cauchy sc rb

7/30/2019 Cauchy Sc Rb

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Cauchys Cours dAnalyse

Rob Bradley

Dept. of Mathematics and Computer ScienceAdelphi University

Garden City, NY 11530

bradley at adelphi dot edu

MAA Short Course, San Francisco, CAJanuary 12, 2010

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Augustin-Louis Cauchy (1789-1857)

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Editions of the Cours danalyse

French: 1821 Original edition, 1897 Oeuvres completes. Both

are available at gallica.bnf.fr.

German: 1828, 1864

Russian: 1885

Spanish: 1994

English: 2009. Springer, joint work with Ed Sandifer

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From the Translators Preface:

We believe that the primary purpose of a translation

such as this one is to make the work available in English,

and not to provide a platform for our opinions on how this

work should be interpreted. Towards this end, we havegenerally limited our commentary to expository remarks

rather than interpretative ones. For those passages that

are controversial and subject to a variety of interpreta-

tions, we try to refer the interested reader to appropriate

entry-point sources and do not try to be comprehensive.

. . . Our ambition is, as much as the very idea of trans-

lation allows, to let Cauchy speak for himself.

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Outline of this Presentation

1. Brief Survey of Continental Analysis to 1821

Foundations of Calculus Algebraic Analysis

2. Cauchys Life and Times

Childhood and Education Early Career Professor at the Ecole Polytechnique Later Years

3. Contents of the Cours danalyse

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Great Continental Analysis Texts

1670s Leibniz discovery

1696 LHospital Analyse des infiniment petits

Geometric Period

1748 Euler Introductio in analysin infinitorum

Algebraic Period

1821 Cauchy Cours danalyse

Arithmetic Period

1850s Weierstrass et al; the modern paradigm

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Foundational Schools for Calculus

1. Infinitesimals the infinitely small

Bernoulli/lHospital, Euler A collection of ingenious fallacies (Rolle)

The Analyst (Berkeley, 1734)

2. Limits

Implicit in Newton Explicit in Maclaurin (1743)

DAlembert, Lhuilier, Carnot

3. Algebra/Power Series

Theorie des Fonctions analytiques (Lagrange, 1797) Derivatives from power series coefficients

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Bernoullis Differentials, from LHospital

Postulate 1. A quantity which is increased or decreased by

another quantity which is infinitely smaller than it is, may be

considered as remaining the same.

Postulate 2. A curved line may be considered as an assemblageof infinitely many straight lines, each one being infinitely small.

Product Rule. Let z = xy. While x becomes x + dx and y

becomes y + dy, z becomes z + dz = (x + dx)(y + dy). Thus

dz = x dy + y dx + dxdy = x dy + y dx by Postulate 1, becausedxdy is infinitely smaller than either dx or dy. In particular,

dx2 = 2x dx and, by induction,

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Snapshots ca. 1800

Silverstre Francois Lacroix (1765-1843) published Traite decalcul differentiel et du calcul integral (1797-1800). He chose

the Langrangian formalism, but explained and occasionally

used all three foundational methods. In 1802, he distilled the

Traite elementaire de calcul differentiel et du calcul integralfrom it, using limits and derivatives.

Francois-Joseph Servois (1768-1847) wrote an essay in 1814defending the Lagrangian formalism, but that was sympa-

thetic to a foundation based on limits. On the other hand, heclaimed that the use of infinitesimals in mathematics would

one day be accused of having slowed the progress of the

mathematical sciences, and with good reason.

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Cauchy was Pivotal, but . . .

The revision of the fundamental principles of the cal-

culus, which was initiated by Cauchy and Abel and carriedthrough by Weierstrass and his followers, led to the devel-

opment of the -proof (early introduced by Cauchy) and

to the precise formulation of definitions and theorems.

William F. Osgood, review of Cours danalyse mathematique by

Goursat, Bull. AMS 9 (1903), p. 547-555.

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Algebraic Analysis?

Klein, Elementary Mathematics from an Advanced Standpoint

(1908, 1924), in the section Concerning the Modern Develop-

ment and the General Structure of Mathematics

Plan A: (most widespread in the schools and in elementary

textbooks) theory of equations logarithms trig functions algebraic analysis which teaches the development of thesimplest functions into infinite series.

Plan B: graphical representation of simple functions slopeand differential calculus integration; log and arcsin as definiteintegrals power series by means of Taylors theorem.

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Power Series Without Calculus

Geometric Series

1

1 x = 1 + x + x2

+ x

3

+ . . .

Derived by long division, which works for any rational function.

Binomial Series for fractional or negative n

(1 + x)n = 1 + nx +n(n

1)

1 2 x2 +n(n

1)(n

2)

1 2 3 x3 + . . .Derived by analogy with the case of natural numbers.

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Exponential Series, following Eulers Introductio. Let a > 0.

Because a0 = 1, a is infinitely close to 1 when is infinitelysmall. Thus, there is an infinitely small such that

a = 1 + = 1 + k,

for a finite constant k. Let e be the value of a that gives k = 1.

Now let x be a (finite) real number and x = n, so that n isinfinitely large. Then

ex = (e)n = (1 + )n

= 1 + n +n(n 1)

1

2

2 +n(n 1)(n 2)

1

2

3

3 + . . .

= 1 + n +n 1

n (n)2

1 2 +n 1

n n 2

n (n)3

1 2 3 + . . .

= 1 + x +x2

2!+

x3

3!+ . . .

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The Generality of Algebra

Leibniz Law of Continuation (or Law of Continuity) seemed to

give license to pass freely from positive to negative, from integer

to fractional, from real to complex. Algebraic formulas seeemed

to possess something of a magical or alchemical character.

ei = 1 + i +i22

1 2 +i33

1 2 3 +i44

1 2 3 4 + . . .

= cos + i sin

The Cours danalsye was in many ways a reaction to free-wheeling

manipulation of series, without regard to consideration of conver-

gence, and to the automatic extension of real-valued formulas to

the cases of complex numbers, the infinitely large and infinitely

small.

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Selected Biographical Sources

C. A. Valson, La Vie et les Travaux du Baron Cauchy, 1868.

I. Grattan-Guinness, The Development of the Mathematical Foun-dations of Analysis from Euler to Riemann, 1970; Convolutionsin French Mathematics, 1990.

H. Freudenthal in Dictionary of Scientific Biography, 1971.

J. V. Grabiner, The Origins of Cauchys Rigorous Analysis, 1981.

U. Bottazzini, tr. W. Van Egmond, The Higher Calculus: A

History of Real and Complex Analysis from Euler to Weierstrass,1986.

B. Belhoste, tr. F. Ragland, Augustin-Louis Cauchy: A Biogra-phy, 1991.

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Cauchys Early Years

8/21/1789 Augustin-Louis Cauchy born in Paris to Louis-Francois

and Marie-Madeleine (Desestre) a few weeks after the storming

of the Bastille.

1794 Family flees to Arceuil in April to escape the terror. Re-

turns to Paris in the fall. Early education by his father, a gifted

and politically savvy administrator.

1/1/1800 Father becomes Secretary General of the Senate.

Laplace (Chancellor of the Senate) and Lagrange were both

Senators; young Cauchy presented to both. Valson attributes

to Lagrange: One day he will replace all of us simple geome-

ters.

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Cauchys Education

Fall 1802 On Lagranges recommendation, enrols in Ecole Cen-

trale du Pantheon.

10/30/1805 Examined by Biot for the Ecole Polytechnique

(est. 1794). Places second of 293 applicants (125 admitted).

2 year course of study.

October 1807 Enters Ecole des Ponts et Chaussees.

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Ecole Polytechnique

Two year course of study, followed by 2-3 years in an engineering

school. The Ecole des Ponts et Chaussees was a select one, to

which almost all the Polytechniciens aspired. [Belhoste]

Year 1 Year 2

Analysis 29% 18%Mechanics 17% 22%Desc. Geometry 26% 3%

Analysis texts in Cauchys first year: Garnier, Cours danalyse

algebrique and Lacroix, Traite elementaire.

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Early Career I

Jan. 1809 Completed program at Ecole des Ponts et Chaussees.

Won 4 academic prizes. Summer internships at Ourcq Canal

in Paris.

1/18/1810 Appointed aspirant-ingenieur in Cherbourg. Two

papers on polyhedra presented to the First Class of the Institut

de France (FCI). Also worked on conic sections; apparently paper

was never submitted.

9/24/1812 Returns to Paris in ill health.

11/20/1812 Submits a paper on symmetric functions to FCI.

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Early Career II

1813 Works for a while on Ourcq Canal, then takes unpaid

leave. Unsuccessfull applications for faculty position at Ecole

Polytechnique, positions at FCI and Bureau de Longitudes.

1814-1815 Time of war and upheaval in Paris.

8/11/1814 Submits paper on definite integrals to FCI; the start-

ing point for his original work on analysis, esp. complex analysis.

11/28/1814 Comes second to Poisson for a vacancy at FCI.

12/31/1814 Elected to Societe Philomathique, waiting room

for FCI.

5/8/1815 Came last for Napoleons position in mechanics FCI.

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Professor at the Ecole Polytechnique I

7/18/1815 Louis XVIII re-enters Paris. Bonapartists purged,

including from Ecole Polytechnique and FCI.

12/2/1815 Appointed assistant professor of analysis at Ecole

Polytechnique, following several forced resignations. (11/2 Lost

to Binet for Poissons chair.)

11/13/1815 Presents a proof of Fermats conjecture on polyg-

onal numbers to FCI.

12/16/1815 Essay on waves wins first Prize of the FCI.

3/21/1816 Louis XVIII re-orgaiznes FCI as Academie des Sci-

ences, appoints Cauchy and Breguet to replace Carnot and Monge

in mechanics.

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Professor at the Ecole Polytechnique II

March 1816 Ecole Polytechnique closed abruptly, students sent

home, Laplace sets up a commmission to reorganize/demilitarize

the school.

9/4/1816 Cauchy appointed full professor of analysis and me-chanics.

11/15/1816 Curriculum committee rejects Cauchys proposal:

analysis and calculus 1st year, mechanics 2nd year. First year:

50 lectures in analysis/calculus, followed by 35 in mechanics.

1/17/1817 1816-17 academic year begins late. Cauchy and

Ampere teach analysis/machanics in perfect alternation through

1829-30. Cauchys research output declines until 1823.

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Professor at the Ecole Polytechnique III

6/15/1820 Cauchy and Ampere are instructed to revise theanalysis/mechanics sequence vol. 1 of the Cours danalyse

is essentially complete at this point. Printing delays cause it

to miss the deadline for the incoming class of fall 1820, and

he probably added the Notes section as a result. Vol. 1 finally

appears in June 1821, but vol. 2 never would.

4/12/1821 students revolt as his 66th lecture goes into over-

time. During the course of an investigation it was discovered that

Cauchy has been short-changing mechanics. First-year analysis

was therefore revised and algebraic analysis mostly abandoned.

1817 56 lectures in analysis (32+24)

1818-19 63 lectures in analysis (33+30)1820-21 revolt on 66th lecture!

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Cauchys Textbooks of the 1820s

1821: Cours danalyse de lEcole Royale Polytechnique; premiere

partie, analyse algebrique (OC. 2.3).

1823: Resume des lecons donnee a lEcole Royale Polytechnique

sur le calcul infinitesimal (OC 2.4).

1826, 1828: Lecons sur les applications du calcul infinitesimal

a la Geometrie, 2 vols (OC 2.5).

1829: Lecons sur le calcul differentiel (OC 2.4).

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Cauchys Later Life

1830-38 Self-imposed exile in Turin and Prague. In Prague he

was made Baron by the exiled Bourbon king Charles X.

1838-48 In Paris with no teaching position, because he wouldnot swear a loyalty oath to King Louis-Philippe.

1848-57 Loyalty oaths not required during Second Republic;

Cauchy was appointed professor of astronomy at the Faculte des

Sciences, where he had held an adjunct position in the 1820s.

5/22/1857 Cauchy dies in Paris at the age of 67.

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Structure of the Cours danalyse I

Introduction

Foundations: Preliminaries; Chapter 1, functions; Chapter 2,

infinitely small and large, continuity.

Brief Topics in real variables: Chapter 3, symmetric, alternat-

ing, homogeneous functions, Cramers Rule; Chapter 4, interpo-

lating polynomials, factoring falling factorials of x + y; Chapter

5, solving functional equations.

Real Series: Chapter 6, convergent and divergent series, tests,

real power series, summing power series.

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Structure of the Cours danalyse II

Complex Analysis: Chapter 7, imaginary expressions (complex

numbers) and algebraic operations on them, including roots;

Chapter 8, complex variables and functions; Chapter 9, complex

series, including power series; Chapter 10 Real and imaginary

roots of equations, fundamental theorem of algebra, cubic andquartic formulas.

Brief topics in complex variables: Chapter 11 Rational func-

tion decomposition; Chapter 12 Recurrent Series.

Notes: 9 Notes in which, I have presented the derivations which

may be useful both to professors and students of the Royal Col-

leges, as well as to those who wish to make a special study of

analysis.

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A Manageable Historical Unit on Elemen-tary Real Analysis

Preliminaries.

Chapter 1: On real functions.

Chapter 2: On infinitely small and infinitely large quantities,

and on the continuity of functions. Singular values of functions

in various particular cases.

Chapter 6: On convergent and divergent series. Rules for the

convergence of series. The summation of several convergent

series.

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Introduction

Cauchy acknowledges the encouragement of Laplace and Poisson

who were so good as to guide the first steps of my scientific

career. Thanks Poisson, Ampere (twice) and Coriolis, who was

appointed his repetiteur on 11/28/1816, for insights and advice.

Cauchys Manifesto

As for the methods, I have sought to give them all the rigorwhich one demands from geometry, so that one need never rely

on arguments drawn from the generality of algebra.

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[Extended quotation from pages 1-3 of Cauchys Cours danalyse

Bradley and Sandifer, Springer, 2009.]

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Preliminaries

This is similar to what is sometimes called Chapter 0 in modern

textbooks, reviewing basic defnitions and facts.

To Cauchy, numbers arise from the absolute measure of a mag-

nitudes; i.e., they correspond to positive real numbers. A quan-

tity is a signed number and the numerical value of a quantity is

that number which forms its basis; i.e., its absolute value.

Upper case letters represent numbers, lower case represent quan-

tities; e.g., Ax, but xa.

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Variables and Limits

We call a quantity variable if it can be considered as able to

take on successively many different values. We normally denote

such a quantity by a letter taken from the end of the alphabet.

On the other hand, a quantity is called constant, ordinarily de-

noted by a letter from the beginning of the alphabet, if it takeson a fixed and determined value. When the values successively

attributed to a particular variable indefinitely approach a fixed

value in such a way as to end up by differing from it by as little

as we wish, this fixed value is called the limit of all the other

values. Thus, for example, an irrational number is the limit of

the various fractions that give better and better approximations

to it. In Geometry, the area of a circle is the limit towards which

the areas of the inscribed polygons converge when the number

of their sides grows more and more, etc.

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Huh? Where are the Epsilons?

When the values successively attributed to a particular variable

[xn] indefinitely approach a fixed value [L] in such a way as to

end up [n

N for some N] by differing [

|xn

L

|] from it by as

little as we wish [< , and its relation N], this fixed value is calledthe limit of all the other values.

When the values successively attributed to a particular variable

[f(x)] indefinitely approach a fixed value [L] in such a way as to

end up [|xx0| < ] for some > 0] by differing [|f(x)L|] fromit by as little as we wish [< , and its relation ], this fixed value

is called the limit of all the other values.

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Epsilons and deltas wouldnt actually appear in the definitionsuntil the 1860s (e.g. Weierstrass 1861). Cauchy is satisfied

with a didactic (or verbal) definition, but he does use epsilons

in his arguments, although he doesnt need s in the Cours

danalyse. However, in the 1823 Resume des lecons (p. 44 in

Oeuvres completes):

Let and denote two very small numbers, the first being

chosen so that, for numerical values of i smaller than and for

any value of x included between the limits x0 and X, the ratio

f(x + i) f(x)i

always remains greater than f(x) and less than f(x) + .

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Infinitely Small, Infinitely Large[Extended quotation from page 7 of Cauchys Cours danalyse


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Preliminaries, Cont.

Review of algebraic and trigonometric operations.

Discussion of multiple-valued functions, e.g. roots, inversetrig functions and logs. Notation:

a = a or ((a))12 = a12 .

Results about averages arithmetic, geometric and weighted with proofs deferred to Note II.

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Chapter 1: On Real Functions

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.

(Immediately followed by the analogous definition for multivari-

able functions, then implicit vs. explicit.)

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Section 1.2: Simple Functions

[Extended quotation from pages 18-19 of Cauchys Cours danalyse


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Section 1.3: Composite Functions

Classification as algebraic, exponential or logarithmic, andtrigonometric or circular.

Algebraic further subdivided in rational and irrational.

Rational futher subdivided into integer and fractional (orrational fraction). Integer (Fr. entiere, but we wanted to avoid

confusion with the modern sense of entire function.) functions

are polynomial functions in the modern sense, but Cauchy usespolynomial in the broader, literal sense.

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Section 2.1: Inifinitely Small

We say that a variable quantity becomes infinitely small when

its numerical value decreases indefinitely in such a way as toconverge 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 areaof the circle.

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Inifinitely Large

We say that a variable quantity becomes infinitely large when its

numerical value increases indefinitely in such a way as to converge

towards the limit

. It is again essential to observe here that

one ought not confuse a variable that increases indefinitely with avariable that increases constantly. The area of a regular polygon

inscribed in a given circle increases constantly as the number of

sides increases, but not indefinitely. The terms of the natural

sequence of integer numbers

1, 2, 3, 4, 5, . . .

increase constantly and indefinitely.

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Section 2.2: Continuity

Among the objects related to the study of infinitely small quan-

tities, we ought to include ideas about the continuity and the

discontinuity of functions. In view of this, let us first consider

functions of a single variable.

Let f(x) be a function of the variable x, and suppose thatfor each value of x between two given limits, the function al-

ways takes a unique finite value. If, beginning with a value of

x contained between these limits, we add to the variable x an

infinitely small increment , the function itself is incremented by

the differencef(x + ) f(x),

which depends both on the new variable and on the value ofx.

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Given this, the function f(x) is a continuous function of x be-

tween the assigned limits if, for each value of x between theselimits, the numerical value of the difference

f(x + ) f(x)decreases indefinitely with the numerical value of . In other

words, the functionf

(x

) is continuous with respect tox

be-

tween the given limits if, between these limits, an infinitely small

increment in the variable always produces an infinitely small in-

crement in the function itself.

We also say that the function f(x) is a continuous function

of the variable x in a neighborhood of a particular value of thevariable x whenever it is continuous between two limits of x

that enclose that particular value, even if they are very close

together.

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Cauchy then determines the intervals of continuity of the eleven

simple functions. He defines continuity for multivariable func-

tions and states the following incorrect theorem:

Theorem I. If the variables x, y, z, . . . have for their respective

limits the fixed and determined quantities X, Y, Z, . . ., and

the function f(x , y , z , . . .) is continuous with respect to each of

the variables x, y, z, . . . in the neighborhood of the system of

particular values

x = X, y = Y, z = Z, . . . ,

then f(x , y , z , . . .) has f(X , Y , Z , . . .) as its limit.

Counterexample: f(x, y) = xyx2+y2

, X = Y = 0.

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Intermediate Value Theorem

Theorem IV. If the function f(x) is continuous with respect

to the variable x between the limits x = x0 and x = X, and if

b denotes a quantity between f(x0) and f(X), we may always

satisfy the equation

f(x) = bby one or more real values of x contained between x0 and X.

Cauchy gives an unsatisfying proof about the graph of y =f(x) and the line y = b meeting, but then notes that we can

prove theorem IV by a direct and purely analytic method, which

also has the advantage of providing the numerical solution to

the equation f(x) = b. He does this in Note III, on pages 309-

311 of Cauchys Cours danalyse Bradley and Sandifer, Springer,

2009.

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Section 2.3: Singular Values

Cauchy examines the behaviour of the 11 simple functions at

and at singular values. He then proves:

Theorem I. If the difference

f(x + 1) f(x)converges towards a certain limit k, for increasing values of x,

then the fraction

f(x)x

converges at the same time towards the same limit.

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The proof Cauchys first -proof in print was carefully ana-lyzed. It is on pages 35-37 of Cauchys Cours danalyse Bradley

and Sandifer, Springer, 2009.

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Cauchy completes the proof of Theorem 1 by considering thecases k = and k = . Then he gives two applications:

Corollary 1: When f(x) = log(x) with base > 1, then k = 0, so

in a system for which the base is greater than 1, the logarithms

of numbers grow much less rapidly than the numbers themselves.

Corollary 2: When f(x) = Ax with A > 1, then k = , so theexponential Ax, when the number A is greater than 1, eventually

grows more rapidly than the variable x.

He also observes that if f(x) remains finite for x = , the ratiof(x)

x evidently has zero as its limit.

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Next, he sets himself up for the Ratio Test (Chapter 6).

Theorem II. If the function f(x) is positive for very large

values of x and the ratio

f(x + 1)

f(x)

converges towards the limit k when x grows indefinitely, then the

expression

[f(x)]1x

converges at the same time to the same limit.

Cauchy observes that this can be deduced by applying Theorem

1 to log(f(x)), but he also give a direct proof, using geometric

instead of arithmetic means.

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Applications

limxx

1x = 1.

For any polynomial p(x),

limxp(x)

1x = 1.

limx log(x)

1x = 1.

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Foreshadowing Differential Calculus

Cauchy considers the limit of

f(x + ) f(x)

.

He mentions that it is 2x when f(x) = x2 and ax2

when f(x) = ax.

Ha laso proves

limsin

= 1,

but because the study of the limits towards which the ratios

f(x+)f(x) and f()f(0) converge is one of the principal objects

of the infinitesimal Calculus, there is no need to dwell any further

on this.

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Chapter 5

In Section 5.1, Cauchy solves the functional equations

(x + y) = (x) + (y) , (1) (x + y) = (x) (y) , (2)

(xy) = (x) + (y) and (3)

(xy) = (x) (y) , (4)

where (x) is to be continuous for x > 0 in (3) and (4), and forall real x in (1) and (2).

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Solution of Problem II

The proof of this problem was analyzed in detail, as an illustration

of Cauchys conception of the real number field. The proof is on

pages 73-75 of Cauchys Cours danalyse Bradley and Sandifer,

Springer, 2009.

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Soutions

(x + y) = (x) + (y) (x) = ax (x + y) = (x) (y) (x) = Ax

(xy) = (x) + (y) (x) = logA(x) (xy) = (x) (y) (x) = xa

where A is an arbitrary number and a is an arbitrary quantity.

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Section 5.2

Functional equation:

(y + x) + (y x) = 2 (x) (y) . (1)

It follows that (0) = 1. If there is a nearby value of x so that

(x) < 1, then

(x) = cos ax

for some quantity a. On the other hand, if (x) > 1 for any

value of x, then1

2

Ax + Ax

for some number A.

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Chapter 6

6.1: General considerations on series. Definitions, including

Cauchy Criterion. Geometric series, series for e. The infamous

Incorrect Theorem.

6.2: On series for which all the terms are positive. Tests forconvergence. Sum and product.

6.3: On series which contain positive terms and negative terms.

Absolute convergence, sum and product.

6.4: On series ordered according to the ascending integer powers

of a single variable. Power series, radius of convergence, sum and

product, applications of section 5.1.

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Series Definitions

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

sn = u0 + u1 + u2 + . . . + un1be the sum of the first n terms, where n denotes any integer

number.

(Although Cauchys insists on confusing the series with theunderlying sequence of its terms, this is in all other ways our

definition of series by means of partial sums. Note that he has

no general treatment of sequences before this chapter.)

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[Many extended quotations from Cauchys Cours danalyse Bradleyand Sandifer, Springer, 2009 were presented at this point. Most

of chapter 6 was presented, which occupies pages 85-115.]

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Cauchys Famous Incorrect Theorem

Theorem I. When the various terms of series (1) are func-

tions of the same variable x, continuous with respect to this

variable in the neighborhood of a particular value for which the

series converges, the sum s of the series is also a continuousfunction of x in the neighborhood of this particular value.

(This theorem as stated is incorrect. If we impose the additional

condition of uniform convergence on the functions sn, then it

does hold. This theorem is controversial. Some have arguedthat Cauchy really had uniform convergence in mind, but in 1852

he admitted it cannot be accepted without restriction.)

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Abels 1826 Counterexample

sin x 12

sin2x +1

3sin3x . . .

sastifies the hypotheses of the theorem for all x, but is discon-tinuous at every x = (2n + 1).

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Tests for Convergence

In Section 6.2, Cauchy considers series of positive terms and

gives 4 test for convergence:

Theorem I Root Test

Theorem II Ratio Test

Theorem III Cauchy Condensation Test:u0, 2u1, 4u3, 8u7, 16u15 . . .

Theorem IV Logarithmic Convergence Testlog(un)

log(1/2)

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Absolute Convergence, but not by Name

Suppose that the series

u0, u1, u2, . . . , un, . . . (1)

is composed of terms that are sometimes positive and sometimesnegative, and let

0, 1, 2, . . . , n, . . . (2)

be, respectively, the numerical values of these same terms, so

that we have

u0 = 0, u1 = 1, u2 = 2, . . . , un = n, . . . .

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The numerical value of the sum

u0 + u1 + u2 + . . . + un1will never surpass

0 + 1 + 2 + . . . + n

1,

so it follows that the convergence of series (2) always entails

that of series (1).

(Thus, in Section 6.3, Cauchy essentially proves that absolute

convergence implies convergence. Theroems I and II that followapply the Root and Ratio Tests of 6.2, respectivley, to n andn+1

n)

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Alternating Series Test

Theorem III. If the numerical value of the general term un

in series (1) decreases constantly and indefinitely for increasingvalues of n, and if further the different terms are alternately

positive and negative, then the series converges.

(The proof is a rare case of Cauchy using proof by example,

with the alternating harmonic series.)

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Section 6.4 Power Series

Let

a0, a1x, a2x2, . . . , anx

n, . . . (1)

be a series ordered according to the ascending integer powers of

the variable x (Such series had not yet been given the modern

name power series), where

a0, a1, a2, . . . , an, . . . (2)

denote constant coefficients, positive or negative. Furthermore,

let A be the quantity that corresponds to the quantity k of the

previous section (see Section 6.3, theorem II), with respect to

series (2). The same quantity, when calculated for series (1), is

the numerical value of the product

Ax.

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Radius of Convergence

As a consequence, series (1) is convergent if this numerical

value is less than 1, which is to say in other words, if the numer-

ical value of the variable x is less than 1A. On the other hand,

series (1) is divergent if the numerical value of x is greater than1

A. We may therefore state the following proposition:

Theorem I. Let A be the limit towards which the nth root

of the largest numerical values of an converge, for increasing

values of n. Series (1) is convergent for all values of x contained

between the limits

x = 1A

and x = + 1A

,

and divergent for all values of x situated outside of these same

limits.

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Imagine now that we vary the value of x in series (1) by insen-

sible degrees. As long as the series remains convergent, that isas long as the value of x remains contained between the limits

1A

and +1

A,

the sum of the series is (by virtue of theorem I, Section 6.1) a

continuous function of the variable x. Let (x) be this continu-ous function. The equation

(x) = a0 + a1x + a2x2 + . . .

remains true for all values of x contained between the limits 1Aand + 1

A, which we indicate by writing these limits beside the

series, as we see here:

(x) = a0 + a1x + a2x2 + . . .

x = 1

A, x = +

1

A

.

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Problem I. When possible, to expand the function

(1 + x)

into a convergent series ordered according to increasing integer

powers of the variable x.

Cauchy lets

() = 1 + 1

x + ( 1)1 2 x

2 + . . . ,

which he has already shown to have A = 1. He shows that( + ) = ()(), it follows from Problem II of Section 5.1that

() = [(1)]

= (1 + x)

.Therefore

(1+x) = 1+

1x+

( 1)1 2 x

2 +. . . (x = 1, x = +1). (20)69


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Euleres Vindicatus!

By carefully passing to the limit in the binomial theorem, Cauchy

succeeds in giving a rigorous version of Eulers derivation of the

power series for ex from the Introductio. He similarly gives a

cleaned-up derivation of

ln(1 + x) = x x2

2+

x3

3 . . . (x = 1, x = +1).

that is still somewhat faithful to the spitit of Eulers derivation.

Problems II and III, to expand Ax and log(1+ x) now involve only

a change of base.

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Chapter 7: Imaginary ExpressionsIn general, we call an imaginary expression any symbolic expres-

sion of the form

+

1,

where and denote real quantities. We say that two imaginary

expressions

+ 1 and + 1

are equal to each other when there is equality between corre-

sponding parts . . . Given this, any imaginary equation is just the

symbolic representation of two equations involving real quanti-

ties.

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Arithmetic of Imaginary Expressions; Po-lar Coordinates

The four basic arithmetic operations are easily defined, but for

roots of imaginary expression, Cauchy needs polar coordinates.

He coined the terms conjugate and modulus as we use themtoday. He uses the name reduced expression for the factor

cos +1sin ,

that remains after the suppression of the modulus.

He closes the chapter with identities for cos mz and sin mz in

terms of powers of cos z and sin z.

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The First Complex Analysis Text?

Chapters 8-9: Cauchy extends his treatment of functions, con-

tinuity and series (Chapters 1, 2 and 6) to the complex case.

He makes the proper definitions and expands the complex power

series for the binomial, exponential, logarithmic series, as well as

sine and cosine. Other highlights:

Section 9.3: Eulers identity and logarithms of complex num-

bers.

Section 10.1: A proof of the FTA, based on [Legendre 1808].

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