Section 5.6 Complex Numbers

Transcription

1 Secto Secto 5.6 Purpose of Secto To troduce the feld (, +, ) C of complex umbers ad ther Cartesa ad polar represetatos the complex plae. We also show how Euler s theorem coects the complex expoetal to the real trgoometrc fuctos. A Itroductory Tale Try to remember back whe you were the secod grade ad Mss Thomas asked you to fd the umber x that made x = 4, ok maybe ot those exact words. You, of course, told her rght away x =, but the the ext year thrd grade Mss Aderso asked for the umber that made x + 3 = 0 ad you were stumped. Of course Mss Aderso used ths problem to troduce you to egatve umbers. But the, just whe you were gettg comfortable wth egatve umbers, Mss Haso fourth grade asked for the soluto of 5x = 13, whch aturally you sad dd t exst, but oce aga you were forced to elarge your thkg about umbers ad clude ratoal umbers to your umerc repertore. By ths tme you were a juor hgh school ad asked to solve x = 0, whch after aother embarrassg terlude upo aoucg there was t ay, you were troduced to rratoal umbers ad the real umber system. It was about the that you were startg to feel your teachers were playg some kd of swtch ad bat scam, makg up ew umbers o the fly smply to fd solutos to problems they could t solve. Ad the fal crucher came whe you were hgh school ad asked to solve x + 1 = 0. By ths tme you were startg to get comfortable wth the real umbers so there was o questo that ths equato had o soluto; everyoe kew t was mpossble for the square of a umber plus oe to be zero. It just ca t happe. But to your amazemet, your teacher pulled aother rabbt out of the hat ad served up two aswers called magary umbers x = ±, where = 1, whch seemed at the tme rdculous sce all your past teachers had taught that t was mpossble to take the square root of a egatve umber. So you protested ad told your teacher somethg lke that s a lot of, But over tme you started to accept those magary umbers ad eve start to call them by a less cotroversal ame complex umbers. Now, by the tme you got to college you were watg for some professor troduce hyper-complex umbers order to solve some hyper-complex equato lke x + 13x x + x 5 = 0 However, your professor told you ot to worry, a youg mad about your age had prove that ay polyomal equato

2 Secto 5.6 a x a x a x a = 0 whose coeffcets a0, a1,..., a are real or complex umbers, always has complex solutos 1, othg more exotc. Hece, your lfetme ordeal of beg troduced to ew ad larger umber systems every tme you tured aroud had reached a clmax. If you thk ths story s rather far-fetched, t s t. It s, fact the hstory of umbers ad how humakd has adapted ts terpretato of ther meag to ft ts eeds. Ad f you thk humakd dd t kck ad scream wth the troducto of each ew umber system, thk aga. It was ot too may hudred years ago that eve the greatest mathematcas dd t beleve complex umbers. Oh yes, ad that youg lad who showed that every polyomal equato wth teger coeffcets always has complex roots, that was Karl Frederck Gauss ( ), arguably the greatest mathematca who ever lved, who proved the fudametal theorem of algebra 1798 at the rpe old age of 1. Marg Note: I 1545 the Itala mathematca Grolamo Cardao the celebrated Ars Maga solved the smultaeous equatos x + y = 10, xy = 40 gettg the soluto x = , y = Cardao dd ot gve ay terpretato of the square root of a egatve umber, although he dd say that f they obey the usual rules of algebra, the the soluto could be verfed. Although hstorcally complex umber were troduced to solve polyomal equatos, today they are crucal the foudatos of may areas of mathematcs, cludg harmoc aalyss, ordary ad partal dfferetal equatos, aalytc umber theory, aalytc fucto theory, as well as may areas of egeerg ad scece, cludg theoretcal physcs where aalytc fucto theory costtutes much of the foudato of quatum mechacs. Someoe oce argued that real umbers are more atural tha complex umbers sce real umbers measure thgs we ca all see ad feel, lke a perso s heght or weght, whereas o oe ca physcally experece complex umbers. The perso who makes such a clam just does t kow where to look. Every egeer, physcst, ad studet of dfferetal equatos kows t s complex umbers whch allows for the descrpto of oscllatory moto, moto so fudametal that our world would be a dfferet place wthout t. The ext tme you hear members of a orchestra tug ther strumets to the stadard A above Mddle C, you are seeg (or hearg) the 1 The real umbers of course are just specal types of complex umbers.

3 Secto complex umber 440, the complex umber whch descrbes 440 oscllatos per secod. So what are complex umbers? We defe the complex umbers C as umbers of the form x + y, where = 1 (.e. = 1) ad x, y are real umbers, x beg called the real part, deoted x Re( z) magary 3 part, deoted y Im ( z) so o. Also Re( 3), Im ( 4 ) = ad y the =. Examples are 3 +, 4, 3,, 1 ad + = =. Addto ad multplcato of two complex umbers x 1 + y 1, x + y are defed by Addto: ( x1 + y1 ) + ( x + y ) = ( x1 + x ) + ( y1 + y ) Multplcato: ( x + y )( x + y ) = ( x x y y ) + ( x y + x y ) where wth these operatos, the complex umbers satsfes the axoms of a algebrac feld, complete wth addtve detty (the real umber 0) ad multplcatve detty (the real umber 1). The complex umbers do ot form a ordered feld, but they do have a very mportat property the real umbers do ot, such as cotag solutos of equatos ot cotaed the real umber system. For example the equato z + 1 = 0. Algebrasts would say that the complex umbers are algebracally closed feld whereas the real umbers are ot. Geometry of We ca gve thaks to o-mathematcas Casper Wessel ad Jea Robert Argad for ther sght represetg complex umbers as pots the plae, whch we call the complex plae, where a complex umber x + y s plotted as the pot ( x, y ). See Fgure 1. I the complex plae, the horzotal axs ( x -axs) s called the real axs, where the real umber 1 s 1,0, ad the vertcal axs ( y -axs), called the represeted by the pot ( ) magary axs, where the ut complex umber s represeted by the pot 0,1. Keep md the complex plae s ot the same as the Cartesa ( ) plae. I the Cartesa plae pots are smply that, pots. I the complex plae pots are represetatos of complex umbers that we ca multply ad dvde. You may ot thk of them as umbers yet but after we lear to add, subtract, multply ad dvde them, you wll. 3 The word magary s ufortuate sce there s othg really magary about complex umbers. The umber was coed the 1500s whe may people dd t thk of complex umbers as umbers. Ufortuately, the ame as stuck.

4 Secto Complex Plae Represetato of x + y ( x, y) Fgure 1 Marg Note: The + x + y s ulke ay + you have see before. It s ot addto the tradtoal sese, but a placeholder whch separates the real ad pure magary parts of the complex umber; you do t actually add x + y together to get somethg else. The absolute value (or modulus) of a complex umber z = x + y s defed as the oegatve real umber z x + y whch s the legth of the le segmet from 0 to z the complex plae. See Fgure. The cojugate of a complex umber z = x + y s defed to be z = x y, whch geometrcally s the reflecto of z through the real axs. The absolute value of a complex umber ca be wrtte terms of ts cojugate by z = zz. Modulus ad Complex Cojugate of a Complex Number Fgure

5 Secto Hstorcal Note: I 1799 Norwega surveyor Casper Wessel publshed hs sole mathematcal paper where he ht upo the dea of represetg complex umbers as pots the plae, but hs result wet uotced. Later 1813 Jea-Robert Argad, a Parsa bookkeeper, happeed to be the rght place at the rght tme ad also ht upo the dea of terpretg complex umbers geometrcally. For ther cotrbuto ther ames wll forever be recorded the hstory of mathematcs.. Polar Coordates Recall that a pot ( x, y ) the Cartesa plae ca be wrtte polar coordates ( r, θ ) where the relatoshp betwee the varables s x = r cos θ, y = r sθ ad so ay complex umber z = x + y ca also be expressed polar coordates as z = x + y = r cosθ + r sθ = r cosθ + sθ (1) ( ) where r = z = x + y s the absolute value of z, ad θ s the argumet of z, wrtte θ = arg ( z), whch measures the agle betwee the postve real axs ad the le segmet from 0 to z. See Fgure 3. Polar Form of a Complex Number Fgure 3 Sce the argumet θ ca wrap aroud the org several tmes, ether clockwse or couterclockwse, the prcple argumet of a complex umber s π, π. Thus, the complex the uque argumet that les the terval ( ] umber has argumet π /, 1 has argumet π, ad (or sometmes we say 3 π / ). has argumet π /

6 Secto Complex Expoetal ad Euler s Theorem 4 x Ay studet of calculus kows the expoetal fucto e bears lttle x relatoshp to the trgoometrc fuctos s x, cos x. Whereas e grows wthout boud as x gets large, the trgoometrc fuctos oscllate betwee plus ad mus 1. I oe of the most mportat results mathematcs, Swss mathematca Leoard Euler showed 1748 that although real expoetal fuctos may be urelated to trgoometrc fuctos, complex expoetals ad trgoometrc fuctos have a tmate relatoshp. Euler does ths by replacg the θ the Taylor seres expaso of e θ wth the complex umber θ, thus defg a ew fucto e θ, called the complex expoetal. Euler s showed ths complex expoetal has a terestg ad extremely useful relatoshp wth the trgoometrc fuctos. Euler s Theorem For ay real umber θ we have Euler s Equato 5. θ e = cosθ + sθ () e θ where s the result of replacg θ wth θ the Taylor seres expaso of e θ. Replacg θ by θ gves the recprocal θ e = cosθ sθ. Proof: Taylor seres expasos of all real values θ, are e θ, s θ, cosθ about θ = 0, whch coverge for θ θ θ sθ = θ + 3! 5! 7! 4 6 θ θ θ cosθ = 1 + +! 4! 6! 3 4 θ θ θ e θ = 1+ θ ! 3! 4! If we replace θ by θ e θ ad use the facts = 1, 3 =, = 1, =,..., we arrve at the complex expoetal There are two ways to troduced the complex expoetal; oe s to defe the expoetal terms of real ses ad coses, the show ths ew amal.e. the complex expoetal, has all the propertes we expect of a expoetal fucto, such as a b a b e e e + e a =, ( ) b ab = e ad so o. Aother way to troduce the complex expoetal s by meas of Euler s theorem, whch shows how the doma of fuctos of a real varable ca be exteded to the complex plae. The problem wth ths approach, however, s that t requres more kowledge of complex fte seres that we have avalable our bref troducto to complex varables. Nevertheless, we wll proceed formally by presetg a proof of Euler s theorem, where several (true) facts about fte seres are take from grated.. 5 The se ad cose fucto are measured radas ths formula. 6 The rearragemet of terms s allowed sce the Taylor seres are absolutely coverget.

7 Secto ( ) ( ) ( ) θ θ θ θ e = 1+ θ ! 3! 4! θ θ θ θ θ θ = θ +! 4! 6! 3! 5! 7! = cosθ + sθ Note: The value of Euler s theorem les the fact t allows us to work wth expoetals ad all ther woderful mapulatve propertes rather tha wth trgoometrc fuctos ad ther less tha desrable mapulatve propertes. Usg Euler s theorem, we ca prove e θ of a expoetal fucto, such as has all the propertes of oe expects θ 1 θ θ θ θ θ, ( 1+ ) ( ) e e = e e = e usg the trgoometrc dettes of the se ad cose fuctos. Also from Euler s equato we have the detty Example 1 (Complex Expoetals) a) x+ 3 y x e = e ( cos3y + s 3y) b) ( 3+ ) t 3t e = e ( cos t + s t ) x c) e = cos x s x Complex Varables Polar Form ( cos s ) z x+ y x y x e e e e e y y = = = + (3) Usg Euler s equato we ca wrte a complex umber z = x + y polar form ( cosθ sθ ) z x y r re θ = + = + = (4) whch makes the vsual aspect of complex umbers much easer. Note that e θ = + = + = cosθ sθ cos θ s θ 1 from whch we coclude the complex expoetal e θ s a complex umber 7 wth argumet θ lyg o the ut crcle the complex plae. As θ goes 7 We get so used to represetg complex umbers as pots the plae that we use them terchageably.

8 Secto from 0 to π the umber z = e θ goes the couterclockwse drecto aroud the crcle 8 startg from the real umber 1. For example 9 π / π 3 π / π / π e =, e = 1, e = e =, e = 1. See Fgure 4. The Complex Expoetal e θ Fgure 4 Marg Note: Euler s Equato cotas a wealth of formato. If we wrte π / e =, the rase both sdes to the th power, we get (rememberg = 1) π / that e =. I other words Example ( Polar Form) There are always two ways to wrte a complex umber. 1+ = cos π / 4 + s π / 4 = = cos π / + s π / = e π e π a) ( ) ( ) ( / 4) b) ( ) ( ) ( / ) 1+ = 5 cos π / 3 + s π / 3 = 5 e π 1 = cos π + s π = e π c) ( ) ( ) ( /3) d) ( ) ( ) Marg Note: The equato e π = 1 Example d), whch ca be rewrtte e π + 1 = 0, s a amazg equato, called Euler s detty ad cotas 8 We really should say θ goes from π to + π sce we have adopted the egatve agle coveto, but people are sloppy about ths so we smply say 0 to π. 9 Note the coveto of sometmes placg the " " frot of the costats the expoet ad sometmes at the ed. Also the argumet of s ofte represeted terchageably by ether 3 / π /. π or

9 Secto (arguably) the fve most mportat umbers all of mathematcs; 0,1,, π, ad e. Basc Arthmetc of Complex umbers are lke real umbers, they ca be added, subtracted, multpled, ad dvded ad have terestg geometrc terpretatos the complex plae. Addto: z + z = ( x + y ) + ( x + y ) = ( x + x ) + ( y + y ) Two complex umbers are added by addg the real ad complex parts of the two umbers. I the complex plae, the sum of two complex umbers correspods to the pot lyg o the dagoal of a parallelogram whose sdes are the two complex umbers. Subtracto: z z = ( x + y ) ( x + y ) = ( x x ) + ( y y ) Subtracto s aalogous to addto but real ad complex parts are subtracted.

10 Secto Multplcato: ( θ 1 ( 1 ) )( θ ) + Multplcato z z = re r e = r r e θ θ or x + y x + y = x x y y + x y + x y ( )( ) ( ) ( ) The product of two complex umbers s best terpreted usg polar coordates. I the complex plae, the product of two complex umbers s a complex umber whose magtude s the product of the magtudes of the two umbers, ad whose argumet s the sum of the argumets of the two umbers. Dvso: θ z re r e z r e r 1 = 1 1 θ = 1 ( θ θ ) or x1 + y 1 x y x1 + y 1 x1 x + y1 y x y1 x1 y = = + x + y x y x + y x + y x + y Dvso of two complex umbers s best terpreted usg polar coordates. I the complex plae, the quotet of two complex umbers s a complex umber whose magtude s the quotet of the magtudes of the two umbers, ad whose argumet s the dfferece of the argumets of the two umbers. If the quotet s wrtte Cartesa form, dvso s accomplshed by the process of ratoalzg the deomator ator, where oe multples both umerator ad deomator by the cojugate of the deomator ad the collectg real ad complex parts. 1

11 Secto Example 3 (Complex Arthmetc Lesso #1) = a) ( )( ) b) e 3e = 6e π π / 3 π / = = + = c) 6e π 3e e θ = e = cos θ s θ z r r π d) π / = e) ( ) Roots ad Powers of a Complex Number A problem you have heard sce chldhood s, ca you fd a umber x that satsfes x = a? I other words what s the th root of the umber a. You probably kow there are two square roots of a postve umber, lke ± are the two square roots of 4. But f you say s the cube roots of 8 you would be wrog; you are oe-thrd rght sce you mssed the other two cube roots. Ad what about the cube roots of 1; what umbers you cube to get 1? Ths leads us to problem, for a gve complex umber a 0 (the real umber zero s also a complex umber) ad a teger, fd the umbers satsfyg z = a? Wrtg z ad the rght-had sde where φ s the argumet of equato yelds reduces to e θ r ( ) z = re = r e = a e θ θ φ a = a e θ polar form, we have a. Takg the absolute value of each sde of ths = a from whch we coclude φ = e from whch we coclude θ φ th root of the complex umber a 1/ φ / 1/ = a e φ s 1/ ( φ ) ( φ ) z = a e = a cos / + s /. However, ths s ot the oly root sce satsfes z k r = a. The equato ow = or θ = φ /. Hece, a 1/ ( φ+ π k )/ ), = 0,1,,..., 1 also z a e k = a, whch ca be see by drect computato

12 Secto ( ) ( 1/ ( ) ) ) / φ + π φ + π φ π φ a e = a e = a e e = a e = a. These results ca be summarzed as follows. Roots of a Complex Number: For each ozero complex umber 10 a = a e φ C there are dstct th roots of a, whch are the pots o a crcle of radus 1/ a. ( φ + π ) 1/ 1/ k / a = a e 1/ φ + π k φ + π k = a cos + s, k = 0,1,,..., 1 Hece, to fd the th root of a complex umber a, frst fd the prcpal root, 1/ whose absolute value s arg a /. The other a ad whose argumet s ( ) 1 roots are equally spaced pots o the same crcle (radus prcple value, the agle betwee them beg π /. 1/ a ) as the Example 4 (Roots) 0 + π k 0 + π k 4 = cos + s = cos k + s k, k = 0,1 whch gves ad respectvely. a) ( π ) ( π ) 10 Whe we say complex umber, we of course mea the real umbers as well.

13 Secto b) π / + π k π / + π k π π = 1 cos + s = cos + kπ + s + kπ, k = 0,1 4 4 whch gves the two roots o the ut crcle 1, /3 1/3 π / + π k π / + π k c) = 1 cos + s, k = 0,1, 3 3 whch gves the three cube roots o the ut crcle π π k π π k ak = cos + + s +, k = 0,1, show Fgure 5. Three cube roots of Fgure 5 Fractoal Powers of a Complex Number p / q The fractoal powers z are foud the same way the roots of a complex umber were foud ad are / p / q p q pφ + π k pφ + π k a = a cos + s, k = 0,1,..., q 1 q q A useful techque to fdg the values of the rase each root to the p th power. p / q z s to frst fd the q roots z 1/ q,

14 Secto Complex Fuctos I calculus you performed aalytcal calculatos, such as dfferetato ad tegrato, o a host of real-valued fuctos f : R R of oe varable, such as x f ( x) = s x, g( x) = e ad so o. We ow troduce complex-valued fuctos of a complex varable f : C C, but ulke fuctos lke f ( x) = x where oe plots x o the horzotal axs ad y o the vertcal axs of the Cartesa plae, we have a problem whe t comes to plottg complex fuctos w = f ( z). The problem les the fact that both doma ad codoma are complex plaes, so effect we eed four dmesos to draw a graph of a fucto, a rather dffcult problem a 3-dmesoal world. To overcome ths problem we draw two copes of the complex plae 11, the doma plae called the z-plae ad the codoma plae called the w-plae plae. See Fgure 1. Graphg a Fucto of a Complex Varable Fgure 1 Graphg the Complex Plae Recall from calculus that fuctos f : R R descrbe curves the plae, such as the ut crcle x( t) = cos t, y( t) = s t, 0 t < π. We ca also graph complex fuctos the complex plae as follows. If we rewrte a complex umber such as z = e θ ca be wrtte terms of ts real ad complex parts terpreted as a complex fucto f : R R wthere s a great smlarty betwee what you have studed calculus ad graphg descrbes the ut crcle. where t s a real varable cuves Complex umbers provdes a way to graph may mportat curves. For example the equato z = R descrbes a 11 It s really the same complex plae, t s smply more llustratve to draw two separate copes.

15 Secto crcle of radus R cetered at the org, whereas z z0 = R descrbes a crcle of radus R cetered at z 0.

16 Secto Problems 1. For the followg complex umbers, fd z, θ = arg( z), w + z, wz, w / z ad plot the locatos of w, z, w + z, wz, ad w / z the complex plae. a) w =, z = 1+ b) w =, z = c) w =, z = d) w = 1, 1+ e) w =, z = 3 f) w = 1 +, 1+ g) w = 3+, 3 h) w = 1 +, z = 1+. (Covert to Polar Form) Covert the followg complex umbers to polar form re θ. a) b) 1+ c) d) + 4 e) f) 3 g) -5 h) 1 3. (Covert to Cartesa Form) Covert the followg complex umber to cartesa form x + y. a) b) c) d) e) f) 3 e π / 6 e π e π 5 e π / 4 e π / e π e π g) 3 h) e Marg Note: There used to be a compay that sold T-shrts wth Mathematcas, we re Number wrtte o them. e π

17 Secto Fd the polar form of ( 1 )( 1 ) Fd the Cartesa form of ( 1+ ) Show that the complex cojugate of the sum of two complex umbers s the sum of the cojugates; that s ( w + z) = w + z. 7. Verfy the detty z = zz for z = Fd the real ad magary parts of 3 z a) b) 1/ z z 1 c) z Compute a) b) c) 1+ d) 3 1 e) (de Movre s Formula) Use Euler s theorem to prove de Movre s formula for ay postve teger. ( ) cosθ + sθ = cos θ + s θ 11.. (Prmtve Roots of Uty) The roots of uty are the roots of the equato z = 1. Fd ad plot the roots whe = 1,,3, 4, ad (Fractoal Powers) Fd the followg. a) b) 1 3/ 5/ 4 c) ( 1) 5/ d) ( 1+ ) 3/ 13. (Hmmmmmmmmm) Show

18 Secto Fd 1 =. 3/. Ht: Frst fd 3/..