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Author : Albert Eagle
Publisher : Unknown
Page : 548 pages
File Size : 51,6 Mb
Release : 1958
Category : Elliptic functions
ISBN : UCAL:B4073978
Author : Derek F. Lawden
Publisher : Springer Science & Business Media
Page : 349 pages
File Size : 53,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475739800
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 : Naum Ilʹich Akhiezer,H. H. McFaden
Publisher : American Mathematical Soc.
Page : 237 pages
File Size : 41,9 Mb
Release : 1990
Category : Mathematics
ISBN : 0821809008
Presents the theory of elliptic functions and its applications. Suitable primarily for engineers who work with elliptic functions, this work is also intended for those with background in the elements of mathematical analysis and the theory of functions contained in the first two years of mathematics and physics courses at the college level.
Author : Serge Lang
Publisher : Springer Science & Business Media
Page : 319 pages
File Size : 46,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461247524
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 : William A. Schwalm
Publisher : Morgan & Claypool Publishers
Page : 67 pages
File Size : 50,5 Mb
Release : 2015-12-31
Category : Science
ISBN : 9781681742304
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 : Andre Weil
Publisher : Springer Science & Business Media
Page : 112 pages
File Size : 45,9 Mb
Release : 1999
Category : Mathematics
ISBN : 3540650369
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 : Komaravolu Chandrasekharan
Publisher : Springer Science & Business Media
Page : 199 pages
File Size : 54,5 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642522444
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 : Paul F. Byrd,Morris D. Friedman
Publisher : Springer
Page : 370 pages
File Size : 50,9 Mb
Release : 2013-11-21
Category : Mathematics
ISBN : 9783642528033
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 : Viktor Vasil_evich Prasolov,I_Uri_ Pavlovich Solov_ev
Publisher : American Mathematical Soc.
Page : 202 pages
File Size : 43,7 Mb
Release : 1997-09-16
Category : Mathematics
ISBN : 0821897802
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 : Arthur Baker
Publisher : Unknown
Page : 136 pages
File Size : 49,8 Mb
Release : 2013-07-30
Category : Electronic
ISBN : 1491233850
The first step taken in the theory of Elliptic Functions was the determination of a relation between the amplitudes of three functions of either order, such that there should exist an algebraic relation between the three functions themselves of which these were the amplitudes. It is one of the most remarkable discoveries which science owes to Euler. In 1761 he gave to the world the complete integration of an equation of two terms, each an elliptic function of the first or second order, not separately integrable.This integration introduced an arbitrary constant in the form of a third function, related to the first two by a given equation between the amplitudes of the three.In 1775 Landen, an English mathematician published his celebrated theorem showing that any arc of a hyperbola may be measured by two arcs of an ellipse, an important element of the theory of Elliptic Functions, but then an isolated result. The great problem of comparison of Elliptic Functions of different moduli remained unsolved, though Euler, in a measure, exhausted the comparison of functions of the same modulus.It was completed in 1784 by Lagrange, and for the computation of numerical results leaves little to be desired. The value of a function may be determined by it, in terms of increasing or diminishing moduli, until at length it depends upon a function having a modulus of zero, or unity.For all practical purposes this was sufficient. The enormous task of calculating tables was undertaken by Legendre. His labors did not end here, however. There is none of the discoveries of his predecessors which have not received some perfection at his hands; and it was he who first supplied to the whole that connection and arrangement which have made it an independent science.The theory of Elliptic Integrals remained at a standstill from 1786, the year when Legendre took it up, until the year 1827, when the second volume of his Trait´e des Fonctions Elliptiques appeared. Scarcely so, however, when there appeared the researches of Jacobi, a Professor of Mathematics in K¨onigsberg, in the 123d number of the Journal of Schumacher, and those of Abel, Professor of Mathematics at Christiania, in the 3d number of Crelle's Journal for 1827.These publications put the theory of Elliptic Functions upon an entirely new basis. The researches of Jacobi have for their principal object the development of that general relation of functions of the first order having different moduli, of which the scales of Lagrange and Legendre are particular cases.It was to Abel that the idea first occurred of treating the Elliptic Integral as a function of its amplitude. Proceeding from this new point of view, he embraced in his speculations all the principal results of Jacobi. Having undertaken to develop the principle upon which rests the fundamental proposition of Euler establishing an algebraic relation between three functions which have the same moduli, dependent upon a certain relation of their amplitudes, he has extended it from three to an indefinite number of functions; and from Elliptic Functions to an infinite number of other functions embraced under an indefinite number of classes, of which that of Elliptic Functions is but one; and each class having a division analogous to that of Elliptic Functions into three orders having common properties.The discovery of Abel is of infinite moment as presenting the first step of approach towards a more complete theory of the infinite class of ultra-elliptic functions, destined probably ere long to constitute one of the most important of the branches of transcendental analysis, and to include among the integrals of which it effects the solution some of those which at present arrest the researches of the philosopher in the very elements of physics.
Author : Patrick Du Val
Publisher : Cambridge University Press
Page : 257 pages
File Size : 42,5 Mb
Release : 1973-08-02
Category : Mathematics
ISBN : 9780521200363
A comprehensive treatment of elliptic functions is linked by these notes to a study of their application to elliptic curves. This approach provides geometers with the opportunity to acquaint themselves with aspects of their subject virtually ignored by other texts. The exposition is clear and logically carries themes from earlier through to later topics. This enthusiastic work of scholarship is made complete with the inclusion of some interesting historical details and a very comprehensive bibliography.
Author : Naum Ilʹich Akhiezer
Publisher : American Mathematical Soc.
Page : 250 pages
File Size : 48,5 Mb
Release : 2024-06-07
Category : Mathematics
ISBN : 0821886770
This book contains a systematic presentation of the theory of elliptic functions and some of its applications. A translation from the Russian, this book is intended primarily for engineers who work with elliptic functions. It should be accessible to those with background in the elements of mathematical analysis and the theory of functions contained in approximately the first two years of mathematics and physics courses at the college level.
Author : Johannes Blümlein,Carsten Schneider,Peter Paule
Publisher : Springer
Page : 509 pages
File Size : 45,6 Mb
Release : 2019-01-30
Category : Computers
ISBN : 9783030044800
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.
Author : K. Venkatachaliengar,Shaun Cooper
Publisher : World Scientific
Page : 185 pages
File Size : 51,6 Mb
Release : 2012
Category : Mathematics
ISBN : 9789814366458
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 : Louis Melville Milne-Thomson
Publisher : Unknown
Page : 152 pages
File Size : 49,6 Mb
Release : 1950
Category : Elliptic functions
ISBN : WISC:89043105261