Cauchy’s Cours d’analyse (eBook)

Translators' Preface 7
Contents 16
Introduction 20
Preliminaries 23
1 On real functions. 34
1.1 General considerations on functions. 34
1.2 On simple functions. 35
1.3 On composite functions. 36
2 On infinitely small and infinitely large quantities, and on the continuity of functions. Singular values of functions in various particular cases. 38
2.1 On infinitely small and infinitely large quantities. 38
2.2 On the continuity of functions. 43
2.3 On singular values of functions in various particular cases. 49
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
3.1 On symmetric functions. 66
3.2 On alternating functions. 68
3.3 On homogeneous functions. 73
4 Determination of integer functions, when a certain number of particular values are known. Applications. 75
4.1 Research on integer functions of a single variable for which a certain number of particular values are known. 75
4.2 Determination of integer functions of several variables, when a certain number of particular values are assumed to be known. 80
4.3 Applications. 83
5 Determination of continuous functions of a single variable that satisfy certain conditions. 87
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
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
6 On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series. 100
6.1 General considerations on series. 100
6.2 On series for which all the terms are positive. 105
6.3 On series which contain positive terms and negative terms. 111
6.4 On series ordered according to the ascending integer powers of a single variable. 117
7 On imaginary expressions and their moduli. 131
7.1 General considerations on imaginary expressions. 131
7.2 On the moduli of imaginary expressions and on reduced expressions. 136
7.3 On the real and imaginary roots of the two quantities + 1 and -1 and on their fractional powers. 146
7.4 On the roots of imaginary expressions, and on their fractional and irrational powers. 157
7.5 Applications of the principles established in the preceding sections. 166
8 On imaginary functions and variables. 173
8.1 General considerations on imaginary functions and variables. 173
8.2 On infinitely small imaginary expressions and on the continuity of imaginary functions. 179
8.3 On imaginary functions that are symmetric, alternating or homogeneous. 181
8.4 On imaginary integer functions of one or several variables. 181
8.5 Determination of continuous imaginary functions of a single variable that satisfy certain conditions. 186
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
9.1 General considerations on imaginary series. 194
9.2 On imaginary series ordered according to the ascending integer powers of a single variable. 201
9.3 Notations used to represent various imaginary functions which arise from the summation of convergent series. Properties of these same functions. 215
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
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
10.2 Algebraic or trigonometric solution of binomial equations and of some trinomial equations. The theorems of de Moivre and of Cotes. 241
10.3 Algebraic or trigonometric solution of equations of the third and fourth degree. 245
11 Decomposition of rational fractions. 253
11.1 Decomposition of a rational fraction into two other fractions of the same kind. 253
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
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
12 On recurrent series. 269
12.1 General considerations on recurrent series. 269
12.2 Expansion of rational fractions into recurrent series. 270
12.3 Summation of recurrent series and the determination of their general terms. 276
Note I -- On the theory of positive and negative quantities. 278
Note II -- On formulas that result from the use of the signs > or <
Note III -- On the numerical solution of equations. 318
Note IV -- On the expansion of the alternating function ( y - x ) ( z - x )( z - y ) …( v - x )( v - y )( v - z ) …( v - u ). 360
Note V -- On Lagrange's interpolation formula. 363
Note VI -- On figurate numbers. 367
Note VII -- On double series. 374
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
Note IX -- On products composed of an infinite number of factors. 392
Page Concordance of the 1821 and 1897 Editions 332
References 409
Index 412