Bradley / Sandifer Cauchy's Cours d'analyse

An Annotated Translation
2009
ISBN: 978-1-4419-0549-9
Verlag: Springer US
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 412 Seiten

Reihe: Sources and Studies in the History of Mathematics and Physical Sciences

In 1821, Augustin-Louis Cauchy (1789-1857) published a textbook, the Cours d’analyse, to accompany his course in analysis at the Ecole Polytechnique. It is one of the most influential mathematics books ever written. Not only did Cauchy provide a workable definition of limits and a means to make them the basis of a rigorous theory of calculus, but he also revitalized the idea that all mathematics could be set on such rigorous foundations. Today, the quality of a work of mathematics is judged in part on the quality of its rigor, and this standard is largely due to the transformation brought about by Cauchy and the Cours d’analyse.

For this translation, the authors have also added commentary, notes, references, and an index.

Autoren/Hrsg.

Bradley, Robert E.

Sandifer, C. Edward

Inhaltsverzeichnis

1;Translators' Preface;7
2;Contents;16
3;Introduction;20
4;Preliminaries;23
5;1 On real functions.;34
5.1;1.1 General considerations on functions.;34
5.2;1.2 On simple functions.;35
5.3;1.3 On composite functions.;36
6;2 On infinitely small and infinitely large quantities, and on the continuity of functions. Singular values of functions in various particular cases. ;38
6.1;2.1 On infinitely small and infinitely large quantities.;38
6.2;2.2 On the continuity of functions.;43
6.3;2.3 On singular values of functions in various particular cases.;49
7;3 On symmetric functions and alternating functions. The use of these functions for the solution of equations of the first degree in any number of unknowns. On homogeneous functions.;66
7.1;3.1 On symmetric functions.;66
7.2;3.2 On alternating functions.;68
7.3;3.3 On homogeneous functions.;73
8;4 Determination of integer functions, when a certain number of particular values are known. Applications.;75
8.1;4.1 Research on integer functions of a single variable for which a certain number of particular values are known.;75
8.2;4.2 Determination of integer functions of several variables, when a certain number of particular values are assumed to be known.;80
8.3;4.3 Applications.;83
9;5 Determination of continuous functions of a single variable that satisfy certain conditions.;87
9.1;5.1 Research on a continuous function formed so that if two such functions are added or multiplied together, their sum or product is the same function of the sum or product of the same variables.;87
9.2;5.2 Research on a continuous function formed so that if we multiply two such functions together and then double the product, the result equals that function of the sum of the variables added to the same function of the difference of the variables.;93
10;6 On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series.;100
10.1;6.1 General considerations on series.;100
10.2;6.2 On series for which all the terms are positive.;105
10.3;6.3 On series which contain positive terms and negative terms.;111
10.4;6.4 On series ordered according to the ascending integer powers of a single variable.;117
11;7 On imaginary expressions and their moduli.;131
11.1;7.1 General considerations on imaginary expressions.;131
11.2;7.2 On the moduli of imaginary expressions and on reduced expressions.;136
11.3;7.3 On the real and imaginary roots of the two quantities + 1 and -1 and on their fractional powers.;146
11.4;7.4 On the roots of imaginary expressions, and on their fractional and irrational powers.;157
11.5;7.5 Applications of the principles established in the preceding sections.;166
12;8 On imaginary functions and variables.;173
12.1;8.1 General considerations on imaginary functions and variables.;173
12.2;8.2 On infinitely small imaginary expressions and on the continuity of imaginary functions.;179
12.3;8.3 On imaginary functions that are symmetric, alternating or homogeneous.;181
12.4;8.4 On imaginary integer functions of one or several variables.;181
12.5;8.5 Determination of continuous imaginary functions of a single variable that satisfy certain conditions.;186
13;9 On convergent and divergent imaginary series. Summation of some convergent imaginary series. Notations used to represent imaginary functions that we find by evaluating the sum of such series.;194
13.1;9.1 General considerations on imaginary series.;194
13.2;9.2 On imaginary series ordered according to the ascending integer powers of a single variable.;201
13.3;9.3 Notations used to represent various imaginary functions which arise from the summation of convergent series. Properties of these same functions.;215
14;10 On real or imaginary roots of algebraic equations for which the left-hand side is a rational and integer function of one variable. The solution of equations of this kind by algebra or trigonometry.;229
14.1;10.1 We can satisfy any equation for which the left-hand side is a rational and integer function of the variable x by real or imaginary values of that variable. Decomposition of polynomials into factors of the first and second degree. Geometric representation of real factors of the second degree.;229
14.2;10.2 Algebraic or trigonometric solution of binomial equations and of some trinomial equations. The theorems of de Moivre and of Cotes.;241
14.3;10.3 Algebraic or trigonometric solution of equations of the third and fourth degree.;245
15;11 Decomposition of rational fractions.;253
15.1;11.1 Decomposition of a rational fraction into two other fractions of the same kind.;253
15.2;11.2 Decomposition of a rational fraction for which the denominator is the product of several unequal factors into simple fractions which have for their respective denominators these same linear factors and have constant numerators.;257
15.3;11.3 Decomposition of a given rational fraction into other simpler ones which have for their respective denominators the linear factors of the first rational fraction, or of the powers of these same factors, and constants as their numerators.;263
16;12 On recurrent series.;269
16.1;12.1 General considerations on recurrent series.;269
16.2;12.2 Expansion of rational fractions into recurrent series.;270
16.3;12.3 Summation of recurrent series and the determination of their general terms.;276
17;Note I -- On the theory of positive and negative quantities.;278
18;Note II -- On formulas that result from the use of the signs > or <, and on the averages among several quantities.;301
19;Note III -- On the numerical solution of equations.;318
20;Note IV -- On the expansion of the alternating function ( y - x ) ( z - x )( z - y ) …( v - x )( v - y )( v - z ) …( v - u ).;360
21;Note V -- On Lagrange's interpolation formula.;363
22;Note VI -- On figurate numbers.;367
23;Note VII -- On double series.;374
24;Note VIII -- On formulas that are used to convert the sines or cosines of multiples of an arc into polynomials, the different terms of which have the ascending powers of the sines or the cosines of the same arc as factors.;382
25;Note IX -- On products composed of an infinite number of factors.;392
26;Page Concordance of the 1821 and 1897 Editions;332
27;References;409
28;Index;412