goursat a course in mathematical analysis

7/28/2019 Goursat a Course in Mathematical Analysis

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A COURSE IN

BYEDOUARD GOURSAT

PROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF PARIS

TRANSLATED BY

EARLE RAYMOND HEDRICKPROFESSOR OF MATHEMATICS IN THE UNIVERSITY OF MISSOURI

VOL. IDERIVATIVES AND DIFFERENTIALS

DEFINITE INTEGRALS EXPANSION IN SERIESAPPLICATIONS TO GEOMETRY

GINN AND COMPANYBOSTON NEW YORK CHICAGO LONDON

ATLANTA DALLAS COLUMBUS SAN FKANCISCO


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AUTHOR S PREFACEThis book contains, with slight variations, the material given in

my course at the University of Paris. I have modified somewhatthe order followed in the lectures for the sake of uniting in a singlevolume all that has to do with functions of real variables, exceptthe theory of differential equations. The differential notation notbeing treated in the " Classe de Mathematiques speciales," * I havetreated this notation from the beginning, and have presupposed onlya knowledge of the formal rules for calculating derivatives.

Since mathematical analysis is essentially the science of the continuum, it would seem that every course in analysis should begin,logically, with the study of irrational numbers. I have supposed,however, that the student is already familiar with that subject. Thetheory of incommensurable numbers is treated in so many excellentwell-known works f that I have thought it useless to enter upon sucha discussion. As for the other fundamental notions which lie at thebasis of analysis, such as the upper limit, the definite integral, thedouble integral, etc., I have endeavored to treat them with alldesirable rigor, seeking to retain the elementary character of thework, and to avoid generalizations which would be superfluous in abook intended for purposes of instruction.

Certain paragraphs which are printed in smaller type than thebody of the book contain either problems solved in detail or else*An interesting account of French methods of instruction in mathematics will

be found in an article by Pierpont, Bulletin Amer. Math. Society, Vol. VI, 2d series(1900), p. 225. TRANS.t Such books are not common in English. The reader is referred to Pierpont,

Theory of Functions of Real Variables, Ginn & Company, Boston, 1905; Tannery,Lemons d arithiiietique, 1900, and other foreign works on arithmetic and on realfunctions.

iii

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iv AUTHOR S PREFACEsupplementary matter which the reader may omit at the first reading without inconvenience. Each chapter is followed by a list ofexamples which are directly illustrative of the methods treated inthe chapter. Most of these examples have been set in examinations. Certain others, which are designated by an asterisk, aresomewhat more difficult. The latter are taken, for the most part,from original memoirs to which references are made.Two of my old students at the Ecole Normale, M. Emile Cotton

and M. Jean Clairin, have kindly assisted in the correction of proofs ;I take this occasion to tender them my hearty thanks.

E. GOURSATJANUARY 27, 1902


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TRANSLATOR S PREFACEThe translation of this Course was undertaken at the suggestion

of Professor W. F. Osgood, whose review of the original appearedin the July number of the Bulletin of the American MathematicalSociety in 1903. The lack of standard texts on mathematical subjects in the English language is too well known to require insistence.I earnestly hope that this book will help to fill the need so generallyfelt throughout the American mathematical world. It may be usedconveniently in our system of instruction as a text for a second coursein calculus, and as a book of reference it will be found valuable toan American student throughout his work.Few alterations have been made from the French text. Slight

changes of notation have been introduced occasionally for convenience, and several changes and .additions have been made at the suggestion of Professor Goursat, who has very kindly interested himselfin the work of translation. To him is due all the additional matternot to be found in the French text, except the footnotes which aresigned, and even these, though not of his initiative, were alwaysedited by him. I take this opportunity to express my gratitude tothe author for the permission to translate the work and for thesympathetic attitude which he has consistently assumed. I am alsoindebted to Professor Osgood for counsel as the work progressedand for aid in doubtful matters pertaining to the translation.The publishers, Messrs. Ginn & Company, have spared no pains to

make the typography excellent. Their spirit has been far from commercial in the whole enterprise, and it is their hope, as it is mine,that the publication of this book will contribute to the advance ofmathematics in America. E R HEDRICK

AUGUST, 1904


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CONTENTSCHAPTER PAGE

I. DERIVATIVES AND DIFFERENTIALS 1I. Functions of a Single Variable 1

II. Functions of Several Variables 11III. The Differential Notation 19

II. IMPLICIT FUNCTIONS. FUNCTIONAL DETERMINANTS. CHANGEOF VARIABLE 35

I. Implicit Functions........ 35II. Functional Determinants ...... 52

III. Transformations ... .... 61III. TAYLOR S SERIES. ELEMENTARY APPLICATIONS. MAXIMA

AND MINIMA ........ 89I. Taylor s Series with a Remainder. Taylor s Series . 89II. Singular Points. Maxima and Minima . . . .110IV. DEFINITE INTEGRALS ........ 134

I. Special Methods of Quadrature . . . . .134II. Definite Integrals. Allied Geometrical Concepts . . 140

III. Change of Variable. Integration by Parts . . .166IV. Generalizations of the Idea of an Integral. Improper

Integrals. Line Integrals ...... 175V. Functions defined by Definite Integrals .... 192VI. Approximate Evaluation of Definite Integrals . .196

V. INDEFINITE INTEGRALS 208I. Integration of Rational Functions ..... 208

II. Elliptic and Hyperelliptic Integrals .... 226III. Integration of Transcendental Functions . . .236

VI. DOUBLE INTEGRALS ........ 250I. Double Integrals. Methods of Evaluation. Green sTheorem 250

II. Change of Variables. Area of a Surface . . . 264III. Generalizations of Double Integrals. Improper Integrals.

Surface Integrals ....... 277IV. Analytical and Geometrical Applications . . . 284


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viii CONTENTSCHAPTER PAGEVII. MULTIPLE INTEGRALS. INTEGRATION OF TOTAL DIFFER

ENTIALS 296I. Multiple Integrals. Change of Variables . . . 296

II. Integration of Total Differentials . . . . .313VIII. INFINITE SERIES . . 327

I. Series of Real Constant Terms. General Properties.Tests for Convergence 327

II. Series of Complex Terms. Multiple Series . . . 350III. Series of Variable Terms. Uniform Convergence . . 360

IX. POWER SERIES. TRIGONOMETRIC SERIES .... 375I. Power Series of a Single Variable . . . . . 375II. Power Series in Several Variables ..... S94

III. Implicit Functions. Analytic Curves and Surfaces . 399IV. Trigonometric Series. Miscellaneous Series . . .411

X. PLANE CURVES 426I. Envelopes 426

II. Curvature 433III. Contact of Plane Curves 443

XI. SKEW CURVES 453I. Osculating Plane ........ 453

II. Envelopes of Surfaces . . . . . . . 459III. Curvature and Torsion of Skew Curves .... 468IV. Contact between Skew Curves. Contact between Curves

and Surfaces ........ 486XII. SURFACES 497

I. Curvature of Curves drawn on a Surface . . . 497II. Asymptotic Lines. Conjugate Lines .... 506III. Lines of Curvature . . . . . . . .514IV. Families of Straight Lines 526

INDEX . 541


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CHAPTER IDERIVATIVES AND DIFFERENTIALS

I. FUNCTIONS OF A SINGLE VARIABLE1. Limits. When the successive values of a variable x approach

nearer and nearer a constant quantity a, in such a way that theabsolute value of the difference x a finally becomes and remainsless than any preassigned number, the constant a is called thelimit of the variable x. This definition furnishes a criterion fordetermining whether a is the limit of the variable x. The necessary and sufficient condition that it should be, is that, given anypositive number e, no matter how small, the absolute value of x ashould remain less than e for all values which the variable x canassume, after a certain instant.Numerous examples of limits are to be found in Geometryand Algebra. For example, the limit of the variable quantityx = (a2 m2) / (a m), as m approaches a, is 2 a ; for x 2 a will

be less than e whenever m a is taken less than e. Likewise, thevariable x = a 1/n, where n is a positive integer, approaches thelimit a when n increases indefinitely ; for a x is less than e whenever n is greater than 1/e. It is apparent from these examples thatthe successive values of the variable x, as it approaches its limit, mayform a continuous or a discontinuous sequence.

It is in general very difficult to determine the limit of a variablequantity. The following proposition, which we will assume as self-evident, enables us, in many cases, to establish the existence of a limit.Any variable quantity which never decreases, and which ahvays

remains less than a constant quantity L, approaches a limit I, whichis less than or at most equal to L.

Similarly, any variable quantity which never increases, and whichalways remains greater than a constant quantity L , approaches alimit l } which is greater than or else equal to L .

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2. DERIVATIVES AND DIFFERENTIALS [I, 2For example, if each of an infinite series of positive terms is

less, respectively, than the corresponding term of another infiniteseries of positive terms which is known to converge, then the firstseries converges also ; for the sum 2n of the first n terms evidentlyincreases with n, and this sum is constantly less than the total sum5 of the second series.

2. Functions. When two variable quantities are so related thatthe value of one of them depends upon the value of the other, theyare said to be functions of each other. If one of them be supposed to vary arbitrarily, it is called the independent variable. Letthis variable be denoted by x, and let us suppose, for example,that it can assume all values between two given numbers a and b(a < b). Let y be another variable, such that to each value of xbetween a and b, and also for the values a and b themselves, therecorresponds one definitely determined value of y. Then y is calleda function of x, defined in the interval (a, b) ; and this dependenceis indicated by writing the equation y =/(z). For instance, it mayhappen that y is the result of certain arithmetical operations performed upon x. Such is the case for the very simplest functionsstudied in elementary mathematics, e.g. polynomials, rational functions, radicals, etc.A function may also be defined graphically. Let two coordinateaxes Ox, Oy be taken in a plane ; and let us join any two points Aand B of this plane by a curvilinear arc .4 CB, of any shape, whichis not cut in more than one point by any parallel to the axis Oy.Then the ordinate of a point of this curve will be a function of theabscissa. The arc A CB may be composed of several distinct portions which belong to different curves, such as segments of straightlines, arcs of circles, etc.

In short, any absolutely arbitrary law may be assumed for findingthe value of y from that of x. The word function, in its most general sense, means nothing more nor less than this : to every value ofx corresponds a value of y.

3. Continuity. The definition of functions to which the infinitesimal calculus applies does not admit of such broad generality.Let y =f(x) be a function defined in a certain interval (a, b), andlet x and x -f h be two values of x in that interval. If the difference f(x -f A) f(xo) approaches zero as the absolute value of happroaches zero, the function f(x} is said to be continuous for thevalue x . From the very definition of a limit we may also say that


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I, 3] FUNCTIONS OF A SINGLE VARIABLE 3a function f(x) is continuous for x x if, corresponding to everypositive number e, no matter how small, we can find a positive number 77, such that

|/(*o + A)-/(*o)| (t) isequal to hfx (x -f- ht, y + kf) + kfy (x + ht, y -f- kt) ; hence the preceding formula may be written in the form

12. Tangent plane to a surface. We have seen that the derivativeof a function of a single variable gives the tangent to a plane curve.Similarly, the partial derivatives of a function of two variables occurin the determination of the tangent plane to a surface. Let

(2) z . F(x, y)be the equation of a surface S, and suppose that the function F(x, ?/),together with its first partial derivatives, is continuous at a point(^o? yo) of the xy plane. Let z be the corresponding value of z,and AT (cr , 7/0 & ) the corresponding point on the surface S. Ifthe equations

(3) *=/(*), z/ = * = ^(9represent a curve C on the surface S through the point M , thethree functions f(f), (t), "A(0> which we shall suppose continuousand differentiable, must reduce to x , y , z , respectively, for somevalue t of the parameter t. The tangent to this curve at the pointM is given by the equations ( 5)x x Y z *Since the curve C lies on the surface S, the equation \j/(t)=F[f(t~), .must hold for all values of t; that is, this relation must be an identity

* Another formula may be obtained which involves only one undetermined number 0,and which holds even when the derivatives/^, and/, are discontinuous. For the application of the law of the mean to the auxiliary function (t) =f(x+ ht,y+ k) +f(x, y+ kt)gives

_d_F_d_u d_F_d_v_ ___dy du dy dv dy dw dy

d_F_d_vdz du dz do dz dw dzdw _ dF du dF dv dF dwdt du dt dv dt dw dt

If these four equations be multiplied by dx, dy, dz, dt, respectively,and added, the left-hand side becomesd(

,& W 7 , ^< 7 ,^ W J.3- dx + -r- dy + -^- dz + -^ dt,dx dy dz d

that is, do* ; and the coefficients ofd_F d]F 0Fdu do dw

on the right-hand side are du, dv, dw, respectively. HencedF dF dF

(9) do) = ^ du + -r dv + ^ dw,cu dv cwand we see that the expression of the total differential of the firstorder of a composite function is the same as if the auxiliary functionswere the independent variables. This is one of the main advantagesof the differential notation. The equation (9) does not depend, inform, either upon the number or upon the choice of the independentvariables ; and it is equivalent to as many separate equations asthere are independent variables.To calculate d2 w, let us apply the rule just found for dta, notingthat the second member of (9) involves the six auxiliary functionsu, v, w, du, dv, dw. We thus find


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26 DERIVATIVES AND DIFFERENTIALS [I, lud2 F dF= -i^- du2 4- -z du dv + - du dw + -^- dzu

Ctr cucu cucw en4- - dudv 4- ^ dv2 + ff dvdw + ^du dv cv2 cu cw cv

d2F d2F d2 F dF+ du dw 4- o Q dv dw -f TT-^ t?w + ^^gw ^y^M> Cw 1 cwor, simplifying and using the same symbolism as above,

7 , ^ , ^ ^ ,d2 w = [7^- du+ ^- dv + dw\ + TT- * + c?2 w 4- ^ .Vc/w ^y CM; / cu Co cw

This formula is somewhat complicated on account of the terms ind2 u, dz v, dz w, which drop out when u, v, w are the independentvariables. This limitation of the differential notation should beborne in mind, and the distinction between d2 w in the two casescarefully noted. To determine ds w, we would apply the same ruleto , noting that d2 w depends upon the nine auxiliary functionsu, v, w, du, dv, dw, d2 u, d2 v,d2 w; and so forth. The general expressions for these differentials become more and more complicated ;dnw is an integral function of du, dv, dw, d2 u, , dn u, dn v, dn w, andthe terms containing dn u, dn v, dnw are

dF 7 dF , dF 7dnu 4- dn v 4- dn w.cu cv cwIf, in the expression for d" w, u, v, w, du, dv, dw, be replaced by

their values in terms of the independent variables, dn t becomes anintegral polynomial in dx, dy, dz, whose coefficients are equal(cf. Note, 15) to the partial derivatives of w of order n, multipliedby certain numerical factors. We thus obtain all these derivativesat once.

Suppose, for example, that we wished to calculate the first andsecond derivatives of a composite function dudx du dx dy du dyAgain, taking the derivatives of these two equations with respectto x, and then with respect to y, we find only the three followingdistinct equations, which give the second derivatives :


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THE DIFFERENTIAL NOTATION 27

(11)dx*

dx dy

du\*ex du

d*U C U C = uv. For the first values of n we have

dw = v dti + u dv, d* a) = v d* u + 2 du dv -f ud* v, ;and, in general, it is evident from the law of formation that

d" w = v d" u 4- r, dr dn ~^u + Cd*v dn ~ 2 n -f +where Clt C2 , are positive integers. It might be shown by algebraic induction that these coefficients are equal to those of theexpansion of (a + &)" ; but the same end may be reached by thefollowing method, which is much more elegant, and which appliesto many similar problems. Observing that C l , C 2 , do not dependupon the particular functions n and v employed, let us take the


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28 DERIVATIVES AND DIFFERENTIALS [I, 17special functions u = e*, v = &, where x and y are the two independent variables,

and determine the coefficients for this case. Wethus find

w = ex+y, dw = ex+y (dx + dy), -, dn dnw = -5- du + -T- dv 4- 5 dw I .

We proceed to apply this remark.18. Homogeneous functions. A function (x, y, z) is said to be

homogeneous of degree m, if the equation

goursat a course in mathematical analysis

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