math117 lecturenotes5

math117 lecturenotes5 - 1.13 Complex NUmbers Their...

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Unformatted text preview: § 1.13 _ Complex NUmbers . Their Arithmehfic and Their ‘ 13.2 Baptzsenl-ahon. (men ) 154 - 195). o Moi-Nation : Read. 13‘159-50 o; MEM , More MoHval-ion laden i—xa—j‘g 5 whore j=m ism imaam on‘d: Peak, "T ‘lmakgnibnarj Kin Ma Has and ’Pkds. Paid: 0 Re (:2) If“ (a) L = F o fifafihical Eepresenbhim (Argand. Diagram) 1 3 (Imagfinwxg axis) 3 “““““ .° 1" =><+3v A Complex no. cam also be, onugkt 01c cos Gun, ordered Lu} (:93) ,sbnd‘mg {or m Car’res‘tah Cooths 01C a, 190%}, P hike Home o Arikkmt‘c 0.} COMP!“ numbers : Given al=xs+jfh )%2:X2+j.'12) (1") Eqflg mew” Hunt 21:22. 05‘? "F‘xz 3% 13:52.) (it) AMHon u .. 2322 =(_X.+x,_)+j(1gl+“gz); @ii) MpRcaHon .. u 31%; = = <X\X2‘U.Uz)+j (“31+ X230) (RV) Division n H E‘El- : xiii-53' = (X.+j3‘2.(X1-3flz? 2. 2934131 Gym) (xz‘ng) :: (X. +5 3‘ )(X1 —j Biz. 1 My: = -_- (X.xz+yp1,)+j (mp-ml) --'::-- -—1- x1+3f x1453: A Wad-ice. “wee. {Webenbies by atom? Mu. exam 5w M W Asfiawmwt. 0 POICW Rzipresembhm . ~ ‘J 1% orolu to represmt q, )poC/wb m~ “We. flaw. rune m4. "mo W W) but HM. when 3 ____ 335813) W W WW Wfis'Tk" {FWD Mose COMM one; am. HM. {ouowiw: (a) M comm; 3*?» {'he Pow: {S fePTeMQd by 093) ’ wkgre 7: Cs “'3‘ abcdssq GM 3 Ha. orcb‘mwte. Caxtesiam CooraL‘s ($02. {tau/w) 393009) 12:” ([3) Palm Coordimi'es 5 Le. Hm. Poi/wt ”.379 M Lxfi l}; reprecmfied 103 0,9) , wlwu. 1" 0% A {5 Ha diat'amce {mm ‘Hae Pole O Pde ‘poku’ “'X‘IS Md, 6 is Hae Mfile Debunked no posik‘i'oc {1mm HAL Polar cud/3 A «Mi rnovmg wmdoclcwlse. {Sea yawn) Rah/r Cooi‘d.’ s I PM: HM.» fwo catapks on man. on ’cop of each oner. C329. 93W) Then we. We (5) { x =vcc?s(e) 3=r5m(e) duct 50 . (e) 2=x+jj :1~(w5(e)+j we»), mm is caQQeoL {’he aw momma“, a; We complex (is. ”—— Wwbelf 2- % Rem/k . 5%.(5') Lg edged CL Ws—form’cw became LE diam w; fie 394; Hay. fantasia/u word‘s (>93) {mm a. {2440le 9;, Hum bola/r cooxoUs (139) of Hie {900% P. NMJ: aJooch Hue am mmm2flmkio,camwe an}: Hm. P016111, avoid): (139) {300w amuledfie cg, (Xflj)? The MSW is Yes, —Fof we can. Solve (5) for T amok [email protected] isCGdeM Hue GwveuSe hmqmmisw : . Fast Squaw. Md, SW m 5‘5 +em)%el£m3 1_ 7. 1. ‘ { in???) -—-—> x‘+3"= “(WWGHWGJ ) = ‘3'; => *= ”1+? "3 =T $‘M.(9) =[2-,\ . Second diviie (8‘) W laj RM“, ) %€qu3 i = V‘SEMG) ' fame — 9 ' - " 3 7° ran(9) ’ a “—7? ’ é G’WGE)’ The awake e cw we pom Yepmsenbakw :Avgm ‘ 0,; 2- is CCLQQBOQ, "HA6 mau/MM” o4 % ,deho’ced '93 Ag ‘ I New ‘rhak 41-352) is the Principal an manhwtvlt/k ms’: 69. ”restricted. In pewttwficw) 3m Em,“ Hm reshcoum Ls -15 imam <1; . m §i.14 — Euler's Ramada , (HEM)’l7l—5) ‘14.: . Om complex We»; W W Whom) u; is haul-Mid {7; conga!“ Wfiex Wles M {motions as: a complex Me As um Ouzt, a”. a; m Caflcuflws mbe ex£wdeolto ’Humz W «gobs. Fm name) we aJZreawlj Lemow {few “Loot about Hui. Ted Exponekkia/Q. M 67‘. N60.) Hum, @KPoMCaQ can. aflso 66 dag/RM {or complex vankablze Lw Hm usual), W 2 2—9 Q/XPQQ’) and a): cam, be shown M M5 {at WOW: H2 (propwhies of Me «real exponential , wak am 6%'\ 6%: =6a,+az ) cqu 50 m. . IV! [740 a“. alga); Euler dLSCOWEd. a, W 'to GKPfBSS WM exponmlciafls 43w W3 9{ $1M, anal, @5th 01; mmtale) am We MomOfWMSfT/Mpwtand: .formuQa/aiwam. 9.} MwEkMics .‘ 55 MEHORlEE AND (1444) Q” = 6%ij SW9) 26mm FOREVER ' Thar? am. sewmfl mugs ’9: {Wave —H~.Ls ¥oqu)bub Maj mauve MWS we. Mme m4: 43ch leamd.<we skcwl cow». gq‘f’l‘ +43 “W; {90% ) F0» ‘Hruz Momma}: ) we slam be W and, mukipkg M ’cw-o amplex mid/r5 ,m 019% ‘141 Q. - Q/ 2 2' £m(el)+jSW(9.)]-[Cm(92)+j94491)]) —_.- [magma — sweowea} j [0%)st . SMBQMBZ = 009(9‘+91)+j SIM/(9362‘) ) )1) :81. (91"92} mm is WOW m gm Prevents mesa by M W exp) “LO/wig €x-6y= 8x”. 30) a1: Hm. vest, [ea/at) Euler’s {om-uh (D mam 7c“ +ke opwfipu elf magmatic». 0 5PM. g MP0?” WETQWWHJK 0-f- a. CO’MHBX WLer Cap. be film m m: of. Hm imaaanut! anfiqflj .For (144.2) a = Tcos(9)+j awe) ._. WWW-Pi $9M») a mi? 1'9 2 =°°3(9) +jSVI¢(9), -‘e 2,1 =m(9)—jcv'n(e)) / \\ Swmin kem bag“? subtrac’cing {new 193—53,... (1.1.4.3) 009(9) =2 ~2L (816+ 240) I Eula/(Is {chL Cam be med. 4:0 deg/fie mm $L¢mol~£ons. Repm 9 [75 +3” W -j>< (X #349.) ‘m «er boxeoL formula, and grab Hm fouows‘m%g _x aux) LEE 3 = 2 = cos(5x)+j SONG") Z” _._ Q1. (33x) = cosux) —J 59“ G") // \\ cosC—jx)=an(5x) Smk (-jx) =46“ij summi n Subhaol-ima, Wm 5:2 M m 53 mm . 7‘ -X ' \ ¥ 7C -—K (1.14.4) Cocax) =Ji(e +2 )) max) _ ”2348 -3 Ms grim/N3 med/wing "to Hut $9342 cud cosu‘wz. o; cw. {Mai’nam‘j M113. Fwdchmore ) Hm. Combinq’cims —li(€x+ 6.x) and icy—Ex) 00am. {{equmflg ,aud. W been aim special nwms)nam.&‘., (1.145) [email protected])=_pl:<€x+5x) 5 SMCK)=Jz—(?x“ef_x) ? kgperbolic Cosine SMPQJLBJCC SWIM (56% HEN ) P. 1.3”- ) {or {he T6080“ {or +ke$e names ) W) He 6‘1?“ (L144) above ch. CLQS'O be mete/n cu: (144.5) cosOx) :coskoc) ,‘ Show) =j 5mm) . I Powwrc 0; complex WWW: -—- De Mo£we‘s Theorw : Wand: 3"”. {or was In . Us‘m% Eulev’s {Lormulq )we Wmecuatdlj {what ‘ “‘- ‘ u . (1414.6) 2“ = (1‘ 619) = 1”“. @3719 = r“ <69’)(’n9)+J $M(n9)>. oThefe axe {ale/«1:3 0(- exwple; 'm MEN )Hp,173_g4) M exmdses 1w Hm assign want - ...
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