2.1. Bilangan Kompleks

Transkripsi

1 II MEMIK UNUK NLISIS SISEM DINMIK uju: Mh mmpu meyuu d meyeleik model memik (perm ked) uu iem (proe) ehigg dp mejelk dimik uu proe Meri:. Bilg Komplek. rormi Lplce: deiii, i-i rormi lplce 3. Peyelei PD deg rormi Lplce: proedur, iverio, peyelei ime dely. Krkeriik Repo Proe: vribel devii, repo oupu, bili 5. Lierii DINPRO / II /.. Bilg Komplek Sebuh bilg diebu komplek jik bilg b idk dp diyk ebgi bilg y (rel); u bilg b dlh khyl (imgier) Bilg Imgier : i Beuk crei : c + i b dim: bgi rel b bgi imgier (..) DINPRO / II /.. Bilg Komplek c + i b Complex Ple I (,b) b r Noi Polr θ Rel xi R r mgiude θ rgume Imgiry xi DINPRO / II / 3

2 .. Bilg Komplek Noi Polr Uuk Meyk Bilg Komplek: mgiude r c + b b b rgume θ rc (...b) mk: r co θ d b r i θ (..3) oi crei cojuge iθ ( co θ + i θ ) r e c r i dim: θ coj. ( co θ i i θ ) e i + ( + i b) ( i b) (...) (..) (..5) DINPRO / II / Operi Bilg Komplek.. Bilg Komplek Perimbgk: c + i b iθ r e d p v + i w Pejumlh & Pegurg: c ± p ( ± v) + i ( b ± w) i β q e (..6) Perkli: cp ( + ib)( v + iw) v + i bw + ibv + iw cp ( v bw) + i( bv + w) iθ i β i ( )( ) ( θ + β r e q e rqe ) Perkli dg cojuge: ( + i b)( i b) + b r (..7) (..8) (..9) DINPRO / II / 5 Operi Bilg Komplek c ( + ib) ( v iw) Pembgi: p ( v + iw) ( v iw) Beuk polr c p v + bw v + w re qe iθ iβ + r i( θ β e ) q ( v + bw) + i( bv w) bv w i v + w.. Bilg Komplek v + w (..) (..) Pgk: c r iθ e (..) kr: c re i( θ kπ )/ r e iθ + dim k, ±, ±,, mpi diperoleh kr (..3) DINPRO / II / 6 BB II Memik Uuk lii Siem Dimik DH - DINPRO -

3 .. Bilg Komplek Cooh Sol..: koveri bilg komplek mejdi polr Bil. komplek: 3 + i b 8 i6 c + i Mgiude (r): 5 b c. rgume (θ): θ 3.97 rd Polr: 97 5e i. 6 b θ 8.63 rd b 5e i.63 θc 3π rd i( 3π / b 5e ) DINPRO / II / 7 Cooh Sol: (lju).. Bilg Komplek Complex Ple 6 I 3 + i c +i R b 8 i 6 DINPRO / II / 8 Cooh Sol: (lju) Perkli: c ( 3 ) + i( 3 ) 7 i bc ( 8 + 6) + i( 8 + 6) i.. Bilg Komplek Beuk polr: i.97 i( 3π / ) i3. 83 c 5e.e 7.7e ( co i i 3.83) i Pembgi: b b ( 3 + i) ( 8 + i6) ( 8 i6) ( 8 + i6) i.97 5e i. 57 i.63 ( ) + i( 8 + 3) Beuk polr:.5e.5( + i) i. 5 e i.5 DINPRO / II / 9 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 3

4 Cooh Sol: (lju).. Bilg Komplek kr: mil 6 6e i i i( + kπ / ) i( kπ / x 6e 6 e e ) dim uuk k k k k kr dri 6 dlh: x e i x e iπ/ ( + i) i x e iπ/ ( i) i x e iπ ( + i) DINPRO / II / Deiii.. rormi Lplce Dlm lii dimik proe, vribel proe d iyl korol dlh ugi wku,. rormi Lplce () dlh: F () L [ () ] () e Dim: F() rormi Lplce dri () Vrible rormi Lplce, ime - (..) DINPRO / II /.. rormi Lplce Jei-Jei Ipu < Fugi hp (ep ucio) u(). L u u e [ ()] () ( ) e FugiPule H L () H [ () ] () <, < e H e H H e ( e ) DINPRO / II / BB II Memik Uuk lii Siem Dimik DH - DINPRO -

5 Jei-Jei Ipu FugiImpule FugiSiu mpliude -.. rormi Lplce Dirc del ucio: δ() <, > δ () L [ δ () ] δ () e i ( ω) Frequecy Period e iω ω π e i iω DINPRO / II / 3 FugiSiu (lju) L e iω e i iω [ i( ω) ] e iω + i ω i [ e ( ) e ( ) ] ( iω ) ( + iω ) e e + i iω + iω + i iω + iω ω + ω.. rormi Lplce iω i + ω DINPRO / II / bel... rormi Lplce Uuk Fugi-Fugi Umum () δ() u() e e F() L [()]! + + ( + ) () e i(ω) co(ω) e i(ω) e co(ω).. rormi Lplce F() L [()]! ( + ) + ω + ω + ω ω ( + ) + ω + ( + ) + ω DINPRO / II / 5 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 5

6 .. rormi Lplce UGS Bukik koveri dri () mejdi F() berdrk bel ormi Lplce Uuk Fugi-Fugi Umum (Lih bel...) DINPRO / II / 6 Si-Si rormi Lplce Lieriy [ ( ) ] L [ ( ) ] F( ).. rormi Lplce L merupk operi lier, hl ii berri, jik dlh ko, mk: L (..) Si diribui: L [ ( ) + b g( ) ] F( ) + bg( ) (..3) Rel Diereiio heorem d L () Pembuki: F d L () () () d () e (..) DINPRO / II / 7 Iegrl pril: u e du e d dv v ( ).. rormi Lplce ( ) d L () [ () e ] ()( e ) [ ( ) ] + () e ( ) ( ) F erbuki F() DINPRO / II / 8 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 6

7 Uuk derivi order : d L ( ) d d ( ) L d L L F ( ) d [ () () ] () () F.. rormi Lplce d d o DINPRO / II / 9.. rormi Lplce Secr umum, uuk derivi: d L () F () () d... (..5) Dlm pegedli proe, kodii wl dlh pd kodii uk. Jdi ime deriviy ol (zero), d vribel dlh devii dri kodii wl, ehigg Lplce derivive dlh: d L ( ) F () (..6) DINPRO / II /.. rormi Lplce Rel Iegrio heorem L () F() Pembukiy m deg cr rel diereiio heorem. (..7) Cob d bukik di Rumh! Rel rlio heorem L D [ ( )] e F( ) D (..8) eori ii berki deg keerlmb wku (ime dely) dlm merepo perubh ipu, d eljuy dikel ebgi ded ime. () D (- D ) DINPRO / II / BB II Memik Uuk lii Siem Dimik DH - DINPRO - 7

8 Pembuki: L[ ( )] ( ) Mil, τ D u D + τ L C: (τ ) uuk τ < < ( D ) erbuki D D.. rormi Lplce e [ ( )] ( ) ( D + τ τ e ) d( + τ ) D e D () τ e D e dτ τ D e D F ( τ ) e dτ ( ) D DINPRO / II /.. rormi Lplce Fil Vlue heorem lim lim F Complex Diereiio heorem d L[ () ] F() d L ( e ( ) ) F( ) lim () ( ) Complex rlio heorem Iiil Vlue heorem () lim F( ) (..9) (..) (..) (..) DINPRO / II / 3.3. Peyelei PD deg L ggp: kodii wl dlh pd ked uk (edy e) d emu vribel diyk dlm erm devii. Proedur Peyelei L. Ubh PD mejdi beuk lplce deg vribel.. Bu hubug r vribel oupu (vribel idk beb/ depede) d vribel ipu. 3. Blik (iver) beuk lplce mejdi beuk wku uuk memperoleh repo oupu. C: dlm iem pegedli proe, PD meujukk hubug r iyl oupu, y(), d iyl ipu, x(). DINPRO / II / BB II Memik Uuk lii Siem Dimik DH - DINPRO - 8

9 Perimbgk: L uuk mig-mig erm: () L d y ( ) dy( ) d y + + y x() diebu vribel ipu (orce ucio) y() diebu vribel oupu (depede vrible),,, d b dlh ko Kodii wl y(), d dy/ L dri PD pgk du: d y() dy( ) L + L + L () y() dy.3. Peyelei PD deg L () b x() [ y() ] bl [ x() ] (.3.) (.3.) DINPRO / II / 5 L uuk mig-mig erm: dy() [ y( ) ] ( ) L [ x( ) ] b X ( ) b L () () L y.3. Peyelei PD deg L Jdi diperoleh: dy ( + + ) () ( + )() y bx () Peyederh (hubug oupu d ipu): erm di dlm kurug diebu FUNGSI RNSFER () X () b + + (.3.3) (.3.) DINPRO / II / 6 () b + + ( + + ) ( r )( r ) r, ±.3. Peyelei PD deg L Keblik dri L Deg Ekpi Pril: Jik ipu berubh ui ugi hp: Pegmbg (ekpi) deomior: kr polyomil kudr: X () dim r d r dlh kr kudr dri: + + (.3.5) (.3.6) (.3.7) DINPRO / II / 7 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 9

10 .3. Peyelei PD deg L Ekpi pril L: () + r r + 3 (.3.8) Uuk kr-kr yg idk berulg, berlku: k lim r k ( r ) ( ) k (.3.9) Berdrk bel L, keblik (iver) dri lplce dlh: y r r () e + e + u( ) 3 DINPRO / II / 8.3. Peyelei PD deg L Uuk kr-kr yg berulg, mily r r, berlku: 3 () + + r r (.3.) Koeiie 3 dihiug eperi ebelumy, d dihiug deg cr: lim y r lim r ( ) Berdrlk bel L, keblik (iver) dri lplce dlh: r r () e + e + u( ) ( r ) ( ) [( r ) ( ) ]! d d 3 (.3.) DINPRO / II / 9.3. Peyelei PD deg L Secr umum, jik r diulg m kli: m () m m r r r Koeiie-koeiie dihiug ebgi beriku: Uuk k,, m, mk Iver lplce dlh y () ( ) ( ) lim r lim m + m ( r ) ( ) ( k )! k r k d d k m ( m )! ( m ) ! m e m [( r ) ( ) ] r (.3.) (.3.3) (.3.) DINPRO / II / 3 BB II Memik Uuk lii Siem Dimik DH - DINPRO -

11 .3. Peyelei PD deg L ime Dely (Ded-ime) Perimbgk ku dim erdp erm ekpoeil D () e (.3.5) Deg () p erm ekpoeil () r r r (.3.6) Iver () meghilk: r r () r e + e +... e y + (.3.7) DINPRO / II / 3 Jdi, Iver () meghilk: () L e ( ).3. Peyelei PD deg L Jdi, deg megguk rel rlio heorem: r () ( D ) r ( D ) r ( D y e + e e ) [ ] y ( ) y D D (.3.8) Jik erdp muli-dely: D D D () () e + ( ) e ( ) e (.3.9) Jdi, deg megguk rel rlio heorem: y () y ( ) + y ( ) y ( ) D D D (.3.) DINPRO / II / 3 Cooh.3. : megi ime dely Dikehui PD beriku:.3. Peyelei PD deg L Deg c(), euk repo oupu c(), jik pd, ipu berubh deg u ui ep: () u( )! Jdi: D d F() e L dri PD d ubiui F() meghilk: C + ( ) dc + c + () F() e () () DINPRO / II / 33 BB II Memik Uuk lii Siem Dimik DH - DINPRO -

12 .3. Peyelei PD deg L mil: () ( ) C C Iver dri C (): () lim e B C ( + ) ( ) lim + ( + ) DINPRO / II / 3 + c () e u().3. Peyelei PD deg L Jdi iver dri C () meghilk (lih bel..): c ()( e ) u pliki rel rlio heorem: () L C ( ) [ ] ( ) ( )[ ( e c u e )] C ui ep u( ) hru diklik deg erm ekpoeil, hl ii meujukk bhw c() uuk <. DINPRO / II / 35.. Krkeriik Repo Proe Beberp pery yg relev erhdp repo:. pkh repo bil? iu repo erjg pd ili ereu.. Jik bil, berp ili uk bru? 3. pkh repoy mooo u beroili?. Jik mooo d bil, berp wku yg diperluk uuk mecpi kodii bil (uk bru)? 5. Jik beroili, berp periode oili d berp wku beroili mpi khiry bil? DINPRO / II / 36 BB II Memik Uuk lii Siem Dimik DH - DINPRO -

13 Vribel Devii () y() y( ) Dim: y() ili vribel ol y() ili vribel pd kodii wl.. Krkeriik Repo Proe. (..) Dri deiii vribel devii, mk vribel devii pd kodii wl ellu ol (): () y() y() Perimbgk PD lier order : () d y( ) d y + + K+ () y m m d x( ) d x( ) bm + bm + + b x() + c m L m. (..) DINPRO / II / 37.. Krkeriik Repo Proe Dim > m, y() oupu, x() ipu, d c ko Pd kodii uk wl, emu ugi derivi wku dlh ol ( ) b x( ) c ehigg: y +. (..3) Per. (..) Per. (..3) : () d ( ) d + + K+ m () m ( ) d X ( ) d X bm + bm b X () m + L +. (..) m Dim: () y() y() d X() x() x() DINPRO / II / 38.. Krkeriik Repo Proe Repo Oupu Uuk meujukk hubug r repo oupu d kr-kr dri deomior ugi rer, mk peyelei L dri per. (..) dlm erm devii: b m + b m m () X () + m + L+ b + L+ Deomior per. (.5) dp dikork mejdi derj beriku: () b m m + b m ( r )( r ) L( r ). (..5) m b + L+ X (). (..6) Dim r, r,, r dlh kr polyomil deomior. Dimpig kor (lih per..6), erdp kor li dri X() yg ergug pd jei ipu (ep, pule, rmp, dll.) DINPRO / II / 39 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 3

14 .. Krkeriik Repo Proe Pegembg dlm rki pril: () + + L+ r r r + erm dri X (). (..7) Keblik lplce per. (.7) meghilk: ( ) e r r r + e + L+ e erm dri X ( ) +. (..8) kr-kr Ny: kr poii : repo ik eirig iky wku IDK SBIL kr egi : meluruh mpi ol SBIL Jik emu kr deomior dri F dlh y: repo moooic (o-ocillory) repo bil jik emu kry egi (lih Gmbr..) DINPRO / II / Gmbr... Repo uuk kr-kr y.. Krkeriik Repo Proe () () k () Sbil, kr y egi kodii uk bru k 5 r k. (..9) (b) idk Sbil, kr y poii DINPRO / II /.. Krkeriik Repo Proe Pg kr Complex Cojuge: r ρ + i ω r ρ i ω dim: ρ bgi rel; ω bgi imgier Pegembg F: ρ iω ρ + iω () + + L ( + )( ρ) i( ) + ( ρ) + ω ( ρ ) + B( ρ ) Cω + ( ρ) + ω ( ρ ) + ω +L ω +L ω. (..) dim: B + d C i ( ) DINPRO / II / BB II Memik Uuk lii Siem Dimik DH - DINPRO -

15 Jdi iver dri per. (..) meghilk (lih bel..): ρ ρ () Be coω + Ce iω + L ρ e [ B coω + C iω] + L Peyederh megguk beuk rigoomeri: ( ω + θ ) iθ coω coθ iω i + ρ ( ) De i( ω + θ ) + L.. Krkeriik Repo Proe meghilk:. (..) dim: D B + C B θ rc C mpliudo wl Phe gle, dlm rdi DINPRO / II / 3.. Krkeriik Repo Proe Berdrk per. (..), diimpulk: Repo beroili Oili mejdi IDK SBIL, jik bilg komplek cojuge mempuyi kr bgi rel poii Perhik erm e ρ : ρ poii mpliudo emki ber deg wku ρ egi mpliudo meluruh Frekuei gelombg iu merupk bgi imgier dri kr, ω dlm rdi/wku. Periode oili: wku yg diperluk uuk meempuh u iklu gelombg. u, wku yg diperluk uuk meikk rgume gelombg iu (ω + θ) eber π rdi. π ω. (..) DINPRO / II /.. Krkeriik Repo Proe Gmbr... Repo uuk kr-kr complex cojuge () () () Sbil, kr y egi (b) idk Sbil, kr y kodii uk bru poii 5. (..3) ρ Decy rio e ρ πρ / ω e. (..) DINPRO / II / 5 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 5

16 Kodii uk Bru.. Krkeriik Repo Proe Kodii uk bru dp dicri deg il vlue heorem umi ipu berubh deg ugi hp dim X() x u() u X() x / ubiui ke per. (..5) bm lim m + b + m m + L+ b + L+ x b x (..5) Krieri Kebil Siem k SBIL jik emu kr deomior dri F dlh NEGIF, yiu: egi uuk kr y d egi uuk bgi rel dri kr complex. Lih Gmbr bidg komplek (Gmbr..3) DINPRO / II / 6 Gmbr..3. Complex Ple I.. Krkeriik Repo Proe SBIL R SBIL DINPRO / II / 7 Megp perlu lierii?.5. Lierii Slh u keuli dlm lii repo dimik uuk proe dlh i keidk-lier proe erebu. Meode rormi Lplce (L) yg elh ki peljri dp meggmbrk dimik iem proe. Sygy, hy iem lier j yg dp dili deg L. D, idk d ekik liy yg dp diguk uuk lii dimik iem o-lier. Lierii diguk uuk medeki repo iem o-lier deg PD lier yg kemudi dp dili deg L Pedek lier erhdp iem o-lier dp dierim (vlid) uuk derh yg dek deg beberp iik dr (be poi) yg dibu. Mk, ki k memilih kodii uk wl ebgi be poi. DINPRO / II / 8 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 6

17 .5. Lierii Beberp ugi o-lier yg umum: Elpi (H), ebgi ugi uhu (): 3 [ () ] H + ( ) + ( ) + ( ) ( ) H dim: H,,,, 3, d dlh ko. Per. oie: ek up (p ) ebgi ugi uhu () [ + C p ] B [ ()] ( e ) 3 + dim:, B, d C dlh ko. (.5.) (.5.) Frki mol up eimbg (y), ebgi ugi rki mol cir (x) [ x() ] αx( ) ( α ) x( ) y (.5.3) + dim: α dlh volili reli, biy diumik ko. DINPRO / II / 9.5. Lierii Lju lir (), ebgi ugi preure drop ( p): [ ()] ( ) p k p dim: k dlh koeii kuduki ko. Lju perpidh p rdii q, ebgi ugi uhu () [ ()] ( ) (.5.) q εσ (.5.5) dim: ε, σ, d dlh ko. Per. rheiu: keergug koe. lju reki (k) erhdp () E [ ()] [ R ( k k e )] (.5.6) dim: α k, E, d R dlh ko. Per. Lju reki (r): ebgi ugi uhu (), d koeri C, C B. r b [ (), c (), c ( ),...] k[ ( ) ] c ( ) c ( )... B dim: k[()] per. (.7.6);, d b dlh ko. B (.5.7) DINPRO / II / 5 Lierii Fugi Su Vribel.5. Lierii Semu ugi dp dikembgk ke dlm dere ylor ekir be poi: d d [ x() ] ( x) + [ x() x] + [ x() x] + L (.5.8) dx x! dx x dim: x dlh be vlue x diekir ugi yg diekpi. Dlm lierii, beuk order du u lebih dri per. (.5.8) dp dibik, ehigg mejdi: d dx [ x() ] ( x) + [ x() x] x (.5.9) Per. (.5.9) dlh ugi dr lierii yg diilurik pd Gmbr.5.. Kre x dlh ko, mk perm diebelh k d m deg dlh lier dlm vribel x() DINPRO / II / 5 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 7

18 .5. Lierii Gmbr.5. Pedek lier dlh ge dri ugi o-lier pd be poi x [ x() ] ( x) Gri ge d dx Fugi o-lier x x x() DINPRO / II / 5 Cooh.5.:Lierii Per. rrheiu Be poi: k ( ) [ ec] Eergi kivi, E kcl/kmol, & R.987 kcl/kmol-k.5. Lierii Perkirk error pd lope dlm reg ± o C di ekir 3 o C Peyelei: pliki Per. (.5.9) ke (.5.6): dk k[ () ] k( ) + [ () ] d Dim: dk d E [ ( R ( ) k e ) ] d d E R E ke k R E ( ) ( ) R DINPRO / II / 53 Slope: dk d o 3 C ( ) 3.37 ( )( ) o C Jdi diperoleh pedek lier: k[ ( ) ] ( ) 9 o C, k Sebgi perbdig: diulik pedek lier beriku: ec Dlm rge 9 3 o C, diperoleh ili cul d lope: 3 o C, k.5. Lierii [ ] o ( ) 7.95ec, dk d.8ec / C o ( ) 39.3ec, dk d.5ec / C k(9 o C) (9 3) 66.3 ec - error 6.6% k(3 o C) (3 3) 33.7 ec - error % DINPRO / II / 5 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 8

19 Lierii Fugi Du Vribel u Lebih Ekpi dere ylor uuk du vribel u lebih: d.5. Lierii [ x (), x (), L ] ( x, x L) + [ x () x ] + [ x () x ] + L, dx dx dim: x k x k ( x, x,l) d x, x,l d (.5.) Cooh.5.: ku ederh lu () egi emp dlh ugi dri pjg (w) d lebr (h): Lierii: [ w(), h( ) ] w( ) h( ) [ w(), h() ] ( w, h ) dlh be vlue dri mig-mig vribel + [ w() w] + h w ( w, h ) h ( w, h ) w, h w, h + h w w + w h h [ ] [ () ( )] ( ) [ ( ) ] ( ) [ () h ] DINPRO / II / 55 Gmbr.5. Error pedek lier dri lu egi emp.5. Lierii w [h() h] error h() h (w,h) wh h [w() w] w w() umi: w m d h m Lu pd be poi: m Icreme: w(). m d h(). m cul. m Lu pedek + (.) + (.). m error... m Lu derh rir (.)(.). m DINPRO / II / Lierii Cooh.5.3: Lierii Per. dei g idel bg ugi ek d uhu Mp p R Fugi dei o-lier ρ[ (), () ] ( ) () Uuk evlui, ki megguk g udr: M ber molekul 9 [kg/kmol] ; IB ek bolu.3 kp uhu bolu [K] ; & R 8.3 kp-m 3 /kmol-k Peyelei: pliki Per. (.5.): [ p(), () ] ρ( p ) + [ p() p] + () ρ, Dim: ρ Mp p dp R ρ p ( ) M () ( ) R p, ρ [ ] DINPRO / II / 57 BB II Memik Uuk lii Siem Dimik DH - DINPRO - 9

20 .5. Lierii ρ d Mp R ( ) Mp () ( ) p, R Jdi pedek ugi dei lier ρ [ p(), () ] + [ p() p] () Secr umerik: ρ Mp R M R Mp R [ ] [ ] [ p(), () ] [ p( ) p]. 393( ) Deg u: ρ [kg/m 3 ], p [kp], [K] DINPRO / II / 58 Lierii Perm Diereil Perimbgk PD Order u deg u ipu beriku: dim: Pd kodii uk wl: Be poi: ( ) ( ), y y( ) x x ( x y) b g, +.5. Lierii dy g[ x(), y() ] + b (.5.) g[x(),y()] dlh ugi o-lier deg ipu x(), oupu y(), d b dlh ko. Per. (.5.) (.5.): () dy [ x(), y() ] g( x, y) (.5.) g (.5.3) DINPRO / II / 59 Lierii ugi muli-vribel dri per. (.5.3): () dy g x x ( x, y) [ () x] + g y y ( x, y) erm devii [ () y].5. Lierii (.5.) Diperoleh PD lier dlm erm devii: d () (.5.5) X () + () dim: g x d ( x, y) g y ( x, y) C:. Ko b di per. (.5.) hilg. idk d uu ko dlm perm yg diyk dlm erm devii (.5.5).. Pd kodii wl: () y() y() DINPRO / II / 6 BB II Memik Uuk lii Siem Dimik DH - DINPRO -

21 Cooh.5.: Lierii PD muli vribel Dri erc m RB, dihilk PD o-lier beriku: dc () V V () c () () c () k[ () ] c () i.5. Lierii k[()] per. o-lier yg elh dilierk (lih cooh.5.) V diggp ko, () lju lir rek, c i koeri rek muk rekor, c koeri rek kelur rekor, () uhu kelur rekor Peyelei: () dc g i, V [ (), c (), () c () ] V () c () () c () k[ () ] c () i DINPRO / II / 6.5. Lierii pliki per. (.5.5): dc () F dim: C() c( ) c Γ () + C () + Γ() C () i 3 + () ( ) ( ) ( ), F, C dlh vribel-vribel devii i ( ) C i ( ) C i,, 3, d diperoleh deg uru pril ugi g beriku: g c c V i g c i V g E g k( ) c k( ) 3 R c V DINPRO / II / 6 Pidh erm C () ke kiri, d bgi deg, diperoleh: dim: K dc τ C () V τ + Vk () K F() + K C () + K Γ() + C i 3 + Vk K τ + ( ) ( ) Deg rormi Lplce, diperoleh: K τ + K τ + 3 () F() + C () + Γ() K K 3 i 3 ci c + Vk Vk R.5. Lierii ( ) ( ) Ec [ + Vk( )] DINPRO / II / 63 BB II Memik Uuk lii Siem Dimik DH - DINPRO -