CHAPTER 1 - ELLIPTIC FUNCTIONS

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. In this paper, we extend Floquet theorem and another theorem (which is mentioned in [1]) related to it, which are dependent on elliptic functions.

In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. In this paper, we extend Floquet theorem and another theorem (which is mentioned in [1]) related to it, which are dependent on elliptic functions.

As a by-product of a finite-size Bethe ansatz calculation in statistical mechanics, Kim has established, by an indirect route, three mathematical identities rather similar to the conjugate modulus relations satisfied by the elliptic theta constants. However, they contain factors such as and , instead of . We show that there is a fourth relation that naturally completes the set, in much the same way as there are four relations for the four elliptic theta functions. We derive all of them directly by proving and using a specialization of Weierstrass' factorization theorem in complex variable theory.

Jacobi elliptic functions are a realization of one of the simplest cases of elliptic functions, as described in Chapter 14: functions with two simple poles in a period parallelogram that are odd around each pole. These functions come up naturally in certain problems of mechanics, such as the motion of an ideal pendulum. In pure mathematics they arise, for example, in connection with maps from the upper half plane to a parallelogram. In this chapter we begin with the pendulum equation and derive the properties of the functions associated to it. The triple of Jacobi functions \({{\,\mathrm{\mathrm{sn\,}}\,}}\), \({{\,\mathrm{\mathrm{cn\,}}\,}}\), \({{\,\mathrm{\mathrm{dn\,}}\,}}\) is closely analogous to the pair of trigonometric functions sine and cosine, and satisfy similar identities.

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.

The 1d family of Poncelet polygons interscribed between two circles is known as the Bicentric family. Using elliptic functions and Liouville's theorem, we show (i) that this family has invariant sum of internal angle cosines and (ii) that the pedal polygons with respect to the family's limiting points have invariant perimeter. Interestingly, both (i) and (ii) are also properties of elliptic billiard N-periodics. Furthermore, since the pedal polygons in (ii) are identical to inversions of elliptic billiard N-periodics with respect to a focus-centered circle, an important corollary is that (iii) elliptic billiard focus-inversive N-gons have constant perimeter. Interestingly, these also conserve their sum of cosines (except for the N=4 case).

The aim of this chapter is to give a very brief review of the basic algebraic techniques which form the foundation of invariant theory and of algebraic geometry generally. Beginning in Section 2.1 we introduce Noetherian rings, taking as our point of departure Hilbert's Basis Theorem, which was discovered in the search for a proof of finite generation of rings of invariants. (This result will appear in Chapter 4). In Section 2.2 we prove unique factorisation in polynomial rings, by induction on the number of variables using Gauss's lemma. In Section 2.3 we prove the important fact that in a finitely generated algebra over a field an element contained in all maximal ideals is nilpotent. As we will see in Chapter 3, this observation is really nothing other than Hilbert's Nullstellensatz. A power series ring in one variable is an example of a valuation ring, and we discuss these in Section 2.4. A valuation ring (together with its maximal ideal) is characterised among subrings of its field of fractions as a maximal element with respect to the dominance relation. This will be used in Chapter 3 for proving the Valuative Criterion for completeness of an algebraic variety. In the final section we discuss Nagata's example of a group action under which the ring of invariants which is not finitely generated – that is, his counterexample to Hilbert's 14th problem. This is constructed by taking nine points in general position in the projective plane and considering the existence and non-existence of curves of degree d with assigned multiplicity m at each of the points, and making use of Liouville's Theorem on elliptic functions. Hilbert's Basis Theorem We begin with a discussion of the Basis Theorem, which is the key to Hilbert's theorem of finite generatedness that we will meet in Chapter 4. In Hilbert's original paper [19] the word ideal is not used; and we would like to state the Basis Theorem in a form close to that expressed by Hilbert.

Weierstrass’s ζ-function is a meromorphic function, which has simple poles, with residues equal to one, at all points which correspond to the periods of Weierstrass’s ℘-function. It is not elliptic. But every elliptic function can be expressed in terms of ζ and its derivatives; in fact ζ’(z)= -℘(z).KeywordsEntire FunctionMeromorphic FunctionElliptic FunctionSimple PolisBasic PeriodThese 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.

Given the complex parameter τ (with positive imaginary part), the quarter-periods K and iK′ of the Jacobian elliptic functions are determined by the equations (2.2.7) and (2.2.8); according to equation (2.2.3), r is precisely the ratio iK′/K of these quarter-periods. Thus it is possible to construct a set of Jacobi functions having a common pair of arbitrary periods 2ω1, 2ω3 in the following manner: First, choose the notation so that ω3/ω1 = τ has positive imaginary part (if this ratio is real, the elliptic functions cannot be defined (see section 8.1)). Secondly, construct Jacobian elliptic functions, sn u,etc., from theta functions with parameter τ. Thirdly, transform from the variable u to a new variable y by the equation u = 2Kv/ω 1. Clearly, the resulting functions of y will have periods 2ω1 and 2ω3. As proved in section 2.8, these functions will have exactly two simple poles in each of their primitive period parallelograms.

8-9 - SPECIAL FUNCTIONS

Analytical solutions for rock stress around square tunnels using complex variable theory

In this study, we proposed an evolutionary complex variable method (ECVM) to find the analytical solution for the stress distribution in an infinite homogeneous, isotropic, and elastic rock mass. This ECVM was a combination of conformal mapping functions, firefly algorithm (FA), and the complex variable theory. Conformal mapping functions were determined by FA to transform arbitrary stope configurations into unit circles. The complex variable theory was then utilized to calculate two complex potential functions, resulting in stress distribution around arbitrary stope configurations solved. A case study involving the analytical solution around rectangular stopes was performed and validated by Abaqus finite-element software. The implementation of the proposed method for arbitrary stope configurations was discussed, and conformal mapping functions for several complex stope configurations were provided. The results showed that there was a good agreement between the analytical solution and numerical modeling. The difference between the analytical solution and Abaqus were mainly around stope corners, which might be because the grid size in Abaqus is not small enough. FA was found to be efficient and advantageous in the determination of conformal mapping functions. The proposed analytical solution has practical significance because it could be used for parameter sensitivity analysis, feasibility studies, and verification of numerical modeling.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804–51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802–29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891. Edited by fellow German mathematician Karl Weierstrass (1815–97), Volume 6 appeared in 1891.

One of the greatest mathematicians of the nineteenth century, Carl Gustav Jacob Jacobi (1804–51) burst into the limelight with his redevelopment, together with Niels Henrik Abel (1802–29), of the theory of elliptic functions. His pioneering work was characterised by the variety of problems tackled and the power of the tools used to tackle them. His lasting influence on rational mechanics, number theory, partial differential equations, complex variable theory and computation is marked by the number of fundamental concepts that bear his name (the Jacobian, the Jacobi sum and the Jacobi symbol, among others). His collected works, comprising treatises, letters and papers written in German, Latin and French, were published in eight volumes between 1881 and 1891, edited chiefly by Karl Weierstrass (1815–97). Published in 1884, this supplementary volume contains Jacobi's 1842–3 lectures on dynamics as compiled by Alfred Clebsch (1833–72) in the revised second edition by Eduard Lottner (1826–87).

In this contribution, heterogeneous micropolar elastic fiber‐reinforced composites (FRCs) with a periodic structure are analyzed using the two‐scale asymptotic homogenization method (AHM). We focus on predicting the antiplane effective properties of micropolar two‐phase FRCs with parallelogram‐like unit cells. The periodic structure is defined by unidirectional, infinitely long, and concentric cylindrical fibers embedded in a homogeneous matrix. Constituent materials are assumed centro‐symmetric isotropic materials, and perfect interface conditions are considered. The AHM allows us to address the local problems on the periodic cell and determine the corresponding effective properties. This is achieved by employing two‐scale asymptotic expansions for the displacement and microrotation fields, which depend on both macro‐ and micro‐scales. The complex variable theory, combined with the complex‐potential method and doubly periodic Weierstrass elliptic functions, is applied to determine the solution of the antiplane local problems. Simple closed‐form formulas are provided for the antiplane stiffness and torque effective properties of two‐phase micropolar elastic FRCs, which depend on the physical properties and volume fractions of constituents. Finally, numerical examples are reported and discussed. Comparisons with other theoretical models are also presented, and good agreements are obtained.