How Euler Did It. Introduction to Complex Variables

Transcription

1 How Euler Did It by Ed Sadifer Itroductio to Complex Variables May 2007 O Moday, March 20, 777 the Imperial Academy of Scieces of St. Petersburg had oe of its regular meetigs. Except for holidays ad occasioal special meetigs, they met twice a week o Modays ad Fridays, a total of 70 or 80 meetigs per year. This particular meetig was t much differet from the other meetigs they had that year, though it was a little shorter tha most. The miutes from that meetig are show i the photograph below. [SPA] It opeed with a report from the Academy s traslator, a Mr. Jaehrig, icludig his accout of some letters betwee the Dalai Lama ad the Sakya Trii Lama, ad it closed with the readig of a letter of thaks from someoe amed Mosieur Messier. I betwee, oe of the Adjoit Members of the Academy, Nicolas Fuss, submitted o behalf of Leohard Euler two articles, De itegratioibus maxime memorabilibus ex calculo imagiariorum oriudis, (E656, O some most memorable itegratios arisig from the calculus of the imagiaries ad its sequel, E657, Supplemetum ad dissertatioem m d praecedetem circa itegratioem formulae casu, quo poitur = v ( cosϕ + si ϕ, m d Supplemet to the precedig article about the itegratio of the formula by settig = v cosϕ + siϕ. Eleve days later, o March 3, Fuss brought five more articles, icludig oe related to these two, Ulterior disquisitio de formulis itegralibus imagiaries, (E694, Later article o imagiary itegral formulas. The Academy s most famous member, Leohard Euler, blid for more tha five years, would tur 70 years old i just a moth ad seldom atteded the regular meetigs ay more. Istead, he stayed at his home a few blocks from the Neva, the river through St. Petersburg, ad his assistats wet there to work with him. His assistats would do the actual writig, ad they would sometimes work out the details of the calculatios, but the articles were almost always published uder Euler s ame. I 777, Euler ad his assistats set more tha 50 articles to the Academy. They would be preseted to the Academy, that is, the mauscript was haded over to the Secretary of the Academy. Euler ad compay wrote articles too fast for the Academy s publishers, but those articles that were to be published without delay also had to be read aloud at a meetig of the Academy. These two articles were t published util the 789 issue of the Academy s joural, ad that issue was t actually prited

2 util 793, sixtee years after it was writte, ad these articles apparetly escaped beig read before the Academy. So, what are these most memorable itegratios of which Euler writes? The mai poit of the articles is to show that calculus with complex umbers is possible, ad that it works a lot like calculus with real umbers. Miutes of the Petersburg Academy from March 20, 777 Euler begis askig us to cosider a differetial Zd, where Z is a fuctio of what he calls a imagiary quatity. He writes its itegral as Zd= :, where : is Euler s fuctio otatio. We would write f( or (. Now we separate everythig i sight ito real ad imagiary parts. We take = x + y. (Euler ad his studets have ot yet adopted the symbol i to deote. They do that later i 777. Also, Z = M + N ad : = P + Q. Euler is very patiet with us here, ad explais that Zd= dx+ dy M + N ad that the real ad the imagiary parts (he uses those words are Mdx Ndy ad ( Ndx+ Mdy respectively, ad that P = ( Mdx Ndy ad Q= ( Ndx+ Mdy. Euler s patiece lapses for a momet here whe he just tells us, without givig details, that because of the itegrability criteria it follows that M N N M = ad = y x y x. 2

3 I fact, it is easy to derive these formulas. We eed oly kow that the mixed partial derivatives of Z have to be equal, but it is t as easy as the last few steps have bee. He does this step i more detail eleve days later i his third article, E694. These are, of course, the Cauchy-Riema equatios, used to such great effect two or three mathematical geeratios later by Augusti-Louis Cauchy ( ad Berhard Riema ( Now, Euler wats to show that certai calculus facts familiar for ordiary fuctios of real + umbers are also true for complex umbers. He begis with d =. As was customary at the + time, Euler eglects the costat of itegratio uless he eeds it. I, Euler substitutes x y = +, the expads the resultig biomial as ( x y x x y x yy x y + = + + etc. 2 3 Euler ad his studets had oly recetly started writig the biomial coefficiets as k. Sometime over the ext few decades, people started to omit the fractio bar, leavig us with the moder otatio, k. We will use Euler s otatio. Separatig his expaded biomial ito M, its real part, ad N, the imagiary part, he gets ad M x x yy x y x y = + + N x y x y x y = + etc. etc. + He does a similar substitutio ad expasio with the right had side,, ad separates it ito + its real ad imagiary parts, P ad Q, the multiplies by +, to get what he thiks P ad Q should be, if his itegral formula is correct: ad P= x x yy+ x y x y etc Q= x y x y + x y etc. O the other had, he kows that P has to be give by P = ( Mdx Ndy expressios for M ad N gives the rather formidable expressio:. Substitutig his 3

4 dx x x y + x y x y + etc P = dy x y x y + x y etc. 3 5 Euler sets out to show that these two expressios for P are equal. He says that he will do this by itegratig by parts, but he really meas that he will itegrate parts of the actual value of P give i the itegral, ad the show that they equal the correspodig parts of the expressio ( + P. For the term that has x + i its itegral, he gets + x xdx =, + ad that agrees with the correspodig part of ( + P. For the terms that have x i their itegral, he gets 2 2 x ydx x ydy 2 ad shows that these itegrate to give their correspodig part of ( + P as well. He cotiues, pairig 4 4 x ydx 4 with 3 3 x ydy, ad the stops, sayig that the patter is clear. He also omits the 3 details i showig that the expressio that follows from Q= ( Ndx+ Mdy give as ( + Q. agrees with the expressio Euler agrees that that was hard work, ad offers us a easier way. It is slightly less geeral, sice the biomial series expasios ca be made to work eve if is ot a iteger ad his easier way oly works whe is a positive iteger. But it is cosiderably shorter ad a good deal more elegat. From x ad y, Euler creates two ew variables, v= xx+ yy ad a agle ϕ chose so that, as Euler writes it, tag. ϕ = y x. This makes Their differetials are x = vcosϕ ad y = vsi ϕ. dx = dvcosϕ vdϕsiϕ ad dy = dvsiϕ + vdϕcos ϕ. Now Euler ca use demoivre s formula to get x y v ( cosϕ siϕ + = +. This has real ad imagiary parts M ad N respectively equal to ad A similar calculatio shows that, for M = v cosϕ N = v si ϕ. + d =, we would have to have + 4

5 v P = + ( + cos ϕ + ad v Q = + ( + si + ϕ. O the other had, we kow that P = ( Mdx Ndy ad Q = ( Ndx + Mdy few steps i his substitutio. We ll skip some, too, but ot as may as Euler did. For P, we get (( v cosϕ( dvcosϕ vdϕsiϕ ( v siϕ( dvsiϕ vdϕcosϕ P = Mdx Ndy = + Now, after a careful expasio ad applicatio of the trigoometric idetities. Euler skips a ad we get that as promised. cos( + ϕ = cos( ϕ+ ϕ = cosϕcosϕ siϕ si ϕ ( + ϕ = ( ϕ + ϕ si si = siϕcosϕ+ cosϕ siϕ + v v P = cos( + ϕ ad Q = + + ( + si + ϕ, Ideed, Euler s secod solutio took him oly a page of calculatios to fid both P ad Q, whereas his first method had take two ad a half pages to fid oly the first three terms of P. Thigs are t quite as rosy as he would have us believe, though, because he does skip a good umber of easy but paper-cosumig calculatios. We have described oly the first five of the 44 pages of Euler s first paper. The secod paper adds 7 pages, ad the third aother 8. Euler goes o (ad o ad o to apply the same methods to d d itegrate ad get, as we would expect, ta, ad. He goes o to itegrate the more + + m d geeral form, which cotais the last two as special cases. + Gradually, he comes to appreciate the power ad coveiece of the substitutio ( cos si = v ϕ + ϕ, ad devotes the secod paper to that substitutio. The third paper gives Euler s derivatio of the Cauchy-Riema formulas i more detail, ad the attacks some more geeral m d itegrals like. λ a± b Over the ext few moths, Euler expads his use of complex umbers i calculus, ad i a paper dϕcosϕ [E67] preseted to the Academy o May 5, 777, as he is studyig the itegral of, he writes cosϕ that he will be usig imagiary umbers ad that I will use i to deote. His otatio caught o. 5

6 O a techical level, we ve see excitig developmets here. We see Euler discoverig the Cauchy-Riema formulas more tha a decade before Cauchy was eve bor, ad almost 50 years before Riema, ad we ve doe calculus with complex umbers. Somethig has happeed o a philosophical level as well. For most of his life, Euler was cotet to use a priciple that Leibi had called the Priciple of Cotiuatio. This said, roughly, that similar thigs ought to behave similarly. This gave Euler reaso to use the same rules of calculatio with ifiite ad ifiitesimal umbers that he used for fiite umbers ad to treat solid bodies as if they were poit masses. The Priciple of Cotiuatio should have allowed Euler to itegrate complex fuctios just like he itegrated real oes. We ca oly speculate why Euler chose to try to be aalytically rigorous whe writig about complex variables. Perhaps he wrote this paper to explai the use of complex umbers to his studets, especially to Nikolas Fuss. That could also explai why Euler was sometimes very careful about icludig details, so that his studets would uderstad, but at other times he skipped them, to leave gaps for his studets to fill i. O the other had, perhaps he or the people he was writig for did ot agree that complex umbers, or, as he called them, imagiary umbers, are similar to real umbers, so he was ot comfortable applyig the Priciple of Cotiuatio. We wo t eve worry about why Cauchy ad Riema got their ames o those equatios istead of Euler. Refereces: [E656] [E657] [E694] [SPA] Euler, Leohard, De itegratioibus maxime memorabilibus ex calculo imagiariorum oriudis, Nova acta academiae scietiarum Petropolitaae 7 ( , pp Reprited i Opera Omia Series I vol. 9, pp Available olie at EulerArchive.org. m Euler, Leohard, Supplemetum ad dissertatioem praecedetem circa itegratioem formulae d casu, quo = v cosϕ + si ϕ, Nova acata academiae scietiarum Petropolitaae 7 ( , pp poitur Reprited i Opera Omia Series I vol. 9, pp Available olie at EulerArchive.org. Euler, Leohard,Ulterior disquisitio de formulis itegralibus imagiariis, Nova acata academiae scietiarum Petropolitaae 0 ( , pp Reprited i Opera Omia Series I vol. 9, pp Available olie at EulerArchive.org. St. Petersburg Academy, Procés-Verbaux des Séaces de l Académie Impériale des Scieces depuis-sa Fodatio jusq à 803. Vol. III St. Petersburg, 900. Ed Sadifer (SadiferE@wcsu.edu is Professor of Mathematics at Wester Coecticut State Uiversity i Dabury, CT. He is a avid maratho ruer, with 35 Bosto Marathos o his shoes, ad he is Secretary of The Euler Society ( His ew book, The Early Mathematics of Leohard Euler, was published by the MAA i December 2006, as part of the celebratios of Euler s terceteial i The MAA will be publishig a collectio of the How Euler Did It colums durig the summer of How Euler Did It is updated each moth. Copyright 2007 Ed Sadifer 6