Elliptic Functions

In classical analysis, one builds the catalog of special functions by repeatedly adjoining solutions of differential equations whose coefficients are previously known functions. Consequently, the properties of special functions depend crucially on the basic properties of ordinary differential equations. This naturally led to the study of formal differential equations, as in the seminal work of Turrittin [165]; this may be viewed retroactively as a theory of differential equations over a trivially valued field. After the introduction of p-adic analysis in the early twentieth century, there began to be corresponding interest in solutions of p-adic differential equations; however, aside from some isolated instances (e.g., the proof of the Nagell–Lutz theorem; see Theorem 3.4), a unified theory of p-adic ordinary differential equations did not emerge until the pioneering work of Dwork on the relationship between p-adic special functions and the zeta functions of algebraic varieties over finite fields (e.g., see [57, 58]). At that point, serious attention began to be devoted to a serious discrepancy between the p-adic and complex-analytic theories: on an open p-adic disc, a nonsingular differential equation can have a formal solution which does not converge in the entire disc (e.g., the exponential series). One is thus led to quantify the convergence of power series solutions of differential equations involving rational functions over a nonarchimedean field; this was originally done by Dwork in terms of the generic radius of convergence [59]. This and more refined invariants were studied by numerous authors during the half-century following Dwork’s initial work, as documented in the author’s book [92].

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.

This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with natural background and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with

5 - Indefinite Integrals of Special Functions

On impulsive semidynamical systems

One of the main tasks in the theory of differential equations and systems is to obtain exact solutions, which lead to serious calculations. However, we do not always succeed in finding actual solutions. Our paper considers a system of nonlinear differential equations that describes electromagnetoelasticity problems for a segmentoelectric and ferromagnetic medium. Recently, different methods have been developed for solving nonlinear differential equations, one of which is the method of decomposition by elliptic Jacobi functions, which we used to obtain the solution in our paper.

This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.

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

In this work, we give approximate expressions for Jacobian and elliptic Weierstrass functions and their inverses by means of the elementary trigonometric functions, sine and cosine. Results are reasonably accurate. We show the way the obtained results may be applied to solve nonlinear ODEs and other problems arising in nonlinear physics. The importance of the results in this work consists on giving easy and accurate way to evaluate the main elliptic functions cn, sn, and dn, as well as the Weierstrass elliptic function and their inverses. A general principle for solving some nonlinear problems through elementary functions is stated. No similar approach has been found in the existing literature.

This paper will be the first part of our works on differential Galois theory which we plan to write. Our goal is to establish a Galois Theory of ordinary differential equations. The theory isinfinite dimensionalby nature and has a long history. The pioneer of this field is S. Lie who tried to apply the idea of Abel and Galois to differential equations. Picard [P] realized Galois Theory of linear ordinary differential equations, which is called nowadays Picard-Vessiot Theory. Picard-Vessiot Theory isfinite dimensionaland the Galois group is a linear algebraic group. The first attempt of Galois theory of a general ordinary differential equations which isinfinite dimensional, is done by the thesis of Drach [D]. He replaced an ordinary differential equation by a linear partial differential equation satisfied by the first integrals and looked for a Galois Theory of linear partial differential equations. It is widely admitted that the work of Drach is full of imcomplete definitions and gaps in proofs. In fact in a few months after Drach had got his degree, Vessiot was aware of the defects of Drach’s thesis. Vessiot took the matter serious and devoted all his life to make the Drach theory complete. Vessiot got the grand prix of the academy of Paris in Mathematics in 1903 by a series of articles.

This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals. Primarily, the elliptic functions stand out as closed solutions to a class of physical and geometrical problems giving rise to nonlinear differential equations. While these nonlinear equations may not be the types of greatest interest currently, the fact that they are solvable exactly in terms of functions about which much is known makes up for this. The elliptic functions of Jacobi, or equivalently the Weierstrass elliptic functions, inhabit the literature on current problems in condensed matter and statistical physics, on solitons and conformal representations, and all sorts of famous problems in classical mechanics. The lectures on elliptic functions have evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. They are for graduate students or for researchers who want an elementary introduction to the subject that nevertheless leaves them with enough of the details to address real problems. The style is supposed to be informal. The intention is to introduce the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. This entre depends upon fewer tools and has seemed less intimidating that other typical introductions to the subject that depend on some knowledge of complex variables. The first three lectures assume only calculus, including the chain rule and elementary knowledge of differential equations. In the later lectures, the complex analytic properties are introduced naturally so that a more complete study becomes possible.

The purpose of this chapter is to provide examples of elliptic functions with prescribed properties of the orbits of critical points (and values). We are primarily focused on constructing examples of various classes of compactly nonrecurrent elliptic functions. All these examples are either Weierstrass elliptic functions or their modifications. The dynamics of such functions depends heavily on the lattice. The first three sections of this chapter have a preparatory character and, respectively, describe the basic dynamical and geometric properties of all Weierstrass elliptic functions generated by square and triangular lattices. We then provide simple constructions of many classes of elliptic functions discerned in the previous chapter. We essentially cover all of them. All these examples stem from Weierstrass $\wp$ functions. Finally, we also provide some different, interesting on their own, and historically first examples of various kinds of Weierstrass $\wp$ elliptic functions and their modifications coming from a series of papers by Hawkins and her collaborators.

Теория дифференциальных уравнений в настоящее время представляет собой исключительно богатый содержанием, быстро развивающийся раздел математики, тесно связанный с другими областями математики и с ее приложениями. При изучении конкретных дифференциальных уравнений, которые возникают в процессе решения физических задач, создаются методы, обладающие большой общностью и применяющиеся к широкому кругу математических проблем. Задачи интегрирования дифференциальных уравнений с постоянными коэффициентами оказали большое влияние на развитие линейной алгебры. В настоящее время задача решения системы линейных обыкновенных дифференциальных уравнений с постоянными коэффициентами x′(t)=A⋅x(t) является одной из важнейших проблем как теории обыкновенных дифференциальных уравнений, так и линейной алгебры. Одним из наиболее известных методов решения системы линейных обыкновенных дифференциальных уравнений с постоянными коэффициентами является метод приведения системы линейных уравнений к одному уравнению высшего порядка, позволяющему находить решения исходной системы в виде линейных комбинаций производных только одной функции. В данной работе исследована следующая задача: для каких матриц A компоненты системы x′(t)=A⋅x(t) при любом начальном условии x(t0​)=x0​ могут быть выражены в виде линейных комбинаций производных только одной заданной компоненты xk​(t). Сформулирован новый простой критерий выразимости и подробно доказана его корректность. Полученный результат может быть также применен при исследовании решений системы x′(t)=A⋅x(t) на периодичность и при изучении линейных систем на полную наблюдаемость. The theory of differential equations is currently an exceptionally content-rich, rapidly developing branch of mathematics, closely related to other areas of mathematics and its applications. When studying specific differential equations that arise in the process of solving physical problems, methods are created that have great generality and are applied to a wide range of mathematical problems. The problem of integrating differential equations with constant coefficients had a great influence on the development of linear algebra. At present, the problem of solving a system of linear ordinary differential equations with constant coefficients x′(t)=A⋅x(t) is one of the most important problems in both the theory of ordinary differential equations and linear algebra. One of the most well-known methods for solving a system of linear ordinary differential equations with constant coefficients is the method of reducing a system of linear equations to a single higher-order equation, which makes it possible to find solutions to the original system in the form of linear combinations of derivatives of only one function. In this paper, we study the following problem: for which matrices A the components of the system x′(t)=A⋅x(t) under any initial condition x(t0​)=x0​ can be expressed as linear combinations of derivatives of only one given component xk​(t). A new simple expressibility criterion is formulated, and its correctness is proved in detail. The result obtained can also be applied in the study of solutions of the system x′(t)=A⋅x(t) for periodicity and in the study of linear systems for complete observability.

Arithmetic differential equations on GLn, I: Differential cocycles