Cauchy - 1967922

7/30/2019 Cauchy - 1967922

1/17

Annals of Mathematics

Cauchy's Paper of 1814 on Definite IntegralsAuthor(s): H. J. EttlingerSource: Annals of Mathematics, Second Series, Vol. 23, No. 3 (Mar., 1922), pp. 255-270Published by: Annals of MathematicsStable URL: http://www.jstor.org/stable/1967922 .

Accessed: 24/03/2013 18:24

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms

of scholarship. For more information about JSTOR, please contact [email protected].

.

Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access toAnnals of

Mathematics.

http://www.jstor.org

This content downloaded from 187.101.182.82 on Sun, 24 Mar 2013 18:24:03 PMAll use subject to JSTOR Terms and Conditions
http://www.jstor.org/action/showPublisher?publisherCode=annalshttp://www.jstor.org/stable/1967922?origin=JSTOR-pdfhttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/stable/1967922?origin=JSTOR-pdfhttp://www.jstor.org/action/showPublisher?publisherCode=annals

7/30/2019 Cauchy - 1967922

2/17

CAUCHY'S PAPER OF 1814 ON DEFINITE INTEGRALS.*BY H. J. ETTLINGER.

Introduction. n 1814 Augustin Louis Cauchy presented before theAcademiedes Sciencesa "memoiron definitentegrals," n whichappearsforthe first ime the essence of his discoverieson residues. The memoirwas first rinted n 1825t with additional notes and again in 18821 withno changesave in thematter ofnotation.Althoughthe kernelof the idea, that the integralof an analyticfunc-tion of a complex variable taken along a closed path depends entirelyupon the behavior of the function t points of discontinuitywithin thepath, is contained in the paper, yet there are several reasons why thereadermightnotice nothing t all like this theorem. In the first lace,although geometricalrepresentations now an essential feature of everypresentationof the theoryof functions,Cauchy used neitherfiguresnorgeometrical anguage. In the second place, the fundamentaltheorem,and, indeed, all the applications in this paper, concernsimple integrals;but the author states the central problemas the determination f thedifferencen the value of an iterated integralaccordingto the order ofintegrationwithrespectto thetwo variables. By the use of thisdifferencehe obtains the residue,therebyobscuringthe relationof the latter to aline integral. Thirdly, he refrainsfromusing complex quantities, in-variablyseparatingan equation into its real and imaginaryparts. Thisnecessitates longer equations, more of them, clumsiernotation, and amuch moreobscuretreatment han would be the case had he used com-plex quantities. Cauchy himself came to appreciate this fact, for hisfootnotesof 1825 are devoted to the simpler complex equations fromwhichhis real ones can be readilydeduced. Finally, all editionsaboundin misprints.For these reasons the discoveries contained in this memoir were notappreciatedeven by the greatmathematiciansof his time. Poisson? sawin the paper merelya means of evaluating integrals nd remarkedthat,at least so faras the first art was concerned,no new formulaewere an-nounced. As for the evaluation of iterated integralsby the so-called

*Presentedo the Amer.Math. Soc., Sept. 2, 1919.t " M6moire ur es integrates efinies,"avantsEtrangers, , p. 509,Academiedes Sciencesde lInstitutde France.$CEuvres ompletes, serie,1, p. 319 ff.? Bulletin e la SocietePhilomatique 3), 1, 1814,p. 185.

255

This content downloaded from 187.101.182.82 on Sun, 24 Mar 2013 18:24:03 PMAll use subject toJSTOR Terms and Conditions
http://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsphttp://www.jstor.org/page/info/about/policies/terms.jsp

7/30/2019 Cauchy - 1967922

3/17

256 H. J. ETTLINGER."singular" integrals whichare equal to the differenceetweenthe valueobtained by integrating irstwithrespectto x and thenwithrespect to yand the value obtained by integratingn the reverseorder) he said that,though the new method was worthyof consideration, t ought not toreplace the old ones! Lacroix and Legendre, n the officialreport onthe paper, stated as the valuable results obtained by Cauchy: (1) theconstruction f a seriesofgeneralformulaeor ransformingnd evaluatingdefinite ntegrals, 2) the pointingout of the fact that the value of aniterated integralmay depend on the order of integration, 3) the dis-coveryof the cause and amountof thisdifferencen value, (4) the deriva-tion of new formulaewhich,to be sure, mighthave been otherwiseob-tained. It seems, then,likelythat the foremostmathematiciansofthattime failed to recognize the contributionsof main importancein thispaper.To appreciate thoroughly he memoir,the followingfacts must benoted in addition: (1) imaginarieshad no secure arithmetical basis in1814, (2) this was the first eductionby rigorousmethods oftheformulae,hithertoobtained by purely formal processes, for evaluating definiteintegrals, 3) while the formhad not yet been cast in the e-mould,whichitselforiginated with Cauchy, neverthelessthe proofs are so conceivedthat they correspondn substanceto the standardsofrigorofthe presentday.

PART I.Continuousntegrand. In discussing the memoir we shall frequentlycombine two separate real equations into one complex equation, asCauchy did in his notes of 1825 and, very likely, n his originalwork.We shall also adopt the language of modern analysis for the sake ofclearnessand accuracy.The first heoremproved in the memoir s, in effect,hat ifa functionofa complexvariable is analytic throughout regionof a certaintypeandcontinuous n and on the boundary,the integralof the functiontakenalong the boundary of the regionis zero.* The regionsconsidered aremapped in a one-to-one manner and continuously,but not in generalconformally, n a rectanglein the real (x, y) plane. The mapping onthe complexM + Ni plane is performed y takingM and N as real andcontinuousfunctions fx and y withderivatives ofall orderswithrespectto x and y, continuous n x and y regarded s independentvariables.*For modern reatmentfthis heorem eeOsgood,Lehrbuch er Funktionentheorie,rsterBand, zweite Auflage, p. 284-285; Pierpont,Functions f a ComplexVariable, pp. 211-214;Goursat, oursd'AnalyseMathematique, ome I, pp. 82-92. These three ext-books ill here-after e referredo as O., P., G., respectively.


7/30/2019 Cauchy - 1967922

4/17

CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. 257Let*(1) f(M + Ni) = P + Qi

be an analyticfunction f M + Ni in a certainregion,S, of the M + Niplane, and let M = (x, y) and N =t(x, y)be single-valued functions, continuous in x and y, in a rectangle,R (0 ! x _ a, 0 c y c b), and on the boundary,r, and possessingcon-tinuouspartial derivatives of all orders with respect to x and y in R andon r.Furthermore,ett(2) S + Ti? f(M + Ni) O(M Ni)

O M + Ni)(3) U + Vi =f(M + Ni) dDifferentiate2) and (3) with regard to y and x respectively:as+i =f (M+Ni) d M + Ni) a(M + Ni)z~ ~~~~~~~~(f \i(M +i)-y Oy Ox Oy+f= f'(M + N 2(M + Ni)au iOVf/(M +Ni) O(M?+Ni) O(M + Ni)Ox Oix Oy ax~

+f(M + Ni)O(M+Ni).But underthe conditions mposed

O2(M+ Ni) -2(M + Ni) $Oxdy OyaxorOS iOT aU iOVOy Oy Ox OxHence OS OU aT = OV(4) @---y and Oy OxMultiplyingthe equations (4) by dydx and integratingfrom x = 0 tox = a and y = 0 to y = b, and notingfurther hat, since the integrandis continuous, he orderofintegration an be reversed,we have:

* The notation fthe original aperhas beenchangedherefrom ' + P"i to P + Qi andy = x + zi to z = x + yi.tHereafter(M + Ni) _ i V(M+ Nvi)willbe designated yd(M + Ni)ax ay d(x + yi)tSeeGoursat-Hedrick,athematical nalysis, ol. I, p. 13.


7/30/2019 Cauchy - 1967922

5/17

258 H. J. ETTLINGER.(5) ddx d = fdy TX dx.

Let S(x, b) = S, S(x, 0) = s, U(a, y) = U, U(O, y) = u; then equa-tion 5) becomes ra r~a b rb(6) fSSdx- sdx = f Udy - udy.

In a similarmanner, ettingT(x, b) = T, T(x, 0) = t andV(a, y) =V,V(O, y) = v,we obtainfa ra fb fb(7) Tdx - tdx= Vdy- vdy.zplane

FIG. 1.Multiplying 7) by -i and (6) by - 1 and adding we have

f(s + ti)dx +Jb (U + Vi) dy - (S + Ti) dx - fb(u + vi)dy = 0,or

ff(M + Ni) d(M + Ni) d(x + yi) = 0,JL ~~d(x yi)whichmeans hat round herectangleeregivenn the plane, nd hencein theAl + Ni planeaboutthecorrespondingurve,* 1,(8) ff(M + Ni)d(M + Ni) = 0.HenceFUNDAMENTALTHEOREM I: Let f(M + Ni) be an analytic functionofM + Ni in a certainregion, , of the M + Ni plane, continuousn S,and on the boundary, , and letM = q(x, y) and N = 41(x,y) be single-valuedcontinuous unctions f x and y in a rectangle, (O _ x _ a, 0 _ y! b), and on the boundary, , possessing ontinuous artial derivativesf

*Cf.equation 3) and equations A), footnote, Euvres ompletes, serie,1, p. 338. Thismemoir illbe referredo hereafters O.C.


7/30/2019 Cauchy - 1967922

6/17

CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 259all orderswithrespect o x and y and mapping theclosedregion, , on theclosed rectangle,R, in a one-to-onemanner and continuously; hen theintegral f f(M + Ni)d(M + Ni) taken around L in the positivedirectionvanishes, r

ff(M + Ni)d(M + Ni) = O.In the applications Cauchy uses the real equations in S and U, T andV respectivelynd does not combinethem as is heredone. The functionsused in this paper for M and N and the correspondingmaps of the rec-tangle,R, on the M + Ni plane are given n figures , 3, 4, and 5..

10. M=x, N=y.

(o~~~o)(

co0)~~~~FIG. 2.2?. M =ax, N = xy;a > O.

,Ao)FIG. 3.

In several of the applications Cauchy allows a, the upper limit of thex-interval, o become infinite. He considered that his conclusions couldbe extended to this case ifthe function (M + Ni) approaches a limit foreach value of y (O < y - b) when a becomes infinite nd the improperintegrals thus introduced converge. These conditions are, of course,


7/30/2019 Cauchy - 1967922

7/17

260 H. J. ETTLINGER.30 M = xcosy, N = x siny.

1-t- N4' la x e(acosi a4 )

X~& 1FIG. 4.

4?. M =ax, N =xy;a > 0.

(aAeb)

(oo)FIG. 5.

insufficient,ut sinceall thefunctions(M + Ni) considered yCauchyapproachtheir imitsuniformlyn (0 - y _ b) and the integrals on-verge, heresultsmay be establisheds correct.The followings an example f the method fapplicationnd oftheresults fPart .Region 0. (See Fig. 2.)M =x, ,N = y.Letf(x+ yi) = P(x, y) + Q(x, y)i, suchthatQ(x, 0) = 0, and S + Ti- P+Qi, U+ Vi = - Q+Pi.Equations 6) and (7) yieldrba tw~a rb rb(6') f (x, b)dx - P(x, O)dx+ b Q(a, y)dy - Q(0, y)dy = 0,(7') JQ( bx-JP yd+b rb(7') Q0 x{hdx -I P(a, y~dy,f P(n- y,4dy n.


7/30/2019 Cauchy - 1967922

8/17

CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. 261Apply hese quations o

f(z) = e-z2 where z = x + yi.*P(x, y) = e-x2ey2 os 2xy, Q(x, y) - e-xey2in 2xy,P(x, 0) = e-2, P(0, y) = ey2, Q(0,y) = 0,so that n thiscase equations 6') and (7') become espectively

(9) fex2eb2 cos 2bxdx e-a2 eY2 sin 2aydy= ,fe2 dx.(10) - e-f'eb2 in 2bxdx ea2f ey2 cos 2aydy= - ey2dy.

Now let a increasewithout imit. The second ntegraln each oftheequations 9) and (10) vanishes, or

fbe-a2ee2 sin 2aydy cf in 2ay ea2ey2dyb_ ea2f ey2dy_ beb2-a2 b > 0.

But lim eb2-a2 = 0,a ahencerblim fbe-X2ei2 sin2aydy O.Similarlyrblim Jbe-a2ey2os 2aydy= 0.

Since t can be readily hown hat theother ntegrals onverge, e arejustifiednwriting auchy's quations:e-2 os2bxdx= e'2 e 2dx 2_ 4LWo 0 ~~~~~~~~2rX r~~~~~~~~~~bf -2 sin 2bxdx e-b2 ey2dy,

ifwe assume e-X2dx - /2.PART II.Integrands ith poles. In the second part of the memoirCauchy

dealswith he ntegrals f functions hich re discontinuoust isolatedpoints. In all theapplicationshese ingularitiesresimplepoles.t* Cf. O., p. 293, Beispiel, 4. G., p. 121, 30.tO.C., p. 413.


7/30/2019 Cauchy - 1967922

9/17

262 H. J. ETTLINGER.Here Cauchy obtains for the first ime a formulawhich contains theessence of the theoremon residues. The true significance f Cauchy'smethodat this point is very obscure. The result s apparentlystated intermsof the evaluation in two different rdters f an iterated integralwhose integrandhas a singularity t a single point.* As a matter offact,the iterated ntegralplays an unessentialrole n Part II, sinceall thetheorems and applications are concerned with simple integrals only.Moreoverno usefulfactsare developed concerningterated ntegrals.The exposition nd criticism fCauchy's methodwe lay aside for themomentand proceed to set forth methodby whichthe resultsof PartII are very simply obtained fromthe fundamentaltheoremof Part I.This method s not so very unlikeCauchy's, as will be pointedout later,

and would probablybe used by him were he writing n the notation ofthe present-day nalyst. It is the methodused in many modern textbooks on the TheoryofFunctions.tFUNDAMENTAL THEOREM II: Let f M + Ni) be an analyticfunctionofM + Ni in a certainregion, , of theM + Ni plane,except ora singlepoleat m + ni, insideofS, and continuousn and on theboundary, , ofS,except t thispole. Let M = +k(x, ) and N = {(x, y) be continuous unc-tionsofx and y in and on theboundary, , ofa rectangle, (O _ x _ a,0 c y _ b), possessing ontinuous artialderivativesf all orders,mappingtheclosedregion, , on theclosedrectangle, , in a one-to-onemanner ndcontinuously,nd such that m = +k(X,Y) and'n = {2(X, Y). Let R' (a'_ x _ a", b' c y c b") be any rectanglenteriort o R and containing(X, Y) withintsboundary ', and letL' be the urven S correspondingoF'.Then the ntegral ff(M + Ni) . d(M + Ni) taken roundL in thepositivesense s equal to the ntegral ff M + Ni) -d(M + Ni) taken n thepositivesense aroundL',ff(M + Ni)d(M + Ni) = f(M + Ni)d(M + Ni).z 4plan C.

4b

-3~_ 4j5IG aFIG. 6.* O.C., p. 388 ff.t 0., p. 331 ff.;P., p. 206ff.;G., p. 114ff.t I.e., 0 < a' < a" < a, 0 < b' < b < b.This content downloaded from 187.101.182.82 on Sun, 24 Mar 2013 18:24:03 PM

All use subject toJSTOR Terms and Conditions

7/30/2019 Cauchy - 1967922

10/17

CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 263The prooffollows mmediatelyfromthe fundamentaltheorem byapplication successivelyto the rectanglesmarked 1 ... 8 in Fig. 6 andaddition of the resulting quations. The equivalent of this equation in

Cauchy's paper givesthefirst tatementof his discoveryon Residues.*We proceedto derivetheformulanecessaryto make the first pplica-tion by evaluating fM + Ni)d(M + Ni)t explicitly or the caseM = x, N = y, or z = M + Ni.

Let: A + Bi =4'(z)dz, nd suppose:Case 1.

f(z) = Z where Z= X + Yi.Then A + Bi CdzLZ - ZLet z - Z = re~i, dz = iresido + e~idr.Then

A + Bi= Cid4+ I drO Lt ~~ror A + Bi = 2 riC,?sincethe initial and finalvalues of r are equal.Case 2.

f(z) =(Z) + -where+(z) is analytic in R and continuous n R and on the boundary,L,and whereZ is withinR,

A + Bi =f(z) dz + Z dzor A+Bi=27riC,since4'(z)dz = 0 bythefundamentalheorem.

Case 3. Iff(z) has a pole on theboundary,L, oftherectangle,R, but* O.C., p. 381, equation 4).tL' is identicalwithr' nthiscase.l The notation as herebeen changed romA' + A"i to A + Bi.?Cf. 0. C., footnote, . 412, equation C).


7/30/2019 Cauchy - 1967922

11/17

264 H. J. ETTLINGER.not at one of the vertices,we constructR" as in figure and denote byL and L" respectivelyhe boundariesof therectanglesR and R", omittingin each case the segmentAB. We apply now thefundamental heoremto therectangles1, 2, 3 and sum the results. In thisway we findff(z)dz+ f (z)dz O.

Z plane

11.~~ 4 RFI G. 7.

If nowa,n particular,we supposef z) =Cl(z Z'), we wirite + Bi= ~(z) dz =- Cl(z Z')dz, and z Z' = refitthen dz = ireside

+ eidr. Hence 33A + Bi =,>Cidq5 + jwCow = 7riC

A~~~~

sinceL" may be chosen n such a mannerthat the initialand finalvaluesofr are equal.Case 4. Letp(z) = C/(z(--Z') + ,(z)where+(z) is analyticthrough-out- and Z' is a point ofL, not and vertexofR. Now q(z)dz =dby thefundamental heorem . Hence herealso we obtainA + Bi = riC.

Case 5. In general et us considera rationalfunctionof ar eqal.f+ W

F(z) z -Z') z - Z )whereF(z) has onlysimpleroots nRa and on the boundary,L, butnotata vertex. Then A + Bi must be computedforeach pole and the resultssummed. Hence


7/30/2019 Cauchy - 1967922

12/17

CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. 265(11 A + Bi = 2wrifCk + 7iECkwhereCk is thecoefficientf1/(z Zk), Zk an interioroint ofR, andwhere k" is the coefficientf1/(z Zk"), Zk" a pointon the boundarynot at a vertex.Let Ck = Xk - iLk andCk" = Xk" - ik". Then(12) A = 2drEuk + drEuk"and(13) B = 27riXk + faXk"'On the basis of the fundamentalheoremI and equations 12) and(13) wemay workout oneof the examples ivenby Cauchy n Part II.Cauchyappliestheseformula o the rectangle oundedby y = 0,y = b> 0, x = - a, x = a, and then llowsa and b to increasewithout imit(Fig.8). f(M + Ni) = f(z) = f(x + yi) = P + Qi,whereQ(x, 0) O.A + Bi = ff(z)dz

pa Ib- J P(x, O)dx + [P(a, y) + iQ(a, y)]idy-asea pb

S n - (X b) + iQ(x, b)]dx - [P(O y) + iQ(O, y)]idy.-aSeparatinghisequation ntoreal and imaginaryarts,we havepa fo ~b Pa rb(14) A = fP(x, O)dx Q(a, y)dy P(x, b)dx+ Q(O,.y)dy,

ob a ob(15) B = f P(a, y)dy- Q(x, b)dx - P(O, y)dy.o ~ ~~~~~aTo be able to eliminate rom he formulaell integralsxceptthosealongtheaxis ofreals, Cauchythinks t sufficiento takef(z) to be afunctionsuch that P and Q vanish when x = so, y = o. This isnot at all sufficient,owever, orstrongeronditions re calledfor toinsure hevanishingfthe ntegralsnquestion. It issufficient,owever,ifa andbincreasendefinitelyna prescribedmanner uchas e.g.blim -= k =Oaco aandthat lim a2+ b2max f(x+ yi)I= 0,Gace0*Cf.O.C., p. 422,footnote,quation G).


7/30/2019 Cauchy - 1967922

13/17

266 H. J. ETTLINGER.the maximum eing akenfor ll points a + yi in the nterval c yc b andall points + bi in the nterval-a c x c a, i.e., for ll pointson three idesofthe rectangle etermined y (- a, 0), (- a, b), (a, b),and (a, 0). That is, we shall assumethat, given positivenumber,,arbitrarilymall,we can find positivenumber, , suchthat

'a2+ b2 max lf(x + yi) Ke,when > X for ll points a + yi in the interval c y c b and allpoints + bi nthe nterval a c x a. ThenQ(-- a, y)dy f( a + yi) dy

fb f~~~~~~~~~~ c j: 4 e dy< a +b < {',Jo~a2 +z2 a2+ b2when > X.Hence, s a andbincreasendefinitelynthisprescribedmanner,blim Q(dI a, y)dy=0.a, b-. cOSimilarly,tmaybe proved hat

r a narblim P(x, b)dx= 0, lim IQ(x, b)dx= 0, lim P(O, y)dy= 0a, ba J*.0a a. baaJ0 baa-PODrb rblim fQ(o0 y)dy = 0, lim JP(a, y)dy = 0.be-O a, b ox=

Moreover, f n addition1ima, af(a) = 0,then faP(x, O)dxconvergesas a increases ndefinitely,or fe is positive nd arbitrarilymall, hereexists positive umber suchthat If(x) < e/afor > x > X, andfP(x, O)dx |f f(x) dx

xa_ (a-X) X.Formula15) tells sthat, or functionulfillinghe boveconditions,B = 0, and theformula14) reduces ofpdx = A,whereA is takenfor ll thepolesoff(z) wherey _ 0. Iff(z) is an evenfunction,16) becomes


7/30/2019 Cauchy - 1967922

14/17

CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 267on= 2fpdx.

Let f(z) = zlm/(1 + Z2n), wheren is thegreater fthe twopositiveintegers,m and n. Then Q(x, 0) - 0 and P(x, 0) = x2m/(1 + X2n), anevenfunction.Hence co x2m(17) A =2 2f2 -+ 1dx.

Wemustnowspecify pathfor + bi suchthat imb = k 4 0 anda- alimala2+ b2max f(x+ yi) = 0 for ll points-4a + yi in the ntervala--*oo0 c y band all pointsx + biinthe nterval a x c a.

FIG. 8.Wewill akea = b (seeFig.8). Thenla2 + b2max f(i a + yi) _ '2a (1 a + ai)2|

2| ( t 1 + i)2m|a2(n-m)-1(a-2n+ 1)C 22m+J 1

a-2n + 1 a2(n-m)-lThe last expressionpproaches eroas a increasesndefinitely.SimilarlyWa2+ b2max f(x + bi) l -'12a a+ aiY?m

1 + (ai)2|a2n + i2n 2(n-m)-iThe last expressionbviouslypproaches eroas a increasesndefinitely.Also a2m+1limaf(a) =ai -+The-p~onitn 21+ a2thThe conditionsufficiento justify17) are thereforeatisfied.


7/30/2019 Cauchy - 1967922

15/17

268 H. J. ETTLINGER.The poles off(z) are to be found where ~n + 1 = 0, or

t2k+ 1,Zk = ek- 2n J k= 0, 1 * 2n - 1.We observethat the poles are on the unit circleand therefore ertainlyinsidethe rectangleR as soon as a > 1.

z2m Co C1 C2n-1- ,, ~~+ C' 3s + + 4-T ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-f4n-11+z2n Z e2n z- e z - e 2n 7rNowCk =im2- - Zk)

2-Zk 1+ Znlim (2m + 1) z2m - 2Mz2m-lZkz-Zk 2nz2n-Zk m 1- ______ _ = -Zk 2(m-n)+l2nZk n-1 2n

- [ 2k+1) 2n]2m+1-2n)2n 1 s= e (2k+1) (2m+1) 2n-.-2nAlso2n-1 1 2n-1 2m+E Ck -- - e(2k+l) 2n 7rk=O ~~n k=O

1 - e(2m+1)7ri 2m+1- - - -~~~en2m+1 7ri2n 1 -2-

i 2= _ _ i2m+1 2m+1 2m + 12ne n -e 2nsin 2 7rAnd 2n-1A = 2rE Msk 22r _ .k~o . 2m+ 12n sin 2nHence rX x2m d 7rJj 21 dx= 2n sin 2n+1 7r

Let 2m + 1 = aand 2n =f. ThenrXxa-1dx=tJO 1 + xI sina 7r

a formulawhichEuler had obtained.


7/30/2019 Cauchy - 1967922

16/17

CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 269In a similarmanner other formulaare obtained by takingother func-tions and other regions and integrating round the correspondingrec-tangle.We return o consider the methodused by Cauchy in the second partto obtain the resultsof the fundamental heorem I and its immediatecorollaries rovedabove. In the first lace, Cauchy adopts the "principalvalue" definition o remove any difficultiesegardingthe evaluation ofsimple integralsdue to singularities n the path of ntegration.tSuppose we have ?'(x), the derivativeof a real function (x) ofa realvariable. Consider ad(19) ft'(x)dx,

where?'(x) has a finite r infinite iscontinuity t (X) betweenc and dbut is continuous n c c x < X and X

7/30/2019 Cauchy - 1967922

17/17

270 H. J. ETTLINGER.It is, however,to be noticedthat these insufficientefinitions o notimpair the value of Cauchy's results,nor do they substantiallyaffectthemethod. If the discontinuity ccursat a cornerofR, themethod s

not applicable in general, as Cauchy's formula tselfshows.* For thedefinition f equation (20) above is an attemptto "cut-out" the singu-larity. But this does not yield a convergentresult forthis case. Wecannot,therefore,ut any real content ntothisparticularresultfrom urmodernpoint ofview and have, therefore,xcluded it in our treatment.Cauchy, himself,makes no use of this equation in any of the numerousapplicationsof the memoir.When the point of discontinuity ccurs inside of R at (X, Y), therectangle s divided into fourpartst by the linesx = X, y = Y, and theiterated integral s separated in a manner corresponding o the doubleintegralsover each of the four rectangles. When these four integralsare added together,we have in ratherobscure formwhat amounts tothemethodwhichwe have set forth bove. The singularpoint has been"cut-out" by a small rectangle, x = X - i, x = X + i, y = Y -77,y Y + a, and a method ofevaluatingthe line integral bout the smallrectangle s givenin the form:(21) A = lim lim J U(X+ ty)dy +l U(X+ y)dy- f U(X -X y) y- U(X - dyThe bracket is substantially he real part ofour equation (11). Thedifferenceetweenourmethod ofevaluation of (11) and Cauchy's methodof evaluating? (21) is a striking xample of the economyof the complexvariable formulation.Whenthe pointofdiscontinuitys on the boundaryofR, thevalue ofA is givenby twoterms I f (21). In bothcases theresults fter ntegra-tionare identicalwiththose of (12) and (13).In thehistorical eviewofthetheory ffunctions yBrilland Noether?a brief reatment fthe historical mportanceofthe memoir s given butnot from criticalpointofviewas is heredone.**UNIVERSITY OF TEXAS,ArTSTIN, TEXAS.

* O.C. Cf.third quationunder 20), p. 412.tO.C., p. 396.t O.C. Cf.p. 397,equation 13).?O.C., pp. 406-412.O.C., p. 400.?JahresberichterdeutschenMathemataken-vereinigung,ol. 3 (1894),p. 165ff.** The abovepaperhas grown utofan investigationfCauchy'swork ndefinitentegralsandresiduesn a Seminar ourse t HarvardUniversity.SomeoftheearlyworkwasdonewiththecollaborationfDr. E. S. Allen.

Documents

Cauchy - 1967922