Complex Numbers. 2 where x represents a root of Equation 1. Note that the ± sign tells us that quadratic equations will have

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1 Complex Numbers I spte of Calv s dscomfture, magar umbers (a subset of the set of complex umbers) exst ad are valuable mathematcs, egeerg, ad scece. I fact, certa felds, such as electrcal egeerg, aeroautcal egeerg ad quatum mechacs, progress has bee crtcall depedet o complex umbers ad ther behavor. I the cotext of mathematcal bolog, a umber of models we ll be workg wth ca eld solutos that volve complex umbers, so we eed to refresh our memores about complex umbers ad how to work wth them. The Quadratc Formula ad Complex Numbers Asde from allowg us to solve dffcult problems such as 9 +??, probabl the most frequet stuato whch most of us have ecoutered complex umbers has bee whe fdg the roots of quadratc equatos of the form f ( x) Ax + Bx + C 0 Equato 1 You ma recall from our hgh-school algebra course that the roots of Equato 1 are the values of x for whch the equato s exactl equal to zero, ad that we ca easl solve for the roots of Equato 1 b meas of the quadratc formula, whch usuall takes the form B ± x B AC A where x represets a root of Equato 1. Note that the ± sg tells us that quadratc equatos wll have B AC two roots that dffer value b a quatt equal to the value of (whch ma equal zero). A Let s work wth Equato 1 ad set A C. Ths leads to the followg expresso for the quadratc formula whch we wll ow solve for calculatos. B ± B 16 x B K. The table o the ext page gves the results of these

2 B B ± B 16 x B AC 9 0., 0 1, 1 0. ± 0. ± ± ± ± ± ± ± , , We thus see that a quadratc equato has two roots (whch ma, depedg o the values of A, B, ad C, be equal to each other). However, the roots for B,, K, probabl look a lttle strage, volvg that 1. We re about to eter the strage ad fabulous world of complex umbers Imagar ad Complex Numbers We frst focus o the etr for B 0, where the result s ± 1. Of course, there s o real umber whose square s 1, so the result s referred to as a magar umber. B coveto, 1 s desgated b the letter. All magar umbers ma the be represeted as b, where b s a real umber. Our quadratc equato calculatos for B 0 thus become ±. But, what f we let B,, K,, B 0? Here the results are a bt more complcated, volvg a combato of real ad magar umbers. Such umbers are referred to as complex umbers, ad are usuall represeted as a + b, where a ad b are real umbers that ma take o a value betwee ad +. A few pots to ote here: 1. If b 0, ou have the real umber a.. If a 0, the umber b s sad to be pure magar, or, more smpl, magar.. If both a, b 0, the the umber s sad to be complex. I ths case o a s referred to as the real part of the complex umber, ad s represeted as Re(x). o b s termed the magar part of the complex umber, ad s represeted b Im(x).. I the cotext of complex umbers, the, the roots of a quadratc equato ma be: o real, o pure magar, or o complex, whch case the occur as the complex cojugates, a + b ad a b.. Fall, pots #1 ad # show that the real ad magar umbers are subsets of the complex umbers, suggestg that the basc mathematcal operatos of addto,

3 subtracto, multplcato, dvso, expoetato, etc. ca be appled to magar ad complex umbers. Ths turs out to be the case, albet some modfcatos are ecessar. Addto ad Subtracto of Complex Numbers Addto ad subtracto of complex umbers are completel trasparet, but serve to llustrate a mportat techque that wll prove useful to us later o. The formula for carrg out addto of complex umbers s smpl ( a + b) + ( c + d) ( a + c) + ( b + d ) where akd are real umbers. The correspodg formula for subtracto of complex umbers s ( a + b) ( c + d) ( a c) + ( b d ) Note what was doe each: the real ad magar parts of the two complex umbers were grouped, after whch summato was carred out separatel wth each group, eldg aother complex umber wth real part equal to ( a ± c) ad magar part equal to ( b ± d ). Also ote that f b d 0, we re just addg or subtractg real umbers. Multplcato ad Dvso of Complex Numbers Multplcato of complex umbers s straghtforward, ad should look famlar to ou from our algebra das: ( a + b)( c + d) ac + ad + bc + bd ( ac + bd ) + ( ad + bc) ( ac bd ) + ( ad + bc) wth the bd comg from the fact that 1. Note that we aga grouped lke terms (real ad magar) to obta the fal result. Complex dvso s smlarl straghtforward. Gve the complex dvso problem ( a + b) ( c + d) we frst ote that we f we multpl both the umerator ad deomator b c d : ( a + b) ( c + d) ( ac bd + bc ad) ( c d ) we ca get rd of that the deomator. (what term do we appl to the combato of c + d ad?) Note the value of the orgal quotet s uchaged sce ( c d c d ) 1.We the group terms to obta ac + bd + ( bc ad ) c + d ( ) f c + d 0. We ca the mmedatel derve the fal result b separatg the real ad magar terms to eld the fal result: ( a b) ( c + d) ( c + d ) ( bc ad ) + ac + bd + ( c d ) +

4 Lke complex addto ad subtracto, complex multplcato ad dvso eld complex umbers of the stadard form, a real part ad a magar part cosstg of a real umber multpled b. Also ote that f b d 0 the above problems reduce to multplcato or dvso of real umbers. I do t expect ou to commt to memor the precedg materal o complex arthmetc. I do, however, wat ou to keep md the techques of groupg terms ad multplg ( a + b ) b ( c + d) to clea up the deomator (ths s frequetl referred to textbooks as ratoalzg the deomator, although that s ot strctl speakg correct usage of the term). Both are useful algebrac trcks that wll come to pla a umber of tmes durg lecture ad lab sessos ext semester. Expoets Ivolvg Complex Numbers We ow arrve at the reaso for our fora to the world of complex umbers. Next semester, ou wll ecouter a umber of smple models that ca eld complex solutos of the form: t ( ) ( a± b) () t f e Equato where e s the base of atural logarthms (.188 ) ad a ± b s a complex cojugate par. How ( a±b)t o earth ca we geerate somethg meagful out of e, a umber rased to a power that volves complex umbers? That seems md-breakgl absurd the extreme. Well, 18 Leohard Euler, oe of hstor s pre-emet mathematcas, showed us the wa. Frst, we take advatage of the fact that, as wth real umbers, a expoetal term volvg a complex sum ca be decomposed to the product of two expoetal terms. For example, f we start wth the a + b member of the cojugate par, we obta: ( a+b) t at bt e e e Equato at That looks promsg, sce a s a real umber ad we kow how to deal wth e. However, there s bt stll that pesk e term remag to be dealt wth, so t mght ot seem as though we ve gaed athg at all. But, we the recall that powers of e ca be represeted b a fte seres: x x x x x x e L+, 1!!!! x 0! ( ) ( ) where! 1 L 1 (.e., factoral). If we substtute bt for x Equato ad recall that 1, 1, 6 1, etc., we ca accomplsh the followg: e bt 1+ bt + +! 6 ( bt) ( bt) ( bt) ( bt) ( bt) ( bt) ( bt) ( bt)! L +!! 6!! 6 ( bt) ( bt) ( bt) ( bt) ( bt) ( ) 6 6 bt 1+ bt !!!! 6!! 6 ( bt) ( bt) ( bt) ( bt) ( bt) 1 + bt + +!!!! 6!! +L.! +L

5 Next, takg a cue from complex arthmetc, we group real ad magar terms to obta: e bt 1 6 ( bt) ( bt) ( bt) ( bt) ( bt) ( bt) L!! 6! L bt!!! 6 ( bt) ( bt) ( bt) ( bt) ( bt) ( bt) 1 + L + bt + L!! 6!!!! Equato whch ma ot seem lke much of a ga, s ce t s stll a com plex umber. Fortuatel ad here s x the cool part Euler recogzed that, lke e, cos x ad s x also have fte seres equvalets: 6 x x x cos x 1 + L!! 6! ad x ( 1) 0! ( ) x x x s x x + L!!! x 1 1 ( 1) 1 ( 1)! Equato a Equato b Take together, Equato a ad b allow us to rewrte Equato as the sum of se ad cose terms: 6 ( bt) ( bt) ( bt) ( bt) ( bt) ( bt) x Ths extremel mportat result s smpl a modfcato of Euler s formula ( e cos x + s x ), the most remarkable formula mathematcs (R. Fema, 19), wth bt substtuted for x. If we had stead started our dervato wth the a b cojugate, we would have arrved at e bt cosbt s bt meag the two solutos for Equato are ad e bt L L bt + +!! 6!! 1!! 1 ( a+ b) t at () t e e ( cosbt + s bt) ( a+ b) t at () t e e ( cosbt s bt) The ext step, whch we wo t detal, combes those two solutos ad leads to the ultmate soluto: cos bt cos bt + s bt at () t e ( c bt c s bt) s bt 1 cos + Equato 6 where a ad b are the real compoets of the complex expoet Equato, ad c1 ad c are costats whose values are determed b the tal codtos specfed the model. The ed result s that a complex expoetal fucto (Equato ) has bee coverted to a purel real-valued fucto that s easl terpreted. Thak ou, Professor Euler!

6 What I Wat You to Take From the Precedg The precedg ma seem a bt much, but I beleve t s mportat to expose ou to the uderlg mathematc s of topcs we ll be studg. What I wat ou to remember s a lttle less tese: Equatos volvg a complex expoetal ca be represeted as real se ad c ose fuctos. Because se ad cose fuctos are perodc, such equatos w ll exhbt perodc (cclc) behavor. Sce a umber of the models we ll be workg wth ths course ca, uder realstc codtos, eld solutos of complex expoetal form, those models wll exhbt perodc behavor. We wll delve extesvel to perodc fuctos ad the ramfcatos of Equato 6 durg the upcomg semester, because models that lead to solutos of the form represeted b Equato have mportat applcatos a wde varet of bologcal models. For ow, we ll cotet ourselves wth a look at graphs of Equato 6 for c c 1, b 6, ad a -0.1, 0, or 0.1: 1 a 0 a 0.1 a Tme The mportat features to ote ths fgure are that () the solutos oscllate, ad () the sg of a determes whether the oscllatos ampltude decreases, remas costat, or creases wth tme. Perodc Fuctos As a asde, t s worth otg that the term perodc has a precse defto mathematcs. A fucto f x s sad to be perodc f ad ol f there exsts some terval, L, such that ad ( ) ( x + L ) f ( x ) f f ( x ) f ( x ), 1,,, K + L where f ( x) represets the th-order dervatve of f ( x). I others words, order for a fucto to be trul perodc, both the value of the fucto ad of all dervatves of the fucto must have the same value at begg of the terval ad at the ed. We ca llustrate ths wth a couple examples. Frst, cosder the se fucto: Agle ( radas )

7 Note that a gve value of the se fucto repeats a umber of tmes over the terval from 8 to +8, but that the value ad the dervatves of the fucto repeat themselves ol at tervals of π ( 6. 8). The se fucto s therefore termed π perodc. Now, tr to determe the perod of the followg fucto: Ca t be determed from the data gve to ou? Agle ( radas )