The natural exponential function

Transcription

1 The atural expoetial fuctio Attila Máté Brookly College of the City Uiversity of New York December, 205 Cotets The atural expoetial fuctio for real x. Beroulli s iequality The fuctio expx) The idetity expx)exp x) = The idetity expx)expy) = expx+y) The umber e ad the fuctio e x The atural expoetial fuctio for complex x 4 2. Extedig the approach to complex umbers The idetity expx)expy) = expx+y) for complex x ad y De Moivre s formula Euler s formula Commets 6 3. Mootoicity of + x ) for real x Boudedess of + x ) for complex x Covergece of + x ) for complex z Euler s formula: is there a ituitive backgroud? The atural expoetial fuctio for real x. Beroulli s iequality For every real x ad every iteger we have ) +x) +x. Equality here holds oly i case = or x = 0. This is easily proved by iductio o. Ideed, ) is true for =, because the it states that +x +x. Let, ad assume ) is true for this. The we have +x) + = +x) +x) +x)+x) = ++)x+x 2 ++)x.

2 Note that the first iequality here results by multiplyig both sides of equatio ) by +x. This is allowed, sice + x is o-egative multiplyig a iequality by a egative umber reverses the iequality). This completes the proof of ). Note that i the last displayed formula, strict iequality holds istead of the last iequality uless x = 0. This also establishes our commet about equality i )..2 The fuctio expx) We write 2) expx) = lim + x. ) To establish the existece of the limit o the right, we first assume that x 0. By the biomial theorem, we have + x ) = ) x k k k = k! xk k j= j ) To see that the series o the right is coverget, oe may observe that for m,, ad x with 0 x < m < we have 3) k=m k! xm x m k=m ) k m, ad the geometric series o the right is coverget. We ca also coclude from the above equatios that + x ) m k k! xk j ) j= for ay positive m. Keepig m fixed while makig, it follows that 4) lim if + x ) m k! for ay x 0. Sice m ca be ay positive iteger here, it follows that 5) expx) = lim + x ) = k! for x 0. I particular, the limit i 2) exists for x 0. From this equatio it follows that 6) +x expx) for x with 0 x. Thus it is easy to show that expx) 7) lim =. xց0 x This equatio will be established for all x i Subsectio 2.. = x k!. 2

3 .3 The idetity expx) exp x) = For x ad with 0 x < we have + x ) x ) ) = x2 2 x2, where the secod iequality holds i view of Beroulli s iequality ). Makig, the existece of the limit i 2) follows also for egative x, ad we obtai 8) expx) exp x) =. This equatio ad equatio 7) allow us to coclude that expx) 9) lim =. x 0 x.4 The idetity expx)expy) = expx+y) Assume x ad y are o-egative ad > xy. We have + x+y ) + x ) + y ) = + x+y + x+y + x+y ) + xy ) 2 ) / xy ), + x+y + xy 2 ) ) / xy 2 ) where the last iequality follows from Beroulli s iequality ). Makig, we obtai 0) expx+y) = expx)expy) for o-egative x ad y. Usig equatio 8), this equatio ca also be established for all real x ad y. With the aid of this idetity ad equatio 9), we ca easily show that ) d expx) = expx). dx Hece it also follows that the fuctio expx) is cotiuous..5 The umber e ad the fuctio e x It is easy to show that expx) ) r = exprx) for ay real x ad ratioal r. All that is ivolved i showig this is chagig the order of takig a limit ad applyig a cotiuous fuctio amely, raisig to a ratioal power), ad otig that a coverget sequece ad ay of its subsequeces have the same limit. 2 Defiig the umber e as exp), we the have e r = expr) for ay ratioal r. As for irratioal x, there is o reasoable way to defie e x other tha approximatig x with ratioals. Asexpx)iscotiuous, itcabeseethatsuchadefiitioissoud, 3 aditfollowsthate x = expx) for all real x with such a defiitio. 2 Alteratively, oe ca also use 9) to show this equatio. 3 Soudess meas that if r are ratioals such that lim r = x the the limit lim expr ) exists, ad if oe takes aother sequece of ratioals s with lim s = x the lim expr ) = lim exps ). 3

4 2 The atural expoetial fuctio for complex x 2. Extedig the approach to complex umbers The above discussio ca be exteded to complex umbers by usig somewhat more about ifiite series. Oe ca establish equatio 5) for all complex x by poitig out that 3) with x replacig x implies the absolute covergece of the series o the right-had side of 5). The iequality 4) eed to be replaced with lim sup + x ) m k! k=m+ x k k!. The use of limsup here ad of limif i 4) ca be avoided at the price of some mior additioal complicatios. Havig established 5) for all complex x, the existece of the limit i 2) ow follows ad we have expx) x for x <. x Istead of usig 5) to establish this iequality, we may also poit out that it directly follows from Lemma below. This iequality will replace the estimates i 6). While the rest of the discussio could be cotiued with makig use of various properties of power series, we will avoid ay further use of expasio 5) i these otes. I particular, we will show how Euler s formula 5) ca be established without the use of power series. The last iequality ca be used to prove 9) directly. We will show how to establish 0) for all complex x ad y ext i Subsectio 2.2. Thus, equatios 5), 9), 0), ad ) ca be exteded to all complex umbers x ad y. 2.2 The idetity expx)expy) = expx+y) for complex x ad y. To exted 0) to complex x ad y, we ca use the followig Lemma. Let > 0 be a iteger ad z be a complex umber such that z <. The Proof. We have +z) z z. +z) = z +z) k z + z ) k z z ) k z k z z where the peultimate iequality 4 follows from Beroulli s iequality ). As a immediate cosequece, we have 4 The oe before the last. z = z z 4

5 Corollary. For let z be complex umbers such that lim z = 0. The lim +z ) =. Let x ad y be arbitrary complex umbers. The 2) + x ) + y ) = + x+y + xy ) 2 = + x+y ) +z ) for large eough, where z = + x+y + xy )/ 2 + x+y ) ; makig sure that is large eough guaratees that the deomiator here is ot 0. It is easy to see that lim z = 0, ad so we have lim + z ) = by Corollary. Makig, equatio 0) follows. 2.3 De Moivre s formula The complex umber a+bi, where a ad b are real umbers, ca be represeted i the coordiate plae as the poit a,b). I this case, the x-axis is called the real axis, ad the y-axis, the imagiary axis. If this same poit is represeted as the pair ρ,θ) i polar coordiates, where ρ 0 ad θ is real, the ρ is called the absolute value a+bi ad θ, the argumet arga+bi) of a+bi; clearly, the argumetisolydetermieduptoaitegralmultiplemultipleof2π. 5 Thatis, arga+bi) = θ 0 +2kπ for some θ 0, where k ca be arbitrarily chose as ay iteger. If ρ = a+bi ad θ = arga+bi), the, clearly a+bi = ρcosθ +isiθ). The right-had side is called the trigoometric form of the complex umber o the left. Whe multiplyig two complex umbers, their absolute values get multiplied ad their argumets get added. That is, for real ρ, ρ 2, θ, ad θ 2, we have ρ cosθ +isiθ ) ρ 2 cosθ 2 +isiθ 2 ) = ρ ρ 2 cosθ +θ 2 )+isiθ +θ 2 )). This equatio easily follows from the additio formulas for si ad cos. A better approach is to verify it directly, usig coordiate trasformatios, ad the use this equatio to prove the additio formulas for si ad cos. The last equatio implies 3) cosx+isix) = cosx+isix for ay iteger ad ay real x. This is called de Moivre s formula. 2.4 Euler s formula Observe that t sit cost lim = lim = 0. t 0 t t 0 t 5 The word itegral does ot have aythig to do with itegratio; is it the adjectival form of the word iteger which ca be positive, egative, or 0). 5

6 Thus, for ay real x we have lim +i x ) cos x +isi x )) = 0. Hece, for a fixed real x, writig +i x z = cos x +isi x, ad otig that cos x +isi x =, we obtai that lim z = 0. Therefore, by Corollary it follows that 4) lim +z ) =. We have +i x ) = cos x ) +isi x +z ) = cosx+isix)+z ), where the secod equatio follows from de Moivre s formula 3). Makig, we obtai 5) expix) = lim for every real x by 4). This is Euler s formula. 3 Commets 3. Mootoicity of + x ) for real x +i x ) = cosx+isix The above approach ca be modified to avoid ay referece to the series i 5); it it, however, ot clear whether these modificatios result i essetial simplificatio. To prove the existece of the limit i 2) for real x, we may start with otig that the sequece o the right-had side is icreasig if x is real. More precisely, 6) + x ) + x ) + if > max{ x,0}; + equality here holds oly i case x = 0. Ideed, otig that + x ) = +x = x +x, this iequality is equivalet to x +x + x ) x ) ) + ; + +x we used the assumptio +x > 0 to make sure that i obtaiig this iequality, we multiplied the startig iequality by a positive umber. The right-had side here equals ) + x x +x)+) +x, where the iequality holds i view of Beroulli s iequality ); equality holds oly i case x = 0 cf. the commet o equality i Beroulli s iequality). 6

7 3.2 Boudedess of + x ) for complex x The sequece o the right-had side of 2) is bouded for all complex x. Ideed, if x <, the + x ) + x ) / x ) x, where the last iequality holds for holds i view of Beroulli s iequality ). If x, let m > 0 be a iteger such that x /m <. The, give ay iteger, let k be a iteger with km >. Usig the mootoicity of +x/) expressed by 6), ad the usig the last displayed iequality with x /m replacig x, we obtai + x ) + x ) + x ) km = + x /m ) km km k x /m Hece, for real x, the existece of the limit i 2) follows from the boudedess ad the mootoicity of the sequece o the right-had side of that formula. 3.3 Covergece of + x ) for complex z Let z = x+iy be a complex umber, where x ad y are real. Usig equatio 2) with iy replacig y, ad modifyig the defiitio of z accordigly, we ca show that lim + x+iy ) = lim + x lim + ) iy ), where we iterpret this equatio i the sese that if the limits o the right-had side exist, the so does the limit o the left-had side, ad the two sides are equal. As for the first limit o the right-had side, its existece ca be justified by what we said above i this sectio, i.e., that we are takig the limit of a bouded icreasig sequece. The existece of the secod limit ca be justified by otig that whe we established 5) above, we proved the existece of the limit directly, ad we did ot rely o the fact that the existece of the limit i 2) had bee established before. Thus, our origial discussio of the existece of the limit i 2) for real or complex x ca be dispesed with etirely. 3.4 Euler s formula: is there a ituitive backgroud? The usual proof of Euler s formula 5) relies o the Taylor expasios of six, cosx, ad expx) also writte as expx; its Taylor expasio is the series i 5)). This proves that Euler s formula is a cosequece of 2) or, more directly, the series expasio of expx) give i 5)), but it leaves oe completely mystified as to what Euler s formula has to do with what oe ormally cosiders expoetiatio. The proof of Euler s formula give above appears to provide a atural ituitive coectio of Euler s formula with multiplicatio of complex umbers. ) m. 7