Cauchy - 1967922

Annals of MathematicsCauchy's Paper of 1814 on Definite Integrals Author(s): H. J. Ettlinger Source: Annals of Mathematics, Second Series, Vol. 23, No. 3 (Mar., 1922), pp. 255-270 Published by: Annals of Mathematics Stable 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 of content 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 to Annals of Mathematics. http://www.jstor.org This content downloaded from 187.101.182.82 on Sun, 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions at least so far as the firstpart was concerned. Thirdly. 185. In the second place. For these reasons the discoveries contained in this memoirwere not appreciated even by the great mathematiciansof his time. Cauchy himself came to appreciate this fact.101. ETTLINGER. that the integralof an analytic function of a complex variable taken along a closed path depends entirely upon the behavior of the functionat points of discontinuitywithin the path. more of them.no new formulae were anAs nounced. Finally. for his footnotes of 1825 are devoted to the simpler complex equations from which his real ones can be readily deduced. yet there are several reasons why the reader might notice nothingat all like this theorem. clumsier notation.In 1814 Augustin Louis Cauchy presented before the Academie des Sciences a "memoir on definite integrals. and a much more obscure treatmentthan would be the case had he used complex quantities.* Introduction. 509. and. 319 ff. J. I serie. Sept. 2. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . the fundamental theorem.Cauchy used neitherfiguresnor geometrical language.1..182. This content downloaded from 187. Althoughthe kernelof the idea. indeed." 1.p. p. By the use ofthis difference integration he obtains the residue. concern simple integrals.82 on Sun. therebyobscuringthe relation of the latter to a line integral. 1814. p. Poisson? saw in the paper merelya means of evaluating integralsand remarkedthat. 1. 255 * Presented to the Amer. invariably separating an equation into its real and imaginaryparts. $ CEuvres Completes. is now an essential featureof every although geometricalrepresentation presentationof the theoryof functions. all editions abound in misprints. This necessitates longer equations. t " M6moire definies."in whichappears forthe firsttime the essence of his discoverieson residues. all the applications in this paper. Soc. Academiedes Sciences de lInstitutde France. ? Bulletin de la SocietePhilomatique (3). is contained in the paper. INTEGRALS. he refrainsfrom using complex quantities. for the evaluation of iterated integrals by the so-called surles integrates Savants Etrangers. The memoir was firstprintedin 1825t with additional notes and again in 18821 with no change save in the matter of notation.Math. 1919. In the firstplace.CAUCHY'S PAPER OF 1814 ON DEFINITE BY H. but the author states the central problem as the determinationof the in the value of an iterated integral according to the order of difference withrespectto the two variables. PART I. We shall also adopt the language of modern analysis for the sake of clearnessand accuracy. J.it ought not to replace the old ones! Lacroix and Legendre. very likely. (4) the derivacoveryof the cause and amount of this difference tion of new formulae which. Goursat. likely that the foremostmathematiciansof that time failed to recognize the contributionsof main importance in this paper. tomeII. (2) the pointing out of the fact that the value of an iterated integral may depend on the order of integration. (2) this was the first deduction by rigorousmethodsof the formulae. the followingfacts must be noted in addition: (1) imaginaries had no secure arithmeticalbasis in 1814. might have been otherwiseobtained.(3) while the formhad not yet been cast in the e-mould. for evaluating definite integrals. continuousin x and y regardedas independentvariables. to be sure.Functions of a ComplexVariable. pp. The first that if a function theoremproved in the memoiris. of a complexvariable is analytic throughout a regionof a certaintype and continuous in and on the boundary. respectively.which itself originated with Cauchy. pp. This content downloaded from 187. zweiteAuflage.* The regions considered are mapped in a one-to-one manner and continuously. hitherto obtained by purely formal processes. In discussing the memoir we shall frequently combine two separate real equations into one complex equation.. To appreciate thoroughlythe memoir. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . 211-214. These threetext-books will hereafter be referred to as O. as Cauchy did in his notes of 1825 and. G. pp. "singular" integrals(which are equal to the difference between the value first with respect to x and then with respect to y obtained by integrating and the value obtained by integrating in the reverseorder) he said that. Continuous integrand. though the new method was worthy of consideration.Lehrbuch der Funktionentheorie.82 on Sun.. stated as the valuable results obtained by Cauchy: (1) the of a seriesof generalformulae construction fortransforming and evaluating definiteintegrals.(3) the disin value. 82-92.256 H. in his original work.but not in general on a rectangle in the real (x.. 284-285. the integral of the functiontaken along the boundary of the region is zero. y) plane. ETTLINGER. Pierpont. P. The mapping on conformally. erster Band. * For modern treatment ofthistheorem see Osgood. in effect. the complex M + Ni plane is performed by taking M and N as real and continuousfunctions of x and y with derivativesof all orderswith respect to x and y. neverthelessthe proofs are so conceived that they correspondin substance to the standards of rigorof the present day.101. It seems. in the officialreport on the paper. Coursd'AnalyseMathematique.182. then. S.82 on Sun.101. lett Furthermore. S + Ti? f(M + Ni) O(M+ Ni) (2) (3) U + Vi =f(M O(M + Ni) + Ni) d (2) and (3) withregardto y and x respectively: Differentiate as+i Oy Oy z~ =f (M+Ni) +f= au Ox f(f ~~~~~~~~ f'(M d(M + Ni) a(M + Ni) Oy Ox Ni)'(M + + + N \i 2(M + Ni) iOVf/(M Oi x +Ni) O(M?+Ni) Oy O(M + Ni) ax~ +f(M But under the conditionsimposed O2(M + Ni) -2(M or Hence + Ni) $ + Ni)O(M+Ni).CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. of the M + Ni plane. and on the boundary. I. Mathematical Analysis. continuous in x and y.182. R (0 ! x _ a. P' + P"i to P + Qi and The notationof the original paper has been changedherefrom t Hereafter a(M + Ni) _ i V(M+ Nvi)willbe designated byd(M + Ni) ax ay d(x + yi) vol. r. since the integrand x = a and y = 0 to y = b. in a rectangle. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . and let and N =t(x.the order of integrationcan be reversed. 0 c y c b).we have: * y = x + zi to z = x + yi. and noting further is continuous. y) M = be single-valued functions. p. 257 Let* (1) f(M + Ni) = P + Qi be an analytic functionof M + Ni in a certainregion. tSee Goursat-Hedrick. y) q(x. and possessingcontinuous partial derivatives of all orderswith respect to x and y in R and on r. Oxdy O yax OS iOT aU iOV Oy Oy Ox Ox OS a @---y (4) OU and aT = OV Oy Ox Multiplying the equations (4) by dydx and integratingfrom x = 0 to that. This content downloaded from 187. 13. U(a. 1. we obtain (7) fa Tdx - tdx = fb Vdy- vdy.101. 0 _ y valuedcontinuous functions of x and y in a rectangle. y) = U. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions .1. y) =V. V(O. L. 0) = s. b) = S. then equafSSdx- ra r~a sdx ra = f b Udy - rb fb udy. U(O.r. of theM + Ni plane. and on the boundary.and hence meansthataround therectangle in the Al + Ni planeabout the corresponding curve. p. footnote. JL ~~d(x+ yi) which heregivenin thez plane. (8) ff(M + Ni)d(M + Ni) = 0. be an analytic function Hence of M + Ni in a certainregion. This CEuvres Completes. continuousin S.182. zplane FIG. y) = v.258 (5) ddx H. and letM = q(x. memoir willbe referred to hereafter as O. In a similarmanner. possessingcontinuouspartial derivatives of * Cf.82 on Sun. ETTLINGER. y) and N = 41(x. and on the boundary. b) = T. d = dy f TX dx. FUNDAMENTAL THEOREM I: Let f(M + Ni) This content downloaded from 187. S(x. I serie. Multiplying(7) by -i f(s + ti)dx +Jb and (6) by - - 1 and adding we have Ti) dx - (U + Vi) dy (S + fb(u + vi)dy = 0. equation (3) and equations(A).y) be singleR (O _ x _ a. y) = u. tion (5) becomes (6) Let S(x.lettingT(x. 0) = t andV(a. ! b). T(x. or d(M + Ni) ff(M + Ni) d(x + yi) = 0.S. J. 338.* L1.C. the upper limit of the x-interval.R.b) when a becomes infiniteand the improper integrals thus introduced converge.Ao) FIG. or ff(M + Ni)d(M + Ni) = O.CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 2.. The functions used in this paper forM and N and the corresponding maps of the rectangle. N = xy. 4. 3. M=x.to become infinite. on the closed rectangle. of course. In several of the applications Cauchy allows a. .101. (o~~~o)( co0)~~~~ FIG. N=y.82 on Sun. 3. These conditions are. a > O. in a one-to-one manner and continuously. 2?. 259 all orderswithrespectto x and y and mapping theclosed region.182. 10. In the applications Cauchy uses the real equations in S and U.S. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . and 5. M =ax. on the M + Ni plane are given in figures 2. T and V respectively and does not combinethemas is here done. He considered that his conclusions could be extendedto this case if the function f(M + Ni) approaches a limitfor each value of y (O < y . This content downloaded from 187.then the integraloff(M + Ni)d(M + Ni) takenaround L in the positivedirection vanishes.R. . la x e N = x siny. P(n- = n. b)dx Q0 b{hdx x JQ( tw~a P(x. and S + Ti . M = xcosy. y) + Q(x. such that Q(x. N =xy. y)dy = 0.82 on Sun.N4' (acosi a4 ) X~& 4?. in (0 . M =ax. (aAeb) (oo) FIG.4dy This content downloaded from 187.) M =x. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions .182. bx-JP +f P(a.Q+Pi. y)dy rb rb Q(0. ETTLINGER. Let f(x + yi) = P(x. 1-t. y~dy. 2. a > 0. 4. O)dx + -I yd+b rb b Q(a.N = y.P+Qi. Equations (6) and (7) yield (6') (7') (7') f rba P(x. 5.y _ b) and the integrals conapproachtheirlimitsuniformly correct. y)i. y.101. as established results may be verge. (See Fig. 1 FIG.260 30 H. Region10. J. 0) = 0. f(M + Ni) considered by Cauchy but sinceall the functions insufficient. U+ Vi = . the and of the is an exampleof the methodof application The following results ofPart I. Q(x. In all the applications t O. 413. withpoles. 0) = e-2. p. 261 Applytheseequationsto z = x + yi. But hence Similarly a limbeb2-a2 a = 0. cos 2aydy = 0.101.CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. In the second part of the memoirCauchy Integrands at isolated of functions whichare discontinuous deals withthe integrals are simplepoles.* where e-x2ey2 sin 2xy. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . y) P(x. The second integralin each of for theequations(9) and (10) vanishes. Q(0. cos 2xy..82 on Sun. e-f'eb2sin 2bxdx - ea2f ey2 ey2dy.182. - 0 e 2dx e_ 2 4LW ~~~~~~~~2 e-X2dx /2. O.fe2 cos 2aydy = - dx. G.. lim fbe-X2ei2 rb rb sin 2aydy= O. in writing Cauchy'sequations: justified f ifwe assume o cos 2bxdx = e-2 e'2 rX r~~~~~~~~~~b = e-b2 e-2 sin 2bxdx ey2dy. Beispiel. = f(z) = e-z2 so thatin thiscase equations(6') and (7') becomerespectively (9) (10) fex2eb2 cos 2bxdx - e-a2 eY2 sin 2aydy = . 4. y) = ey2. PART II. fbe-a2ee2 sin 2aydy cf sin 2ay ea2ey2dy _ ea2f b ey2dy _ beb2-a2 b > 0.e-xey2 P(x. y) . This content downloaded from 187. P(0. Now let a increasewithoutlimit.y) = 0. lim Jbe-a2ey2 we are Since it can be readilyshownthat the otherintegrals converge. 121. p.C.t thesesingularities points.. 30. p. 293. * Cf. mapping of all orders. tions of x and y in and on the boundary. partial derivatives 0 c y _ b).S. p. R. Moreover no useful facts are developed concerning The expositionand criticismof Cauchy's method we lay aside forthe momentand proceed to set fortha method by which the results of Part II are very simply obtained from the fundamentaltheorem of Part I. 4j5 t 0.e. 0 < a' < a" < a. 4b -3~_FIG . and letL' be thecurve (X.S. Y) within L around in the taken positive Then theintegraloff(M + Ni) d(M + Ni) off(M + Ni) -d(M + Ni) takenin thepositive sense is equal to theintegral sense around L'.the iteratedintegralplays an unessentialrole in Part II. ofS.P.. of theM + Ni plane. The result is apparentlystated in of an iterated integral ordters terms of the evaluation in two different whose integrand has a singularityat a single point. _ x _ a". = = {2(X. iterated integrals. It is the method used in many modern text books on the Theory of Functions.. R (O _ x _ a.182. 114ff. p. on theclosedrectangle. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . 6.t FUNDAMENTAL THEOREM II: Let f(M + Ni) be an analyticfunction of M + Ni in a certainregion. ETTLINGER. as will be pointed out later.* As a matter of fact. a FIG. J. 0 < b' < b < b. and continuous funcy) and N = {(x. 388 ff. .101.262 H. Let M = +k(x. The true significanceof Cauchy's method at this point is very obscure. Y).82 on Sun. t I.. 331 ff.. inside of S. y) be continuous exceptat thispole. Here Cauchy obtains for the firsttime a formulawhich contains the essence of the theorem on residues.. in S corresponding its boundaryF'. This method is not so very unlike Cauchy's. ff(M + Ni)d(M + Ni) = f(M + Ni)d(M + Ni).r. b' c y c b") be any rectangleinteriortto R and containing to F'.. possessingcontinuous and manner in a one-to-one theclosedregion. exceptfor a single L. This content downloaded from 187. z 4plan C. Let R' (a' and such that m +k(X. pole at m + ni.C.. and would probably be used by him were he writingin the notation of the present-dayanalyst. p. 206 ff. in and on theboundary. ofa rectangle. Y) and'n continuously. p. since all the theorems and applications are concerned with simple integrals only. * O.G. N = y. t L' is identical withr' in thiscase. p.. C.* Cauchy's paper gives the first We proceed to derive the formula necessaryto make the first applica- tion by evaluating f(M + Ni)d(M + Ni)t explicitlyfor the case M = x. l The notation This content downloaded from 187.C. has herebeen changedfrom A' + A"i to A + Bi.and suppose: Z A + Bi where Z= X + Yi.L. f(z) = Then Let Then or LZ - Cdz Z z - Z = re~i. theorem .L.. 8 in Fig. The equivalent of this equation in statementof his discoveryon Residues. f(z) =(Z) + - where+(z) is analytic in R and continuousin R and on the boundary. and whereZ is withinR. 412.? since the initial and finalvalues of r are equal.82 on Sun. ?Cf. 263 The proof follows immediatelyfrom the fundamentaltheorem I by application successively to the rectangles marked 1 . of the rectangle. = 4' f(z)dz.101. Case 2.182. A + Bi= O Cid4+ I dr ~~r Lt A + Bi = 2 riC. 0. equation (4). dz = iresido + e~idr.CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. equation (C). 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . p.. A + Bi =f(z) dz + Z dz or since * 4' A+Bi=27riC. Let: A + Bi Case 1. If f(z) has a pole on the boundary.. O. or z = M + Ni. 6 and addition of the resultingequations. 381.R.(z)dz = 0 by thefundamental Case 3. but I. footnote. not and vertex of R. the of ar fundamental by eqal.82 on Sun. we wirite in particular. Now q(z)dz = d theoremI. Hence 33 A + Bi =. 4R 7.101. Hence This content downloaded from 187. We apply now the fundamentaltheoremI to the rectangles1. Hence here also we obtain W A +d Bi = riC.f+ Case 5.we constructR" as in figure7 and denote by the boundariesof the rectanglesR and R". A~~~~ = J~(z) dz =Cl(z Z')dz.Z ) Ra and on the boundary.>Cidq5 + jwCow = 7ri C since L" may be chosen in such a mannerthat the initial and finalvalues of r are equal. Then A + Bi must be computed for each pole and the results summed. Case 4. In generallet us considera rational function F(z) z -Z') z .we suppose f(z) =Cl(z If nowa. 3 and sum the results.~~ FI G. J. not at one of the vertices. A + Bi Z'). Let fp(z) = C/(z(--Z') + .264 H. butnotat whereF(z) has only simple roots in a vertex.(z) where+(z) is analytic throughRout and Z' is a point of L. 2. and z Z' = refit then dz = ireside + eidr.182. In this way we find ff(z)dz+ ff(z)dz = O. ETTLINGER.L. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . omitting L and L" respectively in each case the segmentAB. Z plane 11. Then Let Ck = Xk . y)dy. y)]idy. y = o. 265 (11) A + Bi = 2wrifCk + 7iECk pointof R.y)dy. O)dx + [P(a. 8). Gace0 functionsuch that P and Q vanish when x = so.a. This is This content downloaded from 187. x = a. S n thisequationintoreal and imaginary parts. Cauchy thinksit sufficient = f o P(a. however.. y)dy- Pa P(x. x = . f(M + Ni) = f(z) = f(x + yi) = P + Qi. in a prescribed indefinitely ifa and b increase b lim -= k = O aco a and that lim a2 + b2maxIf(x+ yi) I= 0. and thenallowsa and b to increase (Fig. Ib A + Bi = ff(z)dz - J pa [P(O y) + iQ(O. boundedby y = 0. 422. y) + iQ(a. P(O. b)dx - ob conditions are called forto forstronger not at all sufficient.we have Separating iQ(x.C. footnote.101. and of 1/(z-Zk).182. ~ ~~~~~a all integrals from the formulae exceptthose To be able to eliminate to take f(z) to be a along the axis of reals. in question. Zk an interior whereCk is the coefficient of 1/(z -Zk"). y)]idy pb (14) (15) A B = fP(x. y = b to the rectangle Cauchyappliestheseformula without limit > 0. It is sufficient. however.g. O. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . p. where Q(x..82 on Sun.iLk and Ck" = Xk" . * Cf. 0) O.ik". b)]dx - sea -a -a (X b) + O)dx P(x. Zk" a pointon the boundary whereCk" is the coefficient notat a vertex. equation (G). ob pa fo ~~b Q(a. b)dx+ rb Q(O. oftheintegrals thevanishing insure manner suchas e.CAUCHY S PAPER OF 1814 ON DEFINITE INTEGRALS. A = 2drEuk + drEuk" (12) and (13) B = 27riXk + faXk"' theorem II and equations(12) and On the basis of the fundamental (13) we may workout one of the examplesgivenby Cauchyin Part II. y)dy- a Q(x. y)dy= 0 --O be lim fQ(o0 rb y)dy = 0.101. Then Q(-. and number existsa positive fP(x. whena > X. b)dx = 0. whena > X forall pointsi a + yi in the interval0 c y c b and all pointsx + bi in theinterval..266 H. on threesides of the rectangle and (a. ETTLINGER. (16) becomes This content downloaded from 187. indefinitely Hence. givena positivenumber.82 on Sun. then faP(x. X. ba -J*. 0c y beingtakenforall pointsi a + yi in the interval the maximum c b and all pointsx + bi in theinterval-a c x c a. O)dx converges forif e is positiveand arbitrarily small. rb y)dy = 0.as a and b increase a. af(a) = 0. (. (a.we can find 'a2 + b2 max lf(x + yi) Ke. a. theabove conditions. forall points determined by (. {. O)dx |f f(x) dx xa _ (a-X)<e fora > X. cO lim b Q(dI a.e. 0). b-. lim IQ(x.a. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions .if in addition 1ima. baaJ0 baa-P P(O. and theformula (14) reducesto fpdx = A. b).a c x a. y)dy fb f( a + yi) dy c j: 4 e Jo~a2 +z2 f~~~~~~~~~~ dy < a +b a2+ b2 < {'. b -ox= lim JP(a. X suchthat If(x)I < e/afora > x > X. a. 0).for fulfilling B = 0. a function Formula(15) tellsus that.182. A is takenforall the poles off(z) where where y _ 0. a positive arbitrarily small. That is.a. i. b).a.0a lim ra P(x.there as a increases indefinitely. in thisprescribed manner. If f(z) is an even function. narb it maybe provedthat Similarly. y)dy=0. we shall assume that. suchthat number. lim OD a. b)dx = 0. Moreover. J. CAUCHY' S PAPER OF 1814 ON = DEFINITE INTEGRALS. Then la2 + b2max f(i a + yi) _ '2a (1 a + ai)2| 2| ( t 1 + i)2m| a2(n-m)-1(a-2n+ C 22m+J 1) 1 1 a2(n-m)-l a-2n + indefinitely. an integers. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . to justify sufficient The conditions (17) are therefore This content downloaded from 187. Hence (17) A =2 2f2 co x2m -+ 1dx. 8). evenfunction.101. wheren is the greater m and n. of the two positive Let f(z) = zlm/(1 + Z2n).182. approacheszeroas a increases obviously The last expression Also conitn The-p~ limaf(a) =ai a2m+1 21+ a2th + - satisfied. a path fora + bi such thatlim b = k 4 0 and We mustnow specify lim ala2+ b2maxIf(x+ yi) I = 0 forall points -4 a + yi in theinterval a--*oo 0 c y a- a b and all pointsx + bi in theinterval-a x c a.82 on Sun. 0) . FIG.0 and P(x. 0) = x2m/(1 + X2n). 8. Then Q(x. zeroas a increases approaches The last expression Similarly Wa2+ b2max f(x + bi) l -'12a (a+ aiY?m 1 + (ai)2| a2n + i2n 2(n-m)-i indefinitely. We willtake a = b (see Fig. 267 A on 2fpdx. This content downloaded from 187. or 1.e e2n . tJO 1 + xI rXxa-1dx= a formulawhich Euler had obtained.101.IT 7r C1 C2n-1 Now Ck = ]im2 2-Zk 1+ - Zk) Zn 1) z2m 2nz2n2Mz2m-lZk lim (2m + z-Zk ______ Zk2m 2nZk2n-1 _ = 1 -Zk 2(m-n)+l 2n - 2n = -2n (2m+1-2n) [(2k+1) 2n] 1 e (2k+1) (2m+1) 2n-.82 on Sun.1 2n sin . ek- t2k+ 1. The poles off(z) are to be foundwhere z~n Zk = + 1 = 0.182. ETTLINGER. 2n J k= 0. 1 * 2n - certainly We observe that the poles are on the unit circle and therefore inside the rectangleR as soon as a > 1. _r22 2m+ 7r 2n 1 _ . Hence JjX21 rX x2m d dx= 2n sin 2n+1 Then sina 7r 7r Let 2m + 1 = aand 2n =f. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions . z2m Co 1+z2n - Z C' 3s + + 4~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-f4n-1 ~~+ z . J.e 2n z..268 H. s Also 2n-1 Ck -E k=O - 1 2n-1 - ~~n k=O - e(2k+l) 2m+ + 2n 7r 1 1 - 2n 1 i -2- e(2m+1)7ri 2m+1 2n 2m+1 7ri-~~~e 2m+1 2ne n 2= _ -e 2m+1 _ i 2n sin 2m + 1 2 7r And = A = 2rE Msk k~o 2n. p. ? Cf. examplesgivenby Cauchy. 0 c y _ b). Accordingto the moderndefinition conditionfor the existenceof this limit is a necessary but not a sufficient the convergence of (19). Cf.R (O _ x _ a. except at the point (0. of convergence ofan improper integral. 397. Cauchy proceeds as follows... and on the boundary. 394 and p. _2 fU dx dz/z. 01 O.4(X . singularity L.C. 289-295. pp. pp. Let U(x.? ThentI (20) fdy af ) dx = limJI dy where t > 0.182.* We returnto considerthe method used by Cauchy in the second part to obtain the results of the fundamentaltheorem II and its immediate place.p.O- - A = lim [4(X + h) . 269 In a similarmannerotherformula are obtained by taking otherfunctions and other regions and integratingaround the correspondingrectangle. the derivative of a real function+(x) of a real variable. p. In the first regardingthe evaluation of value" definitionto remove any difficulties on the path of integration. 0) whereit possesses a non-removable but does not become infinite.C. - h) - g(X + h) + 4(d)] where -ck(d) 13 -.82 on Sun. at (X) between c and d discontinuity where ?'(x) has a finiteor infinite but is continuous in c c x < X and X <x c d. Cauchy takes it as his working definition.C. G. This definition is totallyinadequate.C. exampleon p. since the simple integralobtained after a firstintegrationdoes not even come under the principal value and may even diverge..h)]. This content downloaded from 187.. 404. Consider (19) f ad ct'(x)dx..O. definition * See O.CAUCHY' S PAPER OF 1814 ON DEFINITE INTEGRALS. 402. y) be a functionwhich is continuousin x and y and possesses a continuous partialderivativewithrespectto x everywhereinside a rectangle.t simple integralsdue to singularities Suppose we have ?'(x). t O. 118-122.A. By the "principal value" of (19) is meant A'd r pX-h l '(x)dx lim 0 q'(x)dx + Iq h-o Lc JX4h - J lim [h-*o q(c) + g(X 4(c) . 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions .101.t Secondly. to definethe improperiterated integralswhich occur. O. 390. Cauchy adopts the "principal corollariesproved above. O. p. Y). and a method of evaluating the line integralabout the small rectangleis given in the form: t A = lim lim J (21) U(X+ y)dy + jl - "cut-out" by a small rectangle. We cannot. In both cases the resultsafterintegration are identical with those of (12) and (13). the rectangleis divided into fourpartst by the lines x = X. and the iterated integral is separated in a manner corresponding to the double integrals over each of the four rectangles. f U(X (X - y)dy- U(X - dy The bracket is substantiallythe real part of our equation (11). J. 397.* For the of equation (20) above is an attempt to "cut-out" the singudefinition larity. y = Y -77.82 on Sun.** UNIVERSITY * OF TEXAS.C. 412. Allen.C. O. equation (13). 165ff. excluded it in our treatment. 396. t O. y = Y. vol. 400. x = X + i. ETTLINGER. ? Jahresbericht der deutschen Mathemataken-vereinigung.182.C. put any real contentinto this particularresultfromour modernpoint of view and have. the value of A is given by two termsIIof (21). 406-412. p.we have in rather obscure form what amounts to the method which we have set forthabove.to be noticed that these insufficient definitions do not impair the value of Cauchy's results.270 H. It is. t O.. But this does not yield a convergentresult for this case. TEXAS.. p. Cf. ? O..makes no use of this equation in any of the numerous applicationsof the memoir. When the point of discontinuity is on the boundary of R. the method is not applicable in general. p. ** The above paper has grown out of an investigation of Cauchy'sworkon definite integrals and residues in a Seminarcourseat HarvardUniversity. Cauchy. When these four integrals are added together.C. The difference betweenour methodof evaluation of (11) and Cauchy's method of evaluating? (21) is a strikingexample of the economy of the complex variable formulation. Some ofthe earlyworkwas donewith the collaboration ofDr. Cf. 3 (1894). If the discontinuity occurs at a cornerof R. This content downloaded from 187. The singularpoint has been y Y + a. therefore. When the point of discontinuityoccurs inside of R at (X. third equationunder(20). ArTSTIN.therefore. as Cauchy's formulaitself shows. E. p. himself. S. x = X - i. In the historicalreviewofthe theoryoffunctions by Brilland Noether? a brieftreatmentof the historicalimportanceof the memoiris given but not froma criticalpoint of view as is here done. nor do they substantially affect the method. however.101. pp. 24 Mar 2013 18:24:03 PM All use subject to JSTOR Terms and Conditions .C. 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