Introduction To Elliptic Functions

Author : William A. Schwalm
Publisher : Morgan & Claypool Publishers
Page : 67 pages
File Size : 46,9 Mb
Release : 2015-12-31
Category : Science
ISBN : 9781681742304

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Lectures on Selected Topics in Mathematical Physics by William A. Schwalm Pdf

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.

Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 319 pages
File Size : 55,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461247524

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Elliptic Functions by Serge Lang Pdf

Elliptic functions parametrize elliptic curves, and the intermingling of the analytic and algebraic-arithmetic theory has been at the center of mathematics since the early part of the nineteenth century. The book is divided into four parts. In the first, Lang presents the general analytic theory starting from scratch. Most of this can be read by a student with a basic knowledge of complex analysis. The next part treats complex multiplication, including a discussion of Deuring's theory of l-adic and p-adic representations, and elliptic curves with singular invariants. Part three covers curves with non-integral invariants, and applies the Tate parametrization to give Serre's results on division points. The last part covers theta functions and the Kronecker Limit Formula. Also included is an appendix by Tate on algebraic formulas in arbitrary charactistic.

Author : B. Schoeneberg
Publisher : Springer Science & Business Media
Page : 244 pages
File Size : 44,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642656637

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Elliptic Modular Functions by B. Schoeneberg Pdf

This book is a fully detailed introduction to the theory of modular functions of a single variable. I hope that it will fill gaps which in view ofthe lively development ofthis theory have often been an obstacle to the students' progress. The study of the book requires an elementary knowledge of algebra, number theory and topology and a deeper knowledge of the theory of functions. An extensive discussion of the modular group SL(2, Z) is followed by the introduction to the theory of automorphic functions and auto morphic forms of integral dimensions belonging to SL(2,Z). The theory is developed first via the Riemann mapping theorem and then again with the help of Eisenstein series. An investigation of the subgroups of SL(2, Z) and the introduction of automorphic functions and forms belonging to these groups folIows. Special attention is given to the subgroups of finite index in SL (2, Z) and, among these, to the so-called congruence groups. The decisive role in this setting is assumed by the Riemann-Roch theorem. Since its proof may be found in the literature, only the pertinent basic concepts are outlined. For the extension of the theory, special fields of modular functions in particular the transformation fields of order n-are studied. Eisen stein series of higher level are introduced which, in case of the dimension - 2, allow the construction of integrals of the 3 rd kind. The properties of these integrals are discussed at length.

Author : Neal I. Koblitz
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 52,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209096

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Introduction to Elliptic Curves and Modular Forms by Neal I. Koblitz Pdf

The theory of elliptic curves and modular forms provides a fruitful meeting ground for such diverse areas as number theory, complex analysis, algebraic geometry, and representation theory. This book starts out with a problem from elementary number theory and proceeds to lead its reader into the modern theory, covering such topics as the Hasse-Weil L-function and the conjecture of Birch and Swinnerton-Dyer. This new edition details the current state of knowledge of elliptic curves.

Author : Andre Weil
Publisher : Springer Science & Business Media
Page : 112 pages
File Size : 46,5 Mb
Release : 1999
Category : Mathematics
ISBN : 3540650369

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Elliptic Functions According to Eisenstein and Kronecker by Andre Weil Pdf

Drawn from the Foreword: (...) On the other hand, since much of the material in this volume seems suitable for inclusion in elementary courses, it may not be superfluous to point out that it is almost entirely self-contained. Even the basic facts about trigonometric functions are treated ab initio in Ch. II, according to Eisenstein's method. It would have been both logical and convenient to treat the gamma -function similarly in Ch. VII; for the sake of brevity, this has not been done, and a knowledge of some elementary properties of T(s) has been assumed. One further prerequisite in Part II is Dirichlet's theorem on Fourier series, together with the method of Poisson summation which is only a special case of that theorem; in the case under consideration (essentially no more than the transformation formula for the theta-function) this presupposes the calculation of some classical integrals. (...) As to the final chapter, it concerns applications to number theory (...).

Author : Frank Bowman
Publisher : Unknown
Page : 124 pages
File Size : 50,7 Mb
Release : 1953
Category : Elliptic functions
ISBN : UCR:31210005075492

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Introduction to Elliptic Functions by Frank Bowman Pdf

Author : J. V. Armitage,W. F. Eberlein
Publisher : Cambridge University Press
Page : 404 pages
File Size : 52,8 Mb
Release : 2006-09-28
Category : Mathematics
ISBN : 0521780780

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Elliptic Functions by J. V. Armitage,W. F. Eberlein Pdf

In its first six chapters, this text presents the basic ideas and properties of the Jacobi elliptic functions as a historical essay. Accordingly, it is based on the idea of inverting integrals which arise in the theory of differential equations and, in particular, the differential equation that describes the motion of a simple pendulum. The later chapters present a more conventional approach to the Weierstrass functions and to elliptic integrals, and the reader is introduced to the richly varied applications of the elliptic and related functions.

Author : Henry McKean,Victor Moll
Publisher : Cambridge University Press
Page : 300 pages
File Size : 42,8 Mb
Release : 1999-08-13
Category : Mathematics
ISBN : 0521658179

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Elliptic Curves by Henry McKean,Victor Moll Pdf

An introductory 1997 account in the style of the original discoverers, treating the fundamental themes even-handedly.

Author : Neal Koblitz
Publisher : Springer Science & Business Media
Page : 266 pages
File Size : 40,5 Mb
Release : 1993-04-29
Category : Mathematics
ISBN : 9780387979663

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Introduction to Elliptic Curves and Modular Forms by Neal Koblitz Pdf

This textbook covers the basic properties of elliptic curves and modular forms, with emphasis on certain connections with number theory. The ancient "congruent number problem" is the central motivating example for most of the book. My purpose is to make the subject accessible to those who find it hard to read more advanced or more algebraically oriented treatments. At the same time I want to introduce topics which are at the forefront of current research. Down-to-earth examples are given in the text and exercises, with the aim of making the material readable and interesting to mathematicians in fields far removed from the subject of the book. With numerous exercises (and answers) included, the textbook is also intended for graduate students who have completed the standard first-year courses in real and complex analysis and algebra. Such students would learn applications of techniques from those courses. thereby solidifying their under standing of some basic tools used throughout mathematics. Graduate stu dents wanting to work in number theory or algebraic geometry would get a motivational, example-oriented introduction. In addition, advanced under graduates could use the book for independent study projects, senior theses, and seminar work.

Author : Viktor Vasil_evich Prasolov,I_Uri_ Pavlovich Solov_ev
Publisher : American Mathematical Soc.
Page : 202 pages
File Size : 48,7 Mb
Release : 1997-09-16
Category : Mathematics
ISBN : 0821897802

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Elliptic Functions and Elliptic Integrals by Viktor Vasil_evich Prasolov,I_Uri_ Pavlovich Solov_ev Pdf

This book is devoted to the geometry and arithmetic of elliptic curves and to elliptic functions with applications to algebra and number theory. It includes modern interpretations of some famous classical algebraic theorems such as Abel's theorem on the lemniscate and Hermite's solution of the fifth degree equation by means of theta functions. Suitable as a text, the book is self-contained and assumes as prerequisites only the standard one-year courses of algebra and analysis.

Author : Sir George Greenhill
Publisher : Unknown
Page : 386 pages
File Size : 45,6 Mb
Release : 1892
Category : Elliptic functions
ISBN : STANFORD:36105032315025

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The Applications of Elliptic Functions by Sir George Greenhill Pdf

Author : Paul F. Byrd,Morris D. Friedman
Publisher : Springer
Page : 370 pages
File Size : 44,9 Mb
Release : 2013-11-21
Category : Mathematics
ISBN : 9783642528033

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Handbook of Elliptic Integrals for Engineers and Physicists by Paul F. Byrd,Morris D. Friedman Pdf

Engineers and physicists are more and more encountering integrations involving nonelementary integrals and higher transeendental functions. Such integrations frequently involve (not always in immediately re cognizable form) elliptic functions and elliptic integrals. The numerous books written on elliptic integrals, while of great value to the student or mathematician, are not especially suitable for the scientist whose primary objective is the ready evaluation of the integrals that occur in his practical problems. As a result, he may entirely avoid problems which lead to elliptic integrals, or is likely to resort to graphical methods or other means of approximation in dealing with all but the siruplest of these integrals. It became apparent in the course of my work in theoretical aero dynamics that there was a need for a handbook embodying in convenient form a comprehensive table of elliptic integrals together with auxiliary formulas and numerical tables of values. Feeling that such a book would save the engineer and physicist much valuable time, I prepared the present volume.

Author : Georgios Pastras
Publisher : Springer Nature
Page : 111 pages
File Size : 48,5 Mb
Release : 2020-10-31
Category : Science
ISBN : 9783030593858

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The Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics by Georgios Pastras Pdf

The field of elliptic functions, apart from its own mathematical beauty, has many applications in physics in a variety of topics, such as string theory or integrable systems. This book, which focuses on the Weierstrass theory of elliptic functions, aims at senior undergraduate and junior graduate students in physics or applied mathematics. Supplemented by problems and solutions, it provides a fast, but thorough introduction to the mathematical theory and presents some important applications in classical and quantum mechanics. Elementary applications, such as the simple pendulum, help the readers develop physical intuition on the behavior of the Weierstrass elliptic and related functions, whereas more Interesting and advanced examples, like the n=1 Lamé problem-a periodic potential with an exactly solvable band structure, are also presented.

Author : Harris Hancock
Publisher : Courier Corporation
Page : 544 pages
File Size : 54,5 Mb
Release : 2004-01-01
Category : Mathematics
ISBN : 0486438252

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Lectures on the Theory of Elliptic Functions by Harris Hancock Pdf

Prized for its extensive coverage of classical material, this text is also well regarded for its unusual fullness of treatment and its comprehensive discussion of both theory and applications. The author developes the theory of elliptic integrals, beginning with formulas establishing the existence, formation, and treatment of all three types, and concluding with the most general description of these integrals in terms of the Riemann surface. The theories of Legendre, Abel, Jacobi, and Weierstrass are developed individually and correlated with the universal laws of Riemann. The important contributory theorems of Hermite and Liouville are also fully developed. 1910 ed.

Author : Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier
Publisher : Springer Science & Business Media
Page : 273 pages
File Size : 43,6 Mb
Release : 2008-02-10
Category : Mathematics
ISBN : 9783540741190

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The 1-2-3 of Modular Forms by Jan Hendrik Bruinier,Gerard van der Geer,Günter Harder,Don Zagier Pdf

This book grew out of three series of lectures given at the summer school on "Modular Forms and their Applications" at the Sophus Lie Conference Center in Nordfjordeid in June 2004. The first series treats the classical one-variable theory of elliptic modular forms. The second series presents the theory of Hilbert modular forms in two variables and Hilbert modular surfaces. The third series gives an introduction to Siegel modular forms and discusses a conjecture by Harder. It also contains Harder's original manuscript with the conjecture. Each part treats a number of beautiful applications.