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Author : J. V. Armitage
ISBN : 1139457497
Genre : Mathematics
File Size : 80.26 MB
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In its first six chapters this 2006 text seeks to present the basic ideas and properties of the Jacobi elliptic functions as an historical essay, an attempt to answer the fascinating question: 'what would the treatment of elliptic functions have been like if Abel had developed the ideas, rather than Jacobi?' 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 then the reader is introduced to the richly varied applications of the elliptic and related functions. Applications spanning arithmetic (solution of the general quintic, the functional equation of the Riemann zeta function), dynamics (orbits, Euler's equations, Green's functions), and also probability and statistics, are discussed.

This book has grown out of a course of lectures on elliptic functions, given in German, at the Swiss Federal Institute of Technology, Zurich, during the summer semester of 1982. Its aim is to give some idea of the theory of elliptic functions, and of its close connexion with theta-functions and modular functions, and to show how it provides an analytic approach to the solution of some classical problems in the theory of numbers. It comprises eleven chapters. The first seven are function-theoretic, and the next four concern arithmetical applications. There are Notes at the end of every chapter, which contain references to the literature, comments on the text, and on the ramifications, old and new, of the problems dealt with, some of them extending into cognate fields. The treatment is self-contained, and makes no special demand on the reader's knowledge beyond the elements of complex analysis in one variable, and of group theory.

Author : Serge Lang
ISBN : 9781461247524
Genre : Mathematics
File Size : 68.68 MB
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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 : Viktor Vasil_evich Prasolov
ISBN : 0821897802
Genre : Mathematics
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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 : Eric Harold Neville
ISBN : 9781483151915
Genre : Mathematics
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Elliptic Functions: A Primer defines and describes what is an elliptic function, attempts to have a more elementary approach to them, and drastically reduce the complications of its classic formulae; from which the book proceeds to a more detailed study of the subject while being reasonably complete in itself. The book squarely faces the situation and acknowledges the history of the subject through the use of twelve allied functions instead of the three Jacobian functions and includes its applications for double periodicity, lattices, multiples and sub-multiple periods, as well as many others in trigonometry. Aimed especially towards but not limited to young mathematicians and undergraduates alike, the text intends to have its readers acquainted on elliptic functions, pass on to a study in Jacobian elliptic functions, and bring a theory of the complex plane back to popularity.

Author : Henry E. Fettis
ISBN : UOM:39015017417604
Genre : Conformal mapping
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The Jacobian elliptic functions sn(w) cn(w) dn(w) are tabulated to five decimals in the complex w = u + iv plane as functions of the nome q = .005(.005).4 for u/K = 0(.1)1. and v/K' = 0(.1)1. where q = e to the minus (pi K'/K) power and K and K' are complete elliptic integrals with moduli k and k' = square root of (l-sq. k), respectively.

Author : Peter L. Walker
ISBN : UOM:39076001868038
Genre : Mathematics
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The theory of elliptic functions represents a high point of classical analysis. Interest in the use of these mathematical tools was recently heightened by John Wile's partial proof of Fermat's Last Theorem. Now this comprehensive guide bridges the gap between elementary texts and the very high level specialist research monographs by demonstrating how the principal results can be derived using relatively modest analytical machinery. In addition to their intrinsic elegance and range, from Circular Functions to Gamma and Related, Basic Elliptic, Theta, Jacobian, Elliptic Integrals, and Modular Functions, they find uses in fields as diverse as number theory and fluid mechanics.

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 : Andre Weil
ISBN : 3540650369
Genre : Mathematics
File Size : 63.83 MB
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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 : Derek F. Lawden
ISBN : 9781475739800
Genre : Mathematics
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The subject matter of this book formed the substance of a mathematical se am which was worked by many of the great mathematicians of the last century. The mining metaphor is here very appropriate, for the analytical tools perfected by Cauchy permitted the mathematical argument to penetra te to unprecedented depths over a restricted region of its domain and enabled mathematicians like Abel, Jacobi, and Weierstrass to uncover a treasurehouse of results whose variety, aesthetic appeal, and capacity for arousing our astonishment have not since been equaled by research in any other area. But the circumstance that this theory can be applied to solve problems arising in many departments of science and engineering graces the topic with an additional aura and provides a powerful argument for including it in university courses for students who are expected to use mathematics as a tool for technological investigations in later life. Unfortunately, since the status of university staff is almost wholly determined by their effectiveness as research workers rather than as teachers, the content of undergraduate courses tends to reflect those academic research topics which are currently popular and bears little relationship to the future needs of students who are themselves not destined to become university teachers. Thus, having been comprehensively explored in the last century and being undoubtedly difficult .

Author : K. Venkatachaliengar
ISBN : 9789814366458
Genre : Mathematics
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This unique book provides an innovative and efficient approach to elliptic functions, based on the ideas of the great Indian mathematician Srinivasa Ramanujan. The original 1988 monograph of K Venkatachaliengar has been completely revised. Many details, omitted from the original version, have been included, and the book has been made comprehensive by notes at the end of each chapter. The book is for graduate students and researchers in Number Theory and Classical Analysis, as well for scholars and aficionados of Ramanujan's work. It can be read by anyone with some undergraduate knowledge of real and complex analysis.

Author : Books, LLC
ISBN : 1156454549
Genre :
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 27. Chapters: Arithmetic-geometric mean, Elliptic integral, Elliptic function, Weierstrass's elliptic functions, Theta function, Jacobi elliptic functions, Carlson symmetric form, Elliptic rational functions, J-invariant, Fundamental pair of periods, Complex multiplication, Theta representation, Dedekind eta function, Jacobi triple product, Landen's transformation, Picard-Fuchs equation, Quarter period, Jacobi theta functions, Modular lambda function, Weierstrass functions, Nome, Ramanujan theta function, Lemniscatic elliptic function, Equianharmonic, Half-period ratio. Excerpt: In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as p-functions and generally written using the symbol (or ) (a stylised letter p called Weierstrass p). The Weierstrass elliptic function can be defined in three closely related ways, each of which possesses certain advantages. One is as a function of a complex variable z and a lattice in the complex plane. Another is in terms of z and two complex numbers 1 and 2 defining a pair of generators, or periods, for the lattice. The third is in terms z and of a modulus in the upper half-plane. This is related to the previous definition by = 2/ 1, which by the conventional choice on the pair of periods is in the upper half-plane. Using this approach, for fixed z the Weierstrass functions become modular functions of . In terms of the two periods, Weierstrass's elliptic function is an elliptic function with periods 1 and 2 defined as Then are the points of the period lattice, so that for any pair of generators of the lattice defines the Weierstrass function as a function of a complex variable and a lattice. If is a complex number in the upper half-plane, then The...

Author : Johannes Blümlein
ISBN : 9783030044800
Genre : Computers
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This book includes review articles in the field of elliptic integrals, elliptic functions and modular forms intending to foster the discussion between theoretical physicists working on higher loop calculations and mathematicians working in the field of modular forms and functions and analytic solutions of higher order differential and difference equations.