special case of meromorphic functions on the complex plane. Inspired by the analogy between the rational (complex) and the elliptic (i.e. doubly periodic meromorphic) functions, a theory on the class of so-called Elliptic Newton flows is developed. With respect to an appropriate topology on the set of all elliptic functions f of fixed order we prove: For almost all functions f, the corresponding Newton flows are structurally stable i.e. topologically invariant under small perturbations of the zeros and poles for f [genericity]. They can be described in terms of nondegeneracy-properties of f similar to the rational case [characterization].

We study the properties of two classes of meromorphic functions in the complex plane. The first one is the class of almost elliptic functions in the sense of Sunyer-i-Balaguer. This is the class of meromorphic functions f such that the family {f(z + h)} h∈ℂ is normal with respect to the uniform convergence in the whole complex plane. Given two sequences of complex numbers, we provide sufficient conditions for themto be zeros and poles of some almost elliptic function. These conditions enable one to give (for the first time) explicit non-trivial examples of almost elliptic functions. The second class was introduced by K. Yosida, who called it the class of normal functions of the first category. This is the class of meromorphic functions f such that the family {f(z + h)} h∈ℂ is normal with respect to the uniform convergence on compacta in the complex plane and no limit point of the family is a constant function. We give necessary and sufficient conditions for two sequences of complex numbers to be zeros and poles of some normal function of the first category and obtain a parametric representation for this class in terms of zeros and poles.

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.

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.

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.

The goal of this article is to introduce double‐periodic elliptic functions on the basis of a “simple” mechanical system, that of the mathematical pendulum. Thereby it is not geometry that is in the foreground, as in Gauß's analysis of the lemniscatian curve, but rather the calculation of the specific attributes of elliptic functions with the aid of a periodic integrable system. Not the spatial degree of freedom, but the time variable is continued into the complex plane. This will make it possible for us to not only identify the known real period of the pendulum oscillation, but also to detect a second imaginary period. Only then does the solution of the equation of motion become a Jacobi‐type elliptic function. Using the Cauchy integral theorem, which Gauß was already familiar with, as well as the simplest Riemannian surface of the function we want to calculate the analytic and topological characteristics of the oscillatory motion of a pendulum. Our intent is to show that elliptic functions could have appeared much earlier than 1796 in the literature. Admittedly, for this the field of complex numbers was necessary, as represented in the Gaußian plane of complex numbers. However, Gauß was unwilling to publish his findings because of his “fear of the cry of the Boeotians”.