THE FUNCTION CONCEPT 405 He includes polynomials,power series,and logarithmic and trigonometric expressions.He also defines a function of several variables.There follows the notion of an algebraic function,in which the operations on the independent variable involve only algebraic operations,which in turn are divided into two classes:the rational,involving only the four usual operations,and the irrational,involving roots.He then introduces the transcendental functions, namely,the trigonometric,the logarithmic,the exponential,variables to irrational powe and some integrals. The principal difference among functions,writes Euler,consists in the combination of variables and constants that compose them.Thus,he adds, the transcendental functions are distinguished from the algebraic function because the former repeat an infinite number of times the combinations of the latter;that is,the transcendental functions could be given by infinite series.Euler and his contemporaries did not regard it as necessary to consider the validity of the expressions obtained by the unending application of the four rational operations Euler distinguished between explicit and implicit functions and between single-valued and multiple-valued functions,the latter being root of higher degree equations in two variables,the coefficients of which are functions of one variable.Here,he says,if a function,such asP,where Pis a one-valued function,has real values for real values of the argument,then most often it can be included among the single-valued functions.From these definitions (which are not free of ontradictions),Euler turns to rational integral functions or polynomials.Such functions with real coefficients,he affirms, can be decomposed into first and second degree factors with real coefficients (see sec.4 and Chap.25,sec.2). By a continuous function,Euler,like Leibniz and the other eighteenth- discontinuity as in y =I/x.6 Other functions were recognized;the curves representing them were drawn. Euler's Introductio was the first work in which the function concept was made primary and used as a basis for ordering the material of the two volumes.Something of the spirit of this book may be gathered from Euler's remarks on the expansion of functions in power series.?He asserts that any function can be so expanded but then states that"if anyone doubts that every function can be so expanded then the doubt will be set aside by actually expanding functions.However in order that the present investigation extend over the widest possible domain,in addition to the positive integral powers 6.In Vol.,Chap.I of his Introductio,Euler intro s“discontinuou uire different analytic expre wdierentdomaingahCiKdnc 7.0pera,(l),8,Chap.4,p.74
406 CALCULUS IN THE EIGHTEENTH CENTURY of z,terms with arbitrary exponents will be admitted.Then it is surely indisputable that every function can be expanded in the form Az+Bz+ Cz+Dz+.in which the exponents B,y,8,.can be any numbers." For Euler the possibility of expanding all functions into series was confirmed by his own experience and the experience of all his contemporarics.And in fact,it was true in those days that all functions given by analytic expressions were developable in serics Though a controversy about the notion of a function did arise in connection with the vibrating-string problem (see Chap.22)and caused Euler to generalize his own notion of what a function was,the concept that dominated the eighteenth century was still the notion of a function given by a single analytic expression,finite or infinite.Thus Lagrange,in his Thdori des fonctions analytiques (1797),defined a function of one or several variables as any expression useful for calculation in which these variables enter in any manner whatsoever.In Lepons sur le calcul des fonctions(1806),he says that functions represent different operations that must be performed on known quantitics to obtain the valucs of unknown quantitics,and that the latter are properly only the last result of the calculation.In other words,a function is a combination of operations. 3.The Technique of Integration and Complex Quantities The basic method for integrating even somewhat complicated algebraic functions and the transcendental functions-the technique introduced by Newton-was to represent the functions as series and integrate term by term. Little by little,the mathematicians developed the techniques of going from one closed form to another. The eighteenth-century use of the integral concept was limited.Newton had utilized the derivative and the antiderivative or indefinite integral, whereas Leibniz had emphasized differentials and the summation of differen- tials.John Bernoulli,presumably following Leibniz,treated the integral as the inverse of the differential,so that if thenyf().That is the Newtonian antiderivative was chosen as the integral,but differentials were used in place of Newton's derivative.According to Bernoulli,the object of the integral calculus was to find from a given relation among differentials of variables,the relation of the variables.Euler emphasized that the derivative is the ratio of the evanescent differentials and said that the integral calculus was concerned with finding the function itself.The summation concept was used by him only for the approximate evaluation of integrals.In fact,all of the eighteenth-century mathematicians treated the integral as the inverse of the derivative or of the differential dy.The existence of an integral was never questioned;it was,of course,found explicitly in most of the applications made in the eighteenth century,so that the question did not occur
INTEGRATION AND COMPLEX QUANTITIES 407 A few instances of the development of the technique of integration are worth noting.To evaluate ∫ James Bernoulli had used the change of variable b9-t2 x=“+2 this converts the integral to the form ∫ which is readily integrable as a logarithm function.John Bernoulli noticed in 1702 and published in the Mimoires of the Academy of Sciences of that year the observation that -++) so that the integration can be performed at once.Thus the method of partial fractions was introduced.This method was also noted independently by Leibniz in the Acta Eruditorum of 1702.10 In correspondence between John Bernoulli and Leibniz,the method was applied to dx Jax?+bx+c However,since the linear factors of ax2+bx +c could be complex,the method of partial fractions led to integrals of the form dx cx+d in which d at least was a complex number.Both Leibniz and John Bernoulli nevertheless integrated by using the logarithm rule and so involved the logarithms of complex numbers.Despite the confusion about complex numbers,neither hesitated to integrate in this manner.Leibniz said the presence of complex numbers did no harm John Bernoulli employed them repeatedly.In a paper published in 170211 he pointed out that,just as adz/(2-z)goes over by means of the 8.Acta Erud,.1699=0pera,2,868-70. 9.0pera,1,393400. 10.Math.Schrifen,5,350-66. 11.Mim.de 1'Acad.des Sci.,Paris,1702,289 ff.=Opera,1,393-400
408 CALCULUS IN THE EIGHTEENTH CENTURY substitution z=b(t-1)/(t+1)into adt/26t,so the differential dz 62+z2 goes over by the substitution z=v-Ib(t-1)/(t+1)to -dt v√-I2bt and the latter is the differential of the logarithm of an imaginary number. Since the original integral also leads to the arc tan function,Bernoulli had thus established a relation between the trigonometric and logarithmic functions. However,these results soon raised lively discussions about the nature of the logarithms of negative and complex numbers.In his article of 171212 and in an exchange of letters with John Bernoulli during the years 1712-13, Leibniz affirmed that the logarithms of negative numbers are nonexistent (he said imaginary),while Bernoulli sought to prove that they must be real Leibniz's argument was that positive logarithms are used for numbers greater than I and negative logarithms for numbers between 0 and 1.Hence there could be no logarithm for negative numbers.Moreover,if there were a logarithm for-1,then the logarithm of v-1 would be half ofit;but surely there was no logarithm forv-I.That Liebniz could argue in this manner after having introduced the logarithms of complex numbers in integration is inexplicable.Bernoulli argued that since (1) 一x then log ()=log *and since log1=0,so is log(-1).Leibniz countered that d(log x)=dx/x holds only for positive x.A second round of corre- spondence and disagreement took place between Euler and John Bernoulli during the years 1727-31.Bernoulli maintained his position,while Euler disagreed with it,though,at the time,he had no consistent position of his own The final clarification of what the logarithm of a complex number is became possible by virtue of related developments that are themselves significant and that led to the relationship between the exponential and the trigonometric functions.In 1714 Roger Cotes (1682-1716)published 13 a theorem on complex numbers,which,in modern notation,states that (2) √-中=loge(cos中+√-1sinφ) 12.Acta Erud.,1712,167-69 =Math.Schriften,5,387-89. 13.Phil.Tra,29,1714,545
INTEGRATION AND COMPLEX QUANTITIES 49 In a letter to John Bernoulli of October 18,1740,Euler stated that y=2 cos x and y=ev-1x+e1*were both solutions of the same differential equation(which he recognized through series solutions),and so must be equal.He published this observation in 1743,14 namely, C085=+ev- sins =-v-T 2V7 In 1748 he rediscovered the result (2)of Cotes,which would also follow from (3). While this development was taking place,Abraham de Moivre (1667- 1754),who left France and settled in London when the Edict of Nantes protecting Huguenots was revoked,obtained,at least implicitly,the formula now named after him.In a note of 1722,which utilizes a result already published in 1707,15 he says that one can obtain a relation between x andt, which represent the versines of two arcs (vers a=1 -cos a)that are in the ratio of I to n,by eliminating z from the two equations I-2zn+z2n =-2z"t and 1-2z+z2=-2zx. In this result the de Moivre formula is implicit,for if one setsx=1-cos and t 1 -cos nd,one can derive (4) (cos中±√-Isinφp)n=cosn吨±V-I sin n For de Moivre n was an integer >0.Actually,he never wrote the last result explicitly;the final formulation is due to Euler1e and was generalized by him to all real n. By 1747 Euler had enough experience with the relationship between exponentials,logarithms,and trigonometric functions to obtain the correct facts about the logarithms of complex numbers.In an article of 1749, entitled"De la controverse entre Mrs.[Messrs]Leibnitz et Bernoulli sur les logarithmes negatifs et imaginaires,"17 Euler disagrees with Leibniz's counterargument that d(log x)=dx/x for positive x only.He says that Leibniz's objection,if correct,shatters the foundation of all analysis,namely, that the rules and operations apply no matter what the nature of the objects to which they are applied.He affirms that d(log x)=dx/x is correct for positive and negative x but adds that Bernoulli forgets that all one can conclude from (1)above is that log(-x)and log x differ by a constant. This constant must belog(-1)because log (-x)=l0g (-1.x)=log (-1)+ log Hence,says Euler,Bernoulli has assumed,in effect,that log ( =0,but this must be proved.Bernoulli had given other arguments,which 14.Miscellanea Berolinensia,7,1743,172-92 =Opera,(1),14,138-55. 15.Phil.Trans.,25,1707,2368-71. 16.Introductio,Chap.8. 17.Hist.dc1'Acad.de Berlin,5,1749,139-79,pub.1751=0pera,(1),17,195-232