Introduction to Analysis of the Infinite

Description

About this book Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art. From this it follows not only that they remain on the fringes, but in addition they entertain strange ideas about the concept of the infinite, which they must try to use. Although analysis does not require an exhaustive knowledge of algebra, even of all the algebraic technique so far discovered, still there are topics whose con­sideration prepares a student for a deeper understanding. However, in the ordinary treatise on the elements of algebra, these topics are either completely omitted or are treated carelessly. For this reason, I am cer­tain that the material I have gathered in this book is quite sufficient to remedy that defect. I have striven to develop more adequately and clearly than is the usual case those things which are absolutely required for analysis. More­ over, I have also unraveled quite a few knotty problems so that the reader gradually and almost imperceptibly becomes acquainted with the idea of the infinite. There are also many questions which are answered in this work by means of ordinary algebra, although they are usually discussed with the aid of analysis. In this way the interrelationship between the two methods becomes clear. From the preface of the author: "...I have divided this work into two books; in the first of these I have confined myself to those matters concerning pure analysis. In the second book I have explained those thing which must be known from geometry, since analysis is ordinarily developed in such a way that its application to geometry is shown. In the first book, since all of analysis is concerned with variable quantities and functions of such variables, I have given full treatment to functions. I have also treated the transformation of functions and functions as the sum of infinite series. In addition I have developed functions in infinite series..." CONTENTS Preface------------------------------------------------------------------------------------------v Translator’s Introduction-----------------------------------------------------------------------xi Book I-----------------------------------------------------------------------------------------1 I. On Functions in General--------------------------------------------------------------------2 II. On the Transformation of Functions--------------------------------------------------------17 III. On the Transformation of Functions by Substitution----------------------------------------38 IV. On the Development of Functions in Infinite Series----------------------------------------50 V. Concerning Functions of Two or More Variables---------------------------------------------64 VI. On Exponentials and Logarithms------------------------------------------------------------75 VII. Exponentials and Logarithms Expressed through Series--------------------------------------92 VIII. On Transcendental Quantities Which Arise from the Circle---------------------------------101 IX. On Trinomial Factors---------------------------------------------------------------------116 X. On the Use of the Discovered Factors to Sum Infinite Series------------------------------137 XI. On Other Infinite Expressions for Arcs and Sines-----------------------------------------154 XII. On the Development of Real Rational Functions--------------------------------------------169 XIII. On Recurrent Series----------------------------------------------------------------------181 XIV. On the Multiplication and Division of Angles---------------------------------------------204 XV. On Series Which Arise from Products------------------------------------------------------228 XVI. On the Partition of Numbers--------------------------------------------------------------256 XVII. Using Recurrent Series to Find Roots of Equations----------------------------------------283 XVII. On Continued Fractions-------------------------------------------------------------------303