ELLIPTIC FUNCTIONS AND THE QUINTUPLE, HIRSCHHORN AND WINQUIST PRODUCT IDENTITIES

In this paper we establish an identity involving logarithmic derivative of theta function by the theory of elliptic functions. Using these identities we introduce Ramanujan’s modular identities, and also re-derive the product identity, and many other new interesting identities.

In the memoir En metod att komma i analytisk besittning af de elliptiska funktionernat we have shown that the method given by Abel in his Recherches sur les fonctions elliptiques,+ which is historically the earliest method of deriving the elliptic functions, is also mathematically sound and leads to the desired result with perfect rigor. The method of Abel has the advantage over all other methods that it does not presuppose any theorems from the general theory of functions. It is consequently the most elementary of all mathematical processes which lead to a real command of the analytical properties of the elliptic functions. A few serious objections can be raised against Abel's own presentation, however. The most important of these is that the passage from the multiplication theorems to the infinite double series and double products is not sufficiently justified; in fact, it is based upon an argument which it is scarcely possible to make mathematically rigorous. This has been pointed out by Broch in his paper Om de elliptiske Funktioners Roekkeudvikling.? In this paper, as well as in his learned and suggestive book Traite Elementaire des Fonctions Elliptiques, ff there is developed a new derivation of this limiting passage which is perfectly distinct from that of Abel.

The object of the memoir is to notice certain forms into which the primary elliptic functions admit of being thrown, and to discuss the identical relations to which they give rise. These latter, it is shown, can be obtained directly by the aid of Fourier’s theorem , or in a less straight-forward manner by ordinary algebra.

The paper is devoted to problems at the intersection of formal group theory, the theory of Hirzebruch genera, and the theory of elliptic functions. In the focus of our interest are Tate formal groups corresponding to the general five-parametric model of the elliptic curve as well as formal groups corresponding to the general four-parametric Krichever genus. We describe coefficient rings of formal groups whose exponentials are determined by elliptic functions of levels 2 and 3.

In this chapter we consider how elliptic function theory and complex variable theory were finally drawn together in the 1830s and 1840s. As the recognition of the importance of the work of Abel and Jacobi grew, mathematicians came to feel that it was unsatisfactory to base the theory of elliptic functions on the inversion of many-valued integrals. One alternative would have been to adopt and develop Cauchy’s theory of complex integrals. By and large this was not done, and it is interesting to examine why. The study of elliptic integrals was felt by many to be fraught with ambiguity because of the square root in the integrand. Moreover, Cauchy’s system of definitions, based on his newly defined concepts of limit, continuity, differentiability, and integrability, was incompatible with talk of many-valued functions—Cauchy did not define continuity for a many-valued function, and indeed a many-valued function cannot be continuous according to Cauchy’s use of the term. Although a doubly periodic function is a meromorphic function defined on the whole of the complex plane, an elliptic integral makes better sense on something like a Riemann surface (a torus in this case). Thus the many-valued nature of an elliptic integral posed a challenge to mathematicians throughout the 1830s and 1840s. So the perceived problem with the foundations did not meet with a ready answer in the newly emerging theory of complex functions. Matters were to be worse with hyperelliptic integrals, because the corresponding inverse functions could not be treated as multiply-periodic functions in the plane.

According to JACOBI, the theory of elliptic functions was born between the twenty-third of December 1751 and the twenty-seventh of January 1752. On the former date, the Berlin Academy of Sciences handed over to EULER the two volumes of Marchese FAGNANO’s Produzioni Matematiche, published in Pesaro in 1750 and just received from the author; Euler was requested to examine the book and draft a suitable letter of thanks. On the latter date, Euler, referring explicitly to Fagnano’s work on the lemniscate, read to the Academy the first of a series of papers, eventually proving in full generality the addition and multiplication theorems for elliptic integrals.KeywordsNumber TheoryArithmetic ProgressionTwelfth CenturyEuclidean AlgorithmTriangular NumberThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Hermite, in the course of his investigations in the theory of elliptic functions, was led to introduce certain related functions which he called doubly periodic functions of the second kind.

A theta function identity of degree 7 is established with the theory of elliptic functions. A multitude of new modular identities are derived from this theta function identity. Some finite trigonometric sums are given.

2 - ARTHUR CAYLEY (1821–1895)

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Introduction Arthur Cayley's 1854 paper On the theory of groups, as depending on the symbolic equation n = 1 inaugurated the abstract idea of a group [2]. I have used this very understandable paper several times for reading, discussion, and homework in teaching introductory abstract algebra, where students are rst introduced to group theory. Many facets of this short paper provide wonderful pedagogical bene ts, not least of which is tremendous motivation for the modern theory. More information on teaching with original historical sources is available at our web resource [8], which includes information on the e cacy of teaching with original sources [3, 4, 5, 9, 10], about courses based entirely on original sources [6, 7], and which provides access to books and other resources for teaching with original sources. Cayley introduces groups as a unifying framework for several critical examples current in the early nineteenth century. These include \substitutions (permutations, with reference to Galois on solutions of equations), \quaternion imaginaries, compositions arising in the theory of elliptic functions, and transposition and inversion of matrices. These phenomena are either familiar to or easily understandable for our students. Cayley observes that they all have common features, which he abstracts in a rst attempt at de ning a group axiomatically. He proceeds to develop initial steps of a theory, including stating and using what Lagrange's theorem tells us on orders of elements, which classi es groups of prime order. Then he provides a detailed classi cation of all groups of order up to six. Cayley gives a brief description of how to form what we call a group algebra, and comments that when applied to the six-element symmetric group on three letters, the group

XLIII. <i>On</i> M. Jacobi's <i>theory of elliptic functions</i>

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DOI: 10.2514/1.41628 This paper explores the analytical solution properties surrounding a hypothetical orbit in an invariant plane perpendicular to the line joining the two primaries in the circular restricted three-body problem. Assuming motion can be maintained in the plane, Jacobi’s integral equation can be analytically integrated, yielding a closed-form expression for the period and path of the third body expressed with elliptic integral and elliptic function theory. In this case, the third body traverses a circular path with nonuniform speed. In a strict sense, the in-plane assumption cannot be maintained naturally. However, the hypothetical orbit is shown to satisfy Jacobi’s integral equation and thetangentialmotionequation exactlyandtheothertwomotionequationsapproximatelyin bounded-averagedand banded sense. More important, the hypothetical solution can be used as the basis for an iterative analytical solution procedure for the three-dimensional trajectory where corrections are computable in closed form. In addition, the in-plane assumption can be strictly enforced with the application of a modulated thrust acceleration which is expressible in closed form. Presented methodology is primarily concentrated on halo-class orbits. Nomenclature a = radius of circular orbit C = Jacobi’s constant G = universal gravitational constant J = Jacobi’s function m1 = mass of first primary m2 = mass of second primary,m2 � m1 r = position vector of third mass relative to center of mass r1 = position vector of first primary relative to center of mass r2 = position vector of second primary relative to center of mass r12 = distance between two primaries � 1 = position vector of third mass relative to first primary � 2 = position vector of third mass relative to second primary ! = angular velocity vector of rotating fame

We produce more elementary algorithms than those of Brent and Salamin for, respectively, evaluating e x {e^x} and π \pi . Although the Gauss arithmetic-geometric process still plays a central role, the elliptic function theory is now unnecessary.