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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

Chapter 1. On real functions.

[31] When variable quantities are related to each other such that the value of one of the variables being given one can find the values of all the other variables, we normally consider these various quantities to be expressed by means of the one among them, which therefore takes the name the

independent variable

. The other quantities expressed by means of the independent variable are called

functions

of that variable.

Robert E. Bradley, C. Edward Sandifer

Chapter 2. On infinitely small and infinitely large quantities, and on the continuity of functions. Singular values of functions in various particular cases.

[37] We say that a variable quantity becomes

infinitely small

when its numerical value decreases indefinitely in such a way as to converge towards the limit zero. It is worth remarking on this point that one ought not confuse a constant decrease with an indefinite decrease. The area of a regular polygon circumscribed about a given circle decreases constantly as the number of sides increases, but not indefinitely, because it has as its limit the area of the circle.

Robert E. Bradley, C. Edward Sandifer

Chapter 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.

[71] A

symmetric

function of several quantities is one which conserves the same value and the same sign after any exchange made among its quantities.

Robert E. Bradley, C. Edward Sandifer

Chapter 4. Determination of integer functions, when a certain number of particular values are known. Applications.

[83] To determine a function when a certain number of particular values are taken to be known is what we call to

interpolate

. When it is a matter of a function of one or two variables, this function can be considered as the ordinates of a curve or of a surface, and the problem of

interpolation

consists of fixing the general value of this ordinate given a certain number of particular values, that is to say, to make the curve or the surface pass through a certain number of points. This question can be solved in an infinity of ways, and in general the problem of interpolation is indeterminate. However, the indeterminacy will cease if, to the knowledge of the particular values of the desired function, we add the expressed condition that this function be integer, and of a degree such that the number of its terms becomes precisely equal to the number of particular values given.

Robert E. Bradley, C. Edward Sandifer

Chapter 5. Determination of continuous functions of a single variable that satisfy certain conditions.

[98] When, instead of integer functions we imagine any functions, so that we leave the form entirely arbitrary, we can no longer successfully determine them given a certain number of particular values, however large that number might be, but we can sometimes do so in the case where we assume certain general properties of these functions.

Robert E. Bradley, C. Edward Sandifer

Chapter 6. On convergent and divergent series. Rules for the convergence of series. The summation of several convergent series.

[114]We call a series an indefinite sequence of quantities, u0, u1, u2, u3, …, which follow from one to another according to a determined law. These quantities themselves are the various terms of the series under consideration. Let be the sum of the first

n

terms, where

n

denotes any integer number. If, for ever increasing values of

n

, the sum

s

n

indefinitely approaches a certain limit

s

, the series is said to be

convergent

, and the limit in question is called the sum of the series. On the contrary, if the sum

s

n

does not approach any fixed limit as n increases indefinitely, the series is

divergent

, and does not have a sum. In either case, the term which corresponds to the index

n

, that is

u

n

, is what we call the

general term

. For the series to be completely determined, it is enough that we give this general term as a function of the index

n

.

Robert E. Bradley, C. Edward Sandifer

Chapter 7. On imaginary expressions and their moduli.

[153] In analysis, we call a

symbolic expression

or

symbol

any combination of algebraic signs that do not mean anything by themselves or to which we attribute a value different from that which they ought naturally to have. Likewise, we call

symbolic equations

all those that, taking the letters and the interpretations according to the generally established conventions, are inexact or do not make sense, but from which we can deduce exact results by modifying and altering either the equations themselves or the symbols which comprise them, according to fixed rules. The use of symbolic expressions or equations is often a means of simplifying calculations and of writing in a short form results that appear quite complicated. We have already seen this in the second section of the third chapter where formula (9) gives a very simple symbolic value to the unknown x satisfying equations (4).

1

Among those symbolic expressions or equations which are of some importance in analysis, we should distinguish above all those which we call

imaginary

. We are going to show how we can put them to good use.

Robert E. Bradley, C. Edward Sandifer

Chapter 8. On imaginary functions and variables.

When the constants or variables contained in a given function, having been considered real are later supposed to be imaginary, the notation that was used to express the function cannot be retained in the calculation except by virtue of new conventions able to determine the sense of this notation under the new hypotheses. Thus, for example, by virtue of the conventions established in the preceding Chapter, the values of the notations.

Robert E. Bradley, C. Edward Sandifer

Chapter 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.

Be the sum of the first

n

terms of this series. Depending on whether or not

s

n

converges towards a fixed limit∈dex{limit} for increasing values of

n

, we say that series (3) is

convergent

and that it has this limit as its sum, or else that it is

divergent

and it does not have a sum. The first case evidently occurs if the two sums.

Robert E. Bradley, C. Edward Sandifer

Chapter 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.

where

n

represents the degree of this equation and

a

0

,

a

1

,

a

2

, …,

a

n−1

, an, are constant coefficients, real or imaginary. A root of this equation is any expression,real or imaginary, that when substituted in place of the unknown value

x

, makes the left-hand side equal to zero.

Robert E. Bradley, C. Edward Sandifer

Chapter 11. Decomposition of rational fractions.

Imagine now that we separate the function F(

x

) into two factors where the first, instead of being linear, corresponds to several roots of the equation F(

x

) = 0. For example, take for the first factor the factor of second degree.

Robert E. Bradley, C. Edward Sandifer

Chapter 12. On recurrent series.

[321]A series ordered according to the ascending integer powers of the variable

x

, is called

recurrent

when in this series, starting after a given term, the coefficient of any power of the variable is expressed as a linear function of a fixed number of the coefficients of lesser powers, and consequently it suffices to

run back

1

to the values of these last coefficients to deduce the one we are seeking.

Robert E. Bradley, C. Edward Sandifer

Backmatter

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