On the Elliptic Function Arising from the Theta Functions and Dedekind’s η-Function

In this appendix we provide some basic definitions and relations for functions that play an important role in Section 2.3. We shall not provide the related mathematical theory nor discuss theta functions in generality here. Starting from the (usual) theta functions, which were introduced by Jacobi, we follow the path taken by Rosenhain and define hyperelliptic theta functions of two variables, sometimes called ‘ultra-elliptic theta functions’ (Rosenhain 1850). Moreover, we list some useful relations between theta functions of each type. Besides the definition of the well-known elliptic integrals of the first and second kinds and the Jacobian elliptic functions, we include two less well-known functions, which can be constructed from elliptic integrals, namely Heuman's lambda function and Jacobi's zeta function. Furthermore, the relations between the Jacobian theta functions and the Jacobian elliptic functions that are important for our purpose in Subsection 2.3.3 are given. Finally, we list derivatives for some of the above functions that were used in Subsection 2.3.4. Note that the notation in the literature (especially concerning theta functions) is not standardized. Throughout this book we comply strictly with the definitions presented here.

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

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].

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.

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan’s \({}_{1}\psi _{1}\) summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved.

The logarithmic capacity (also called Chebyshev constant or transfinite diameter) of two real intervals [−1, α] ∪ [β, 1] has been given explicitly with the help of Jacobi's elliptic and theta functions already by Achieser in 1930. By proving several inequalities for these elliptic and theta functions, an upper bound for the logarithmic capacity in terms of elementary functions of α and β is derived.

CONTENTS Introduction Chapter I. Reduction of Abelian integrals and theta functions § 1. Abelian integrals and Riemann theta functions § 2. Reduction of Abelian integrals and theta functions of genus 2 § 3. Normal coverings and the reduction of theta functions Chapter II. Multiphase (finite-zone) solutions, expressed by Jacobi theta functions, of non-linear equations of KdV-type of genus § 4. Solutions of the “sine-Gordon” equation by elliptic functions § 5. Two-zone Lamé potentials and the associated reduction of hyperelliptic integrals § 6. On a periodic solution of a problem of Kovalevskaya § 7. Solutions of the Landau-Lifschitz equation References

Based on the theories of Ramanujan′s elliptic functions and the (p, k)‐parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins′ cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins′ cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

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.

A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2).

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.

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.

In the chapter we will examine theta functions, elliptic, and modular functions. We will study theta functions in great detail. Since the elliptic and modular functions are simply rational expressions in theta functions, they can be disposed of quite quickly.