Malmsten's Proof of the Integral Theorem

Malmsten's Proof of the Integral Theorem– an Early Swedish Paper on Complex AnalysisKajsa BråtingU.U.D.M HPM Project Report 2002:P1Examensarbete i matematik, 10 poängHandledare och examinator: Gunnar BergMars 2002Department of MathematicsUppsala University

Malmsten’s Proof of The Integral Theorem-an Early Swedish Paper on Complex AnalysisKajsa Bråting21st February 2002

AbstractIn this report we consider the Swedish mathematician C. J. Malmsten’sarticle “Om definita integraler mellan imaginära gränsor” from1865. In his article, Malmsten gives a proof of the Cauchy integral theoremwhere the limits of integration are complex numbers. The aimsof this report is to try to catch the techniques that Malmsten made useof in his proof and to get a glimpse of the mathematical concepts atthis time, especially those which weren’t fully investigated and therebygave rise to some problems for the mathematicians.1

Contents1 Introduction 32 Biography of Malmsten 33 A short History of Complex Analysis 53.1 Development of the Eighteenth Century . . . . . . . . . . . . 53.2 The Geometric Representation of a Complex Number . . . . 63.3 Complex Integration . . . . . . . . . . . . . . . . . . . . . . . 74 Cauchy and the Integral Theorem 84.1 Biography of Cauchy . . . . . . . . . . . . . . . . . . . . . . . 84.2 The Cauchy Integral Theorem . . . . . . . . . . . . . . . . . . 95 Malmsten´s Article: “Om definita integraler mellan imaginäragränsor” 125.1 The Insufficiencies in Cauchy’s Proof According to Malmsten 135.2 How Malmsten Proves the Existence of the Integrals . . . . . 145.3 The Various Properties of the Integral . . . . . . . . . . . . . 196 Final Remarks 202

1 IntroductionDuring the eighteenth century the mathematicians in Uppsala were mainlyastronomers, physicists or theologians. In fact, it wasn’t until the 1820sthat Uppsala, as well as Lund, got their first pure mathematicians. Maybethis was a natural ongoing process toward specialization. It could also havebeen due to the fact that the rumors of great works of Euler, d’Alembert,Lagrange and Laplace now had reached the universities of Scandinavia [3]. InUppsala, at the time, a new generation of talented mathematicians appeared.Among the most prominent were Adolf Fredrik Svanberg (1806-1857), CarlJohan Malmsten (1814-1886) and Emanuel Gabriel Björling (1808-1872).In this report we discuss the article “Om definita integraler mellan imaginäragränsor”, which was written by C. J. Malmsten in 1865. In his work,Malmsten gives a proof of Cauchy’s integral theorem, i.e. that the integralis independent of the path of integration (where the limits of integrationare complex numbers). The aims of this report is to try to catch the techniquesthat Malmsten made use of in his proof and to get a glimpse of themathematical concepts at this time, especially those which weren’t fully investigatedand thereby gave rise to some problems for the mathematicians.The intention is also, to some extent, to become acquainted with the mathematicalarena in Sweden at this time.To give a background to Malmsten’s article and to see how far the developmentof complex analysis had reached at this time, we begin with ashort history of this field. Subsequently, a summary of Cauchy’s proof ofthe integral theorem will be presented, followed by a biography of Cauchy,before the exposition of Malmsten’s article. However, the report begins witha biography of Malmsten.2 Biography of MalmstenCarl Johan Malmsten (1814-1886) grew up in Uddetorp, not far from Skara.Shortly after the death of his father (1827) Malmsten, who at the time wasin ’Skara skola’, obtained a favourable position as a tutor. This post made itpossible for Malmsten to study without interruptions and he was able to keepthe post during his upper secondary studies as well as his university studies.In 1833 Malmsten began to study mathematics at Uppsala university wherehe 1839 received a doctoral degree.In competition with E. G. Björling (1808-1872) Malmsten became professorof mathematics in 1841 and thereby succeeded his own teacher Jöns3

Svanberg (1771-1851) [3]. In the same year Malmsten was one of the initiatorsof Frey, a journal of art and science, which was published in ten volumes[9].During the 1840s Malmsten wrote many remarkable articles and becameafter Samuel Klingenstierna (1698-1765) the first Swedish mathematicianwho not only followed the development of mathematics of his time, butalso contributed to it [3]. He had a gift for languages and his lectures wereappreciated also by pure humanists. He was a friend of Geijer and took partin the social life of ’Malla Silfverstolpes salong’. One can say that Malmstencontributed to the fact that mathematics became a rather popular topicduring the 1840s [9].In his own mathematical research, Malmsten took an active part in analysis,algebra and probability theory. In a summary of Malmsten’s mathematicalresearch, Gösta Mittag Leffler (1846-1927) emphasizes his calculationof the remainder term in Euler’s “summation” formula as well as hissolutions of certain differential equations [9].Malmsten had a genuine interest in probability theory, which mainlyresulted in his essays and popular lectures about the insurance business. Itis said that one of the motives was his sympathy for the misfortunate [9]. In1855 Malmsten was one of the founders of the insurance company ’Skandia’,where he was a director for many years.In 1858 Malmsten became a member of a committee, whose task was toinvestigate the pensions system. In 1859 he was appointed minister withoutportfolio and in 1866 county governor of ’Skaraborgs län’. In the course oftime Malmsten did regret that he had given up science for politics. One ofthe reasons that he didn’t return to his professorship was consideration forhis substitute Daug, whose financial position was poor [9].In public life Malmsten became famous for a brilliant speech concerningthe freedom for the press at the fiftieth anniversary of the constitution andfor his excellent dinners at the county governor’s house in Mariestad [3].The article “Om definita integraler mellan imaginära gränsor” was writtenafter Malmsten’s years as professor (1865). A few years later the Cauchyintegral theorem, as well as the whole theory of analytic functions, wouldprovide a lot of problems for the mathematicians in Uppsala. It also gaverise to their own interpretations of the theorem.4

3 A short History of Complex AnalysisIt took quite a while before the mathematicians began to accept the complexnumbers as independent mathematical objects. One of the reasons mighthave been the relatively late entrance of negative numbers on the mathematicalscene [6]. In fact, mathematicians didn´t accept negative numbersas roots of equations until the sixteenth century.As soon as mathematicians were able to solve the third- and fourthdegreeequations, another problem arose. In Cardano´s (1501-1576) formula(for solving third-degree equations) square roots of negative numberssometimes appeared, which confused mathematicians of the time. RafaelBombelli (1526-1573) was one of the first who tried to operate with thesenumbers. But his methods were not very successful and he was also verysceptical of these numbers.During the seventeenth century the development of the complex numbersdidn´t make much progress and mathematicians such as Isaac Newton(1642-1727) and Wilhelm Leibniz (1646-1716) displayed a sceptical attitudeto these numbers. Leibniz considered them to lie somewhere in between existenceand non-existence and compared them to The holy spirit [6]. Althoughhe made some calculations with complex numbers, he didn’t understand themeaning of them.3.1 Development of the Eighteenth CenturyDuring the eighteenth century the development of complex analysis accelerated.An entirely new theme of this time was the use of complex values ofan independent variable. The leading man of this work was Leonhard Euler(1707-1783).In the very first pages of the first volume of his Introductio in analysininfinitorum (1748), Euler asserted that a variable can assume any valuewhatsoever, including an imaginary one. For example, in order to clarifythat “a function of a variable quantity is itself a variable quantity,” heasserted that there is no value which the function is not capable of assuming,“since a variable quantity also includes imaginary values” [2]. In the workof Euler the theory of elementary functions of a complex variable received afull development and was perfected by the middle of the eighteenth century[4].Euler established that the equation ∂u/∂x = ∂v/∂y is the condition underwhich v dx+u dy is the exact differential of some function. In connectionwith problems in fluid mechanics Jean d´Alembert (1717-1783) was the first5

to arrive at the system of equations∂u∂x = ∂v∂y ,∂u∂y = − ∂v∂x(today known as the Cauchy-Riemann equations). d´Alembert, and afterhim Euler, obtained pairs of solutions of this system as the real and imaginaryparts of (what we today would call) an analytic function f(z) = u + ivof a complex variable.In Eulers work on integral calculus he established the equality∫∫∫f(z)dz = (udx − vdy) + i (vdx + udy),where f(z) = u + iv. Only the concept of integrals of functions of a complexvariable computed along some curve was lacking to here see the Cauchyintegral theorem that the integral is independent of the path of integration.In the work of Euler, as well as d´Alembert, complex numbers enter thediscussion usually as pairs of real numbers having a certain specific meaning,for example as pairs formed from the abscissa and ordinate of a pointin a plane, the projections of the velocity of a particle of fluid on the coordinateaxes, or pairs of functions. Thus the various particular interpretationsof complex numbers (in particular, geometrical interpretations) were incorporatedinto mathematics (in some sense) starting in the mid-eighteenthcentury [4]. However a visual representation of the complex variable as avector or a point moving over the plane was still missing, even though Eulerwas quite close to such a concept in his paper “On the representation of aspherical surface on a plane” (1777) [4].3.2 The Geometric Representation of a Complex NumberThe first explicit and systematic geometric representation of complex numbersas directed line segments and the corresponding interpretation of operationson them occur in a 1799 paper by the Danish surveyor Caspar Wessel(1745-1818). But Wessel´s work went completely unnoticed until it was rediscovereda hundred years later and republished in a French translation[2]. Other roughly contemporary publications had also gone unnoticed, forexample, by abbot Adrien Buée (1748-1826) in 1806, and that of the SwissJean-Robert Argand (1768-1822), who published an anonymous pamphleton the geometric interpretation of complex numbers at his own expense [2].However, in the first quarter of the nineteenth century many mathematicianswere very close to the geometric representation of complex numbers.The concept of complex numbers as points of a plane received general6

ecognition after 1831, when Carl Friedrich Gauss (1777-1855) published thepaper “Theoria residuorum biquadraticorum”. This paper included a justificationof the complex numbers and a geometric interpretation of them.The term “complex numbers” itself is due to Gauss [4].Cauchy had a completely different approach explaining the foundationsof the theory of complex numbers. In his Analyse algébrique (1821) complexquantities were introduced in a formal manner. He classifies most “imaginaryexpressions” (i.e., complex numbers) and “imaginary equations” (i.e.,equalities containing complex numbers) as symbolic expressions. His viewof the meaning of these expressions when taken literally is...imprecise or devoid of meaning, but from which one candeduce precise results by modifying and altering according to definiterules, either the equations themselves or the symbolic containedin them.It wasn´t until 1847 in the paper “Mémoire sur les quantités géométriques”(1847) that Cauchy replaced the imaginary expressions by the geometricalinterpretation. Although, it is curious that when Cauchy had proved thetheorem that the integral is independent of the path of integration (seebelow) he still emphasized that imaginary numbers “do not refer to anythingand have no meaning. The sign √ −1 is in a way only a device, a tool forcomputation...” [4].3.3 Complex IntegrationAt the beginning of the eighteenth century mathematicians began to operatewith complex-valued functions. These were always separated into a real andan imaginary part, so they could be treated like real-valued functions. Asan example of an early use of complex-valued functions Leibniz as well asJohann Bernoulli (1667-1748) evaluated the real integral∫dxax 2 + bx + cby splitting up the integrand into two partial fractions and then doing theintegration. The fact that if the roots to the equationax 2 + bx + c = 0are complex, the denominator will also be complex, didn´t stop them toperform the integration in the same way as if the denominators were real7

[6]. However, the integration of the partial fractions resulted in logarithmsof complex numbers, which gave rise to some problems. Eventually, theseproblems were solved by the well-known formula of Euler, namely;log z = log r + i(ϕ + 2πn)(n ɛ Z).But another problem arose, when the mathematicians had to deal withintegrals as∫ z1f(z)dz,z 0where f is a complex-valued function and the limits of integration are complex.Cauchy was the one who came up with the theory of complex integrationin his pamphlet “Mémoire sur les intégrales définies, prises entre des limitesimaginaires” from 1825 (see below). Nevertheless, it seems that the theoryhad already been known to Gauss. In a letter he had written to his friendFriedrich Wilhelm Bessel (1784-1846) in 1811, but not published until 1880,it can be read that Gauss already knew about the “integral theorem”. Atthe end of the letter he says “This is a very beautiful theorem, for which Iwill give a not difficult proof at a suitable opportunity.” But Gauss neverproved the theorem.4 Cauchy and the Integral TheoremCauchy, along with his contemporaries - Gauss, Abel (1802-1829) and Bolzano(1781-1848) - belong to the pioneers of the new insistence on rigor in mathematics.The eighteenth century had been essentially a period of experimentation.When they worried about rigor, as Euler and Lagrange occasionallydid, their arguments were not always convincing [8]. Cauchy, Gauss, Abeland Bolzano are often called the first truly modern mathematicians [1].In this report a biography of Cauchy will be presented and then a summaryof Cauchy’s proof of the integral theorem from his paper “Mémoiresur les intégrales définies, prises entre des limites imaginaires” (1825).4.1 Biography of CauchyAugustin-Louis Cauchy (1789-1857) was born in Paris about five weeks afterthe fall of the Bastille. His mathematical talent was discovered early andhis father introduced him to Laplace (1749-1827) and Lagrange (1736-1813),who both were visitors in the Cauchy family home. Lagrange in particular8

seems to have taken an interest in Cauchy and advised his father “not toallow his son to open a mathematics book nor write a single number before hehas completed his studies in literature” [1]. Otherwise, Lagrange continued,“the boy might be a great mathematician but not capable of writing inhis own native language” [6]. On Lagrange´s advice, Cauchy entered theÉcole Centrale du Panthéon where he spent two years of studying classicallanguages.From 1804 Cauchy attended classes in mathematics and he took theentrance examination for the École Polytechnique in 1805. Two years laterhe graduated from the École Polytechnique and entered the engineeringschool École des Ponts et Chaussées. After that Cauchy spent a few yearsas an engineer in Cherbourg.An academic career was what Cauchy wanted and he applied for a postin the Bureau des Longitudes. He failed to obtain this post and insteadLegendre was appointed. He also lost in 1815, to Binet, a mechanics chairat the École Poytechnique, but then was appointed assistant professor ofanalysis there. In 1816 he won the Grand Prix of the French Academy ofScience for a work on waves. Later he also became professor at Sorbonne.During this time, until 1830, Cauchy´s mathematical activity was incredible.Because of political events Cauchy gave up his professorship and emigratedto Switzerland and Italy for some years. He returned to Paris in 1838 andregained his position at the Academy.Cauchy´s many contributions to the theory of light and to mechanicshave been obscured by the success of his work in analysis [8]. His crowningachievement in analysis was unquestionably his theory of complex functions,a branch of mathematics that commanded his attention from 1814 until hisdeath. Few mathematicians concerned themselves with Cauchy´s theorybefore the late 1840s. Almost all the progress in this area until that timewas due to Cauchy [1].Cauchy´s first significant paper in the theory of functions was “Mémoiresur les intégrales définies”, presented for the Academy of Sciences in 1814.Subsequently his studies on integration expanded in his lessons at the ÉcolePolytechnique, and culminated in his pamphlet “Mémoire sur les intégralesdéfinies, prises entre des limites imaginaires” from 1825. Its object was thedefinite integral whose limits of integration were complex numbers.4.2 The Cauchy Integral TheoremAt the beginning of his pamphlet “Mémoire sur les intégrales définies, prisesentre des limites imaginaires”, Cauchy presents the definition of integrals9

etween complex limits. He defines∫ X+iYf(z)dz (1)x 0 +iy 0as the limit (or one of the limits) of sums of the formS = [(x 1 − x 0 ) + (y 1 − y 0 )i]f(x 0 + iy 0 )+[(x 2 − x 1 ) + (y 2 − y 1 )i]f(x 1 + iy 1 )+ . . . +[(X − x n−1 ) + (Y − y n−1 )i]f(x n−1 + iy n−1 ),where the sequences(x 0 , x 1 , . . . , x n−1 , X),(2)(y 0 , y 1 , . . . , y n−1 , Y ),are both monotonic (either increasing or decreasing) and the differencesx k − x k−1 ,y k − y k−1converge to zero as n increases indefinitely.One can note at this stage that Cauchy is not constructing the integralalong a definite path, but merely has successive sequences of points runningfrom x 0 + iy 0 to X + iY ; presumably this is one reason why he uses thephrase ’one of the limits’ [7]. The requirement that the sequences should bemonotonic seems to have been carried over almost automatically from hisdefinition for functions of a real variable [7].Cauchy then suggests a procedure for constructing such sequences. Hewritesx = ϕ(t), y = ψ(t)where ϕ(t) and ψ(t) are functions that are monotonic and continuous fort 0 ≤ t ≤ T satisfying the conditionsϕ(t 0 ) = x 0 , ψ(t 0 ) = y 0 ,ϕ(T ) = X, ψ(T ) = Y.Observe, that by choosing ϕ and ψ he has constrained himself to one class ofsequences (i.e. what today will be called a path) and hence only one limit.10

After that, Cauchy indicates the integral (1) by A+iB and substitutes for xand y the functions of t that were defined above. Thus, he gets the integralof the formwhich, by settingcan be writtenA + iB =∫ TA + iB =t 0[ϕ ′ (t) + iψ ′ (t)]f[ϕ(t) + iψ(t)]dtϕ ′ (t) = x ′ , ψ ′ (t) = y ′ ,∫ Tt 0(x ′ + iy ′ )f(x + iy)dt.At this stage Cauchy states his integral theorem as follows:“We now suppose that the function f(x + iy) remains finiteand continuous whenever x remains between the limits x 0 andX. In this particular case we can easily prove that the value ofthe integral (1), that is to say, the imaginary expression A + iB,is independent of the nature of the functions x = ϕ(t) and y =ψ(t).”Thus, we can see that Cauchy has reduced an integral between complexlimits to an integral of a complex-valued function between real limits. Theusual formulation can easily be obtained if we observe that the functionsx = ϕ(t) and y = ψ(t) can be thought of as the parametrical equationsof a curve in the complex plane joining the points (x 0 + iy 0 ) and X + iY.But, as mentioned above, Cauchy didn’t have a geometrical interpretationof complex numbers at this time.The hypotheses for f(z) - that it is finite and continuous - is howeverinsufficient. In fact, Cauchy tacitly assumes and implicitly uses both theexistence and the continuity of f ′ (z). This is due to Cauchy’s conviction thata continuous function is always differentiable and that its derivative can bediscontinuous only at points where the function itself is discontinuous [4].Later Cauchy considers this proof to be not rigorous enough. Instead heuses the techniques of calculus of variations. Given a curveCauchy considers a second curve(x(t), y(t)) (3)(x(t) + u(t), y(t) + v(t)), (4)11

where u(t) and v(t) vanish when t = t 0 , T and takes an infinitesimal steptowards it;(x(t) + εu(t), y(t) + εv(t)). (5)Now he wants an expression for the difference between (3) and (5). As theincremental integrals can be seen as functions of ε he considers all but thefirst term in an expansion in a series of powers and finds that the coefficientof ε disappears as seen below. The coefficient of ε is∫ Tt 0[(u + iv)(x ′ + iy ′ ) + (u ′ + iv ′ )f(x + iy)]dt.Then, by integrating by parts, Cauchy shows that the first term of theintegrand is equal to the second term taken negative.From this he concludes that if we give x and y successive increments offirst order whose sum is a finite non-zero increment, then the correspondingincrement of the integral must be of the second order and therefore vanish[7].5 Malmsten´s Article: “Om definita integralermellan imaginära gränsor”In his article “Om definita integraler mellan imaginära gränsor”, Malmstenbegins with a summary of Cauchy’s proof of the integral theorem (wherethe limits of integration are complex numbers). Then Malmsten finds someinsufficiencies in that proof and says that he wants to prove it analogously toCauchy’s proof of the integral theorem for definite integrals taken betweenreal limits. Cauchy’s motivation when constructing this proof was “it seemedto me necessary to demonstrate in general the existence of the integrals orprimitive functions before making known their various properties.” And thisclosely matches Malmsten’s approach, while he first proves the existence ofthe integral and then its various properties. To prove the existence of theintegral it is of course necessary to show path-independence.This section begins with a survey of the insufficiencies in Cauchy’s proofaccording to Malmsten. After that, an exposition of Malmsten’s proof willbe presented as well as comments.12

5.1 The Insufficiencies in Cauchy’s Proof According to MalmstenAt the beginning of his article, Malmsten is questioning some parts ofthe proof of Cauchy’s integral theorem. The first insufficiency is concernsCauchy’s substitution of x and y. Malmsten writes“Onekligen förefaller det också något oegentligt, att, på sammagång man bör fasthålla x och y : s fullkomliga oberoende afhvarandra, supponera dem vara functioner af en och samma variabelt, nämligenx = ϕ(t), y = ψ(t).Härigenom, äfven om man sedermera lyckas bevisa att värdet påintegralen∫ X+iYx 0 +iy 0f(z)dzär af functionsformerna ϕ och ψ helt och hållet oberoende, kommerdock icke sjelfva bevisningen att framträda med den styrkaoch tydlighet, som vore behöfligt.”Malmsten criticizes Cauchy’s method of considering only all sequences followinga given parameterization, i.e. the two functions ϕ and ψ. Although,Malmsten acknowledges that the resulting integral won’t depend on thesefunctions, he doesn’t think that this method of proof is rigorous enough.Instead he considers all sequences (i.e. all sequences of all possible paths)of the Riemann sum S n (see below) in the limit at once.Then, Malmsten explains the way Cauchy uses the techniques of calculusof variations to prove the theorem. However, Malmsten points out someinsufficiencies in this part as well. He writes“...det kan icke förnekas, att äfven här beviset icke framstårså klart och bindande, som det vore att önska. Först och främstförutsätter vid detsamma variationskalkylen såsom känd, hvilketgör att framställningen af läran om definita integraler mellanimaginära gränsor svårligen kan ske på det ställe och i det sammanhang,som för densamma vore mest passande.”Malmsten continues13

x = X, y = Y ;and letsx 1 , x 2 , . . . , x n−1be values betweenThen Malmsten constructs the sumy 1 , y 2 , . . . , y n−1x 0 and X resp y 0 and Y.S n = (x 1 − x 0 ) · p(x 0 , y 0 ) + (y 1 − y 0 ) · q(x 0 , y 0 )+(x 2 − x 1 ) · p(x 1 , y 1 ) + (y 2 − y 1 ) · q(x 1 , y 1 )+ . . . +With the condition+(X − x n−1 ) · p(x n−1 , y n−1 ) + (Y − y n−1 ) · q(x n−1 , y n−1 ).∂p(x, y)dyMalmsten claims that as n → ∞ and the differencesdecrease indefinitely,=x 1 − x 0 , x 2 − x 1 , . . . , X − x n−1 ,∂q(x, y), (7)dxy 1 − y 0 , y 2 − y 1 , . . . , Y − y n−1(8)lim S n =∫ Xx 0p(x, Y )dx +∫ Yy 0q(x 0 , y)dy,=∫ Xx 0p(x, y 0 )dx +∫ Yy 0q(X, y)dy,i.e lim S n is independent of n as well as of the choice of the values in (8).In short, the last expression says that the limit of any sequence, followingthe conditions described above, is equal to the integrations along two specificpaths, i.e. from one corner of a rectangle to the opposite via the contours(clockwise respectively counter clockwise).Malmstens approach is to treat the terms of the sum S n by replacingthem with approximate integrals and error terms. This new sum has the15

property of being two telescoping sums which in the limit becomes the clockwiseintegration above. The counter clockwise integral is reached via a similarapproach.In more detail, the sum can be written asS n =r=n ∑r=1[(x r − x r−1 )p(x r−1 , y r−1 ) + (y r − y r−1 )q(x r−1 , y r−1 )] ,and each term can be rewritten as∫ xrx 0(x r − x r−1 )p(x r−1 , y r−1 ) + (y r − y r−1 )q(x r−1 , y r−1 ) =∫ xr−1∫ yrp(x, y r )dx− p(x, y r−1 )dx+ q(x 0 , y)dy+(x r −x r−1. )λ r +(y r −y r−1 )σ r ,x 0y r−1where λ r and σ r are error terms approaching 0 in the limit. To prove thisequality Malmsten makes several calculations with Riemann-sums, makesuse of the mean value-theorem for integrals and uses expression (7) in differentialform. He does use the fact that the functions are continuous, althoughhe doesn’t mention (or realize) it. For instance he writes“Men nu är tillika(y r − y r−1 )[q(x 0 , y r−1 ) + γ r ] =∫ yry r−1q(x 0 , y)dy + (y r − y r−1 )σ ′ r(der σ ′ r convergerar mot 0 på samma gång som differenserna(17)), hvilket insatt...”This equality can be achieved by applying the mean value-theorem for integralson the right hand side integral and using continuity for the functionq when substituting a value c (from the mean-value theorem for integrals)between y r−1 and y r with y r−1 . Consequently this gives rise to an error termthat is added to σ ′ r producing a new error term γ r which converges to 0 inthe limit. This is only one of several passages where ’common knowledge’theorems together with the conditions of the proof is used.Although Malmsten had come up with the condition (7), he didn’t useit in Greens formula. Probably he didn’t know about it, even though it waspublished by W. Thompson (Lord Kelvin) in Crelle’s Journal in 1850 [2].In fact, George Green (1793-1841) had published it, at his own expense, asearly as 1828 [2]. But the fact that Green was a physicist together with alacking communication may have been some of the reasons why the theoremremained unknown for more than 20 years.16

In the next part Malmsten uses the Theorem II twice in a proof of asimilar theorem for complex valued functions. I.e. he shows that the sumS n = (z 1 − z 0 )f(z 0 ) + (z 2 − z 1 )f(z 1 ) + . . . + (Z − z n−1 )f(z n−1 ),where z = x + iy and(f synectic) as n → ∞ and the differencesf(z) = ϕ(x, y) + iψ(x, y), (9)z 1 − z 0 , z 2 − z 1 , . . . , Z − z n−1i.e. the differencesx 1 − x 0 , x 2 − x 1 , . . . , X − x n−1 ,y 1 − y 0 , y 2 − y 1 , . . . , Y − y n−1decrease indefinitely, is independent of n as well as of the valuesz 1 , z 2 , z 3 , . . . , z n−1 .To prove this Malmsten constructs the sumS n = P n + iQ nwhereP n = (x 1 − x 0 ) · ϕ(x 0 , y 0 ) − (y 1 − y 0 ) · ψ(x 0 , y 0 )+(x 2 − x 1 ) · ϕ(x 1 , y 1 ) − (y 2 − y 1 ) · ψ(x 1 , y 1 )+ . . .+(X − x n−1 ) · ϕ(x n−1 , y n−1 ) − (Y − y n−1 ) · ψ(x n−1 , y n−1 )andQ n = (x 1 − x 0 ) · ψ(x 0 , y 0 ) + (y 1 − y 0 ) · ϕ(x 0 , y 0 )+(x 2 − x 1 ) · ψ(x 1 , y 1 ) + (y 2 − y 1 ) · ϕ(x 1 , y 1 )+ . . .+(X − x n−1 ) · ψ(x n−1 , y n−1 ) + (Y − y n−1 ) · ϕ(x n−1 , y n−1 )17

and applies Theorem II.It is notable that Malmsten calls the function (9) synectic, which isanother expression for analytic. This term was sometimes used by Cauchy.Malmsten’s defines a synectic function as to be continuous, monodromic, andmonogenic. By reading Malmsten’s article one can understand the meaningof the expression monogenic. Malmsten writes“...hvadan således∂φ(x, y)dx=∂Ψ(x, y),dy∂φ(x, y)dy= −∂Ψ(x, y),dxhvilka båda ekvationer innehålla det fullständiga kriteriet att F (z)är monogen inom samma gränsor som f(z).”Thus, the expression monogenic seems to be the same as satisfying theCauchy-Riemann equations. However, the expression monodromic can’t beunderstood by reading Malmsten’s article but it seems to have been a relativelyfrequently used term at this time. In Étude des fonctions d´une variableimaginaire (1856), where Charles Briot (1817-1882) and Jean-ClaudeBouquet (1819-1885) give the first systematic exposition of the theory offunctions of a complex variable as it had been set out by Cauchy, the definitionof the term monodromic is“...si la fonction u prend la même valeur au même point,quel que soit le chemin suivi pour y arriver, sans sortir de laportion du plan considérée, M. Cauchy dit que la fonction estmonodrome dans cette portion du plan.” 1An analogous interpretation of a monodromic function is “a (single-valued)function defined in a finite portion S of the complex plane where the functionalways takes the same value at one and the same point P in S, whateverpath z may take within S to arrive at P [2].It is notable that Briot and Bouquet made no use of the expressionsynectic as Malmsten, as well as Cauchy, did. Instead they said1 Briot, C. Bouquet, J. (1856). Etude des fonctions d´une variable imaginaire. Journalde l´école impériale polytechnique, Tome XXI, p. 85-131, Paris.18

“Le rayon du cercle de convergence étant infini, ces trois fonctionssont finies, continues, monodromes et monogènes, danstoute l’étendue du plan.”Thus, Briot and Bouqet did not have a generic term of what we today callan analytic function. Even though the enunciations of the theorems anddemonstrations in Briot’s and Bouget’s volume are sometimes incompleteand unsatisfactory, their work played a decisive role in spreading Cauchy’spoint of view and his results to European mathematicians [2]. Actually,Malmsten mentions Briot’s and Bouqet’s exposition in his article.5.3 The Various Properties of the IntegralIn the next section of the article, Malmsten wants to show the various propertiesof the integral (1), analogous to what Cauchy had shown for definiteintegrals taken between real limits. Malmsten writes“Sedan vi nu i föregående theorem visat, att limes för denbildade summan, dvs S n endast är beroende af functionsformenf och af gränsvärdena x 0 , y 0 och X, Y, eller, hvilket är detsammaafz 0 = x 0 + iy 0 och Z = X + iY ;kalla vi - analogt med hvad som skett för definita integraler mellanreella gränsor - nämde limes för definita integralen af f(z)mellan gränsorna z 0 och Z, och betecknalim S n =∫ Zz 0f(z)dz =∫ X+iYx 0 +iy 0f(z)dz. ′′From this definition Malmsten shows that the following properties holds;∫ z0z 0f(z)dz = 0, (10)∫ Zz 0f(z)dz =∫ z0z 1(n∑ ∫ zkk=1f(z)dz = −z k−1f(z)dz∫ z1), (11)z 0f(z)dz. (12)It is notable that Malmsten doesn’t say anything about the property thatthe integral depends linearly on the integrand. Maybe this is due to the fact19

that Cauchy didn’t mention this property either in his treatment of definiteintegrals between real limits.Now Malmsten calls∫ zF (z) = f(z)dz (13)z 0and shows that F (z) is synectic in the same domain as f(z). This is analogousto what Cauchy showed for definite integrals between real limits,namely that F (z) is differentiable in the same domain as f(z).The last property to show isHere Malmsten uses expression (13) and putsF ′ (z) = f(z). (14)F (z + δ) − F (z) =∫ z+δzf(z)dzand then uses an earlier corollary (about weighted mean values) to getF (z + δ) − F (z) = δf(z + α) · θ · e pi ,where α is a ’mean quantity’ to 0 and δ (and therefore converges to 0 at thesame time as δ) and θ · e pi converges to 1 at the same time as δ convergesto 0. From this formula he getsF (z + δ) − f(z)lim = F ′ (z) = f(z).δHence, by proving the existence of the integral (1), and then showingthat the properties (10)-(14) holds, Malmsten has now completed his proofof the integral theorem (for integrals where the limits of integration arecomplex numbers) analogous to how Cauchy proved the integral theoremfor integrals between real limits.6 Final RemarksMalmsten left Uppsala at the end of the 1850s to get into politics (seeabove). His successor Herman Daug (1828-1888), whose mathematical researchmostly concerned differential geometry, didn’t have much influence[3]. Instead, in the 1870s, the mathematical education and research in Uppsalacame to be characterized by Göran Dillner (1832-1906), who had agreat interest in Cauchy’s works of analytic functions. But Dillner’s lacking20

knowledge of analysis resulted in a relatively unprofessional way to do research[3]. However, along with Mittag-Leffler, who came under influence ofKarl Weierstrass (1815-1896), the theory of analytic functions got clearedup. In fact, it was in Germany, with Bernhard Riemann (1826-1866) andWeierstrass, the theory of complex functions followed an autonomous pathof development that went far beyond what had been set out by Cauchy [2].Thus, the intuitive apprehension of the analytic functions that had characterizedthe nineteenth century now became rigorous.Malmsten wrote his proof of the integral theorem some years before thebreakthrough of Weierstrass’ strict analysis and therefore his statement ofthe proof doesn’t reach present-day standards. Although, it seems it keptup to the standards of his time. However, the merit of Malmsten is to havesorted out condition (7), even if he didn’t use it in Green’s formula. Theproof also shows Malmsten’s ability to treat inequalities as well as Riemannsums. Furthermore, it is said that Mittag-Leffler, who at the beginningof his career had been one of Malmsten’s students, was complaining thatMalmsten’s, as well as his own, proof didn’t end up in “Enzyklopädie derMathematischen Wissenschaften” [3]. Maybe this was due to the fact thatMalmsten’s successor Daug didn’t keep up the research in the theory ofanalytic functions.References[1] Belhoste, B. (1991). Augustin-Louis Cauchy. A Biography. New York:Springer-Verlag.[2] Bottazzini, U. (1986). The Higher Calculus: A History of Real and ComplexAnalysis from Euler to Weierstrass. New York: Springer-Verlag.[3] Gårding, L. (1994). Matematik och matematiker. Matematiken i Sverigeföre 1950. Lund: Lund University Press.[4] Kolmogorov, A.N., Yushkevich, A.P. (1996). Mathematics of the 19thCentury. Basel: Birkhäuser Verlag.[5] Malmsten, C.J. (1865). Om definita integraler mellan imaginära gränsor.Kungliga Svenska Vetenskaps-akademiens Handlingar. Bandet 6. N:o 3.[6] Sjöberg, B. (1998). Från Euklides till Hilbert: historien om matematikensutveckling under tvåtusen år. Åbo: Åbo Akademis förlag.21

[7] Smithes, F. (1997). Cauchy and the Creation of Complex Function Theory.Cambridge: Cambridge University Press.[8] Struik, D. (1987). A Concise History of Mathematics. New York: DoverPublications, Inc.[9] Svenskt Biografiskt Lexikon, Band 25. (1987). Stockholm: NorstedtsTryckeri.22