An Essay on Continued Fractions

English translation of the paper: "De Fractionibus Continuis Dissertatio" by Leonhard Euler.Article previously published in the journal: Mathematical Systems Theory, Vol. 18, pages 295-328.The paper translated here, "De Fractionibus Continuis Dissertatio", represents Euler's first published work on the theory of continued fractions, a subject to which he often returned during his long career. The emphasis is on basic identities and analytic theory, culminating in the development of the continued fraction expansion for e. A discussion of the periodic continued fractions associated with quadratic irrationalities is also included, but applications to number theory come in later publications. Readers interested in mathematical system theory will be concerned primarily with Euler's treatment of the Riccati equation, Sections 28-33. This aspect of the paper sewed as the chief motivation for the translation project. We are grateful to C. I. Bymes for suggesting that we translate this paper. and for his continuing support. Thanks for comments and help with references also to C. Bernlohr, P. Fuhrmann, W. Gragg, A. Lindquist, C. Martin, L. F. Meyers. A. Weil, H. Wimmer, and especially to 1. Burckhardt and G. Mislin of the Euler Commission. As far as the translators know, no translation into English or another modern language has been published

SMARANDACHE FUNCTION JOURNAL, 1

This journal is yearly published (in the Spring or Fall) in a 300-400 pages volume, and 800-1000 copies. SNJ is a referred journal: reviewed, indexed, cited, concerning any of Smarandache type functions, numbers, sequences, integer algorithms, paradoxes, Non-Euclidean geometries, etc

An Infinity Of Unsolved Problems Concerning A Function In The Number Theory

W.Sierpinski has asserted to an international conference that if mankind lasted for ever and numbered the unsolved problems, then in the long run all these unsolved problems would be solved

Complex numbers from 1600 to 1840

This thesis uses primary and secondary sources to study advances in complex number theory during the 17th and 18th Centuries. Some space is also given to the early 19th Century. Six questions concerning their rules of operation, usage, symbolism, nature, representation and attitudes to them are posed in the Introduction. The main part of the thesis quotes from the works of Descartes, Newton, Wallis, Saunderson, Maclaurin, d'Alembert, Euler, Waring, Frend, Hutton, Arbogast, de Missery, Argand, Cauchy, Hamilton, de Morgan, Sylvester and others, mainly in chronological order, with comment and discussion. More attention has been given tp algebraists, the originators of most advances in complex numbers, than to writers in trigonometry, calculus and analysis, who tended to be users of them. The last chapter summarises the most important points and considers the extent to which the six questions have been resolved. The most important developments during the period are identified as follows: (i) the advance in status of complex numbers from 'useless' to 'useful'. (ii) their interpretation by Wallis, Argand and Gauss in arithmetic, geometric and algebraic ways. (iii) the discovery that they are essential for understanding polynomials and logarithmic, exponential and trigonometric functions. (iv) the extension of trigonometry, calculus and analysis into the complex number field. (v) the discovery that complex numbers are closed under exponentiation, and so under all algebraic operations. (vi) partial reform of nomenclature and symbolism. (vii) the eventual extension of complex number theory to n dimensions

The calculus according to S. F. Lacroix (1765-1843)

Silvestre François Lacroix (Paris. 1765 - ibid., 1843) was not a prominent mathematical researcher, but he was certainly a most influential mathematical book author. His most famous book is a monumental Traité du calcul différentiel et du calcul intégral (three large volumes, 1797-1800; a second edition appeared in 1810-1819) - an encyclopaedic appraisal of 18th-century calculus. He also published many textbooks, one of which is closely associated to this large Traité: the Traité élémentaire du calcul différentiel et du calcul intégral (first edition in 1802; four more editions in Lacroix's lifetime; four posthumous editions). Although most historians acknowledge the great influence of Lacroix's large Traité in early 19th-century mathematics it has not been thoroughly studied. This thesis is a contribution for correcting this omission. The focus is on its first edition, but the second edition and the Traité élémentaire, are also addressed. The thesis starts with a short biography of Lacroix, followed by an overview of the first edition of the large Traité. Next corne five chapters where particular aspects are analyzed in detail: the foundations of the calculus, analytic and differential geometry, approximate integration and conceptions of the integral, types of solutions of differential equations (singular/complete/general integrals, geometrical interpretations, and generality of arbitrary functions), and three aspects related to finite differences and series (the use of subscript indices, types of solutions of finite difference equations, and mixed difference equations); for all these aspects Lacroix's treatment is compared to the 18th-century background, and to his likely sources. Then we examine how the large Traité was adapted to a textbook - the Traité élémentaire, we take a look at the second edition of the large Traité, and conclude the body of the thesis with some final remarks

On Dirichlet's L-functions.

Fung Yiu-cho.Bibliography: leaves 93-114Thesis (M.Phil.)--Chinese University of Hong Kong, 198