C h G\p fer 7: L T I.. Disc r~te.- T; IVI~ S" yS1-etMS 1"\ l?H ....

+'" ~ \' rIA ~ .. s -Pt,y W'\ '"Dowt a"V\ ' "

CD :LAetA~ Lcn..vr Ct

@) .I.de&t.l EC/~ rqs ~ \St~o S'1-~ble..

® .IJ~\ t3~",~.(+op F; l+e.r :

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~ \J{Jt r~I'"2:Ct ~, e..-

~ ~ot calAS:o... f

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~ tJo+ 13 \ 'EO st-ct1fe .

..

lTv-.

s tor/'tJ~ "; tVc j

;..f \-!. \S a caused) s-t£lble d,)crek L T~ 's'l5f~ witL ~ ~, "IM(JlA\~ r-e

- L-e+ 1)"" [e) ~e alll " •. :~Y OV'J~ POIYVlCNa1I'ei1 iV\ ~-I ~

D ... l~) = \ -T" el, i;-I + dz 'f-~ + '" -t- dM._, -ilM-1) -.-d ~-1111 .. . -..,."'~

D", (i-"") ~ \ + d\-2- -td~~l.. -t ... +dM - l ~"-l + J M ~ M

«VlJ ~ ... tA f\ (:l.-I) : ~-M""" t.\ i:-lt~'-\) j. :a.-(tI\"''2.) d :1.-\+ I z. ~ t;. ,.;-0'1 -c; -+ ... , + .... , c- d M •

-:r.~ 'ttl ~e d~ Qfe r~IJ ~e.n

\ A", te. i"") 12. = AWl te~"") A~ le,I"'» -= A", (e- \~) A ~ Le -,tow)

-= fl..M. t~) A", (i-'J I· • ..,)::: i. 'd LV e {"£ • • =e'"

~ ~V\ ~er words, \ Alo\te G"') I ~ i 'V w G-~. )1

-) TLt,~ is; c~ Ha:i a.lII I{ fA I/plfss ~ if ter. 1. 4

_ All f~sS +1 '-f--e~5 d~i ck.tA~ -rite Maj",;tl(J~ spec+rt.(.., of- t"L.e i t\ ptA t ..

_ \5~-r ~ey ~ ckt(J1jL- ~e f ~etSe.

---- INe vV~ \ see t~+er ~Ot+ it is dtff;'cut./t or "lNfpo~~,\'It. -to dec; ij" I'I.(t +\' l+er5 .v;+h I i~ e6 r F t,~ Ie "

~ !:t () .J: U +,. Iter- w,~ lAV\d~~l'(q b Ie p k4t!e Ca", be Co:fic~tle.J wi+f., .-rI ai/petrS .pt ~er 1z, WtA/l.e f~e f)Ver411 pk&l!'e e:tp, r()~ i~It+ely ItVle()r (~ +k~ ,p.~~~ b~~ I

~ Tltt,~s is CA I (eel t( r ~ll!.e ~ptt" ra

- ~e. V\\A~~ftr- 'tV\~ o~,~~-k 0+ AM. L~) ~V.) be ~+O(eJ +0 ob+-~\n

M'1\. -"') d' -l d -1'1""\ -t =1..- lJ", L~ =- 1" '" 1-~_-, ~ +. H +- I~ 1- ~-~ A't/\ (~) ;: - "".,. ,~) -

lJ' .... , \..:~ l + d\~-f of • + cl :2. - "'+( cI .:::a--'\ • • 14 - rc;;.. ~ '" "'"

= ~ -rr (-~! ;- ~-' ) - k=l -, - A~ i-I II

:9 ~ pole! avJ ~~ C)~ A", (-l:) ()cctA..r- 1V1 V\ltl'rror- [W\efje FaPrs ..

~ J:~ ~~ =, rt.e~¢'" I~ a. pole (~otof ~"~Ct{,-J.J

~ev'\ th~ \5 a... ~ to cC(W fA-t --.>..r + Z -\ -::::. 0

- r\l;. e-i ($" T c .... , ::- 0 -Z-' = r'~ e-~¢IC..

~ MOl{eovtPr) sfnce -t~ V\~W\~.ft.r (l~ d~~f\v{~

o~ A".. L~) Ovte pOIYlldWt iQ/s tv\ '2-1 wif~ reCt t coe .f.frc i ~ +s .J ~ 1\1\ C6'W\p le)( ~~ oCCIJ~f'll CNIJlJj~~ paIrs ~ ,\1\ CoWIpl!¥ po\e~ ()ec 1,I·r ivI aMit..lj4t+e pa,'fS'.

'rJ~ - So J h4A \li~j 4 Co WtpIQ)t" .are-r&;l t:;tt .A~ = ('~ e. ~ \W\bl'es·. .~

r *' -a'(.J~ _ T"'e~ ,'S etl.so ~ ~ a:t ).~ ':: r~e

I \ 'cPr-- T~ er -t- IS ~ f' c; Ie c:l t ~I:.. -:: r;. e. II

\ -d¢~ -\~e,re 's Q. fQle 0.+ ...L ;;: - e ~ ~ r)(.

o I / -,AK

- It' the C\(I],Q5>' ~tlter- AM L~) () Ct1I(Std a.nd Sk-&lb~ h~ GLl t tM.e role:. M\,!!1- be. 1'II),'de ~e lA~i+ ctrde. # I

-- Tt.-elII (ttl t\t..o. =be.ros MIJ!1- ~e. OtA.+~,·Je. 1"~ e lAV\ 'tt C(' rc fe,

--- T~\s \W\phe,s ~'tf a.J\ ~l(pC((~ f't tkr +1A~+ f~ c~II~1 ar..d. S"4-,. ble cA NAirn ~/ave- Q.

CIL\.A. 511.. \) S.fA his. t\ ~ Ve("s e II

- 0,2 1'" O"8~-1 ,. Of 4~--'a.+ ~-~ ~ A 3 (:a-):; ... ow "

t -r 0 .. 4~-t -r ". tBa---Z. -C), ~ ~,...,

o

o

-:- BecC{lJse... I AM{e~)\ =- i v W J i{n a.lIp~s

- For ~ CtwSQlj 5-+ ... \,1e. Glllpc.~s SI

, -

- ThiS €-)(.preS5es -tke .M~ or-clef"" a/lpCtSS Sec-t,oV) GiS a. sex-ieS . Co 1'1 r1 ec-/1'oY\ Q{- M t'irSf- order cd IpGtsS ~ec-+tov\' s:,

- The 6 Ve..a.t f MCA~ rtl-tu.c\e res?"",!€. +~e M4jlAi+u.J-e. \e~p01/\S'e.s t!)-f. c:tl\ S-ec+rCV\s.

- To See +hAt tkey a,y-e aJ.l e~ \.tAl cO~ ) A\ (e(>GJ) , :;:. i \I IN M

!A .... (e b ... )! :0 111 --k :::.1

- Now we. will prove. +'ke fActS 011\ . fct~e I.?

- A 11\ Le;~) i~ 0- ~ ~tl\.l.c-+ o.f. .i 51-/ order aJ lpt..ss 5 e c-t

7.12.

\ d rIC:. S\V\(~ -G6l'-)

dw 1- r\{~s(~ -¢l'-) - l + 2 I + r~sln-t(.',)-4i~)-

[I - r~ cos(w-¢~) Jt. I = 1 +2 t + ,n: .st~'2.(W-¢~ ) • [t - t" \( CoS (

--

--

--

--

\ - 2.r\

- ~lA.ppO~ -tkAt H t5 QVl L-TI dtscrefe-f'W1e syS~ IN t tt. t:\ ('~t i W\ P'4Jse re~ l'C+1!e h L~ ] "

- TIAe¥l i"1Ae- Ilo relatf,"CV1 k ft"e S'is+~ ts" ct d , ~ {er ~ c:e e"~ fAA t,f'~ .W} #1 r-eo.l CO€-·fft C 11fYt'f-S' or

~ TIAe ,p('-e~t.Ae.v1j reSfCf'(\.f.e b-tLeo).y) (S ~ r~+tQ O~ +~O retA\-WePh~C/€4l\t- fO{y.,.~rcJs In e-{~

~ Th -e -brc. rtJ t-er- -P\A VtC -hI' Ct'l l-t(.~:) ts Q reL t-t 0 af 4-wo ('~'_coeffrCi~i potyl1P1Nl/aJs ttl ~-I,

- Moree veri t\- (eli"') I s ~\j IA. j"de ~y WI lilt e fr Ie eu'I J

~ f t.t [e6~ ) f (tJr F1) he" [11-:> . ()-rp-r I

J rL i I-\-Le~)"3 ~ ~ I'IoJ [It J W~M~ ~C"1 -::;. ~~IIOtJ + ~~D

-:Lt\ o-#er wfl«:is, it re'()fJ" r-e~ Hie'~J to k reA-I: ((J'I)J ~utI-n-e~(;I h Vl.. ) . -

- t3u:+ -t\i\ I) "'ee< tf)5~d [II J ~ us.f be -! ero . ~ ~ ()~ W(;IfJs/ ~Gt] JM\)~+ ~e e Veil ,

-~ 1here-kre) ~e o{\~ ~u~1 L--r.:c sy' 5+e..wJ \vItIA q ~I l~rlA.-ke 'teSF7's-e e:{rtd :ce-rv r~~s:e is ~ Id~ti-tJ -Alter wi1Lt htlt)=-I&J.

- NOIltrivft':t ( rea.\ -\JCl\UeJ +1 (fer:s Wf~ ~erv P"4Se.. ~\.tjt 6e nMlQLA(etf CbecC(~e ~Cv1J MlUf ~~even).,

~ 11,er-efc.v-e I ft.ete ~{ Iks- are avt G Ltie~ I 1'vI t\liv\ - reA I h~kII e of l( off I tv. e 'J e:{, r / i Ct:l~ (G1 S .

I

I -[here IS t:t. -it-ie k +or reql;=t;-;"j o+{-lihe...('; 1f-er.5 "'\t\!\ ~~ r~~-e.

~ S'vppt'se f..J Ci) IS a. +'t tter- wlJ-!" a fre~ UeP1l:j re

-\ T~en ""/Vitos-f ~e StAwwe reslAlf could b~ ()b-+-6.i~ecA bj IItPfl!1i~ ~ -P-r!+er f-wl'ce.

- \-\ow -tfAe ~r,c( wo-(l:.c; ~ .... 4

CD 4pp~ -(i.e A {fer +0 'tOt), ul/-fi1e CIt,{ +rt-tf V[.1J :

~~) )~ ~ v~J

(j) L.e:t u~j = v C -t1J C+iWle rever5'~I).

(J) App~ n.e .trlfe.r 4jlllIV/ fv UCIIJ/ Gil ti.e OlA.+f~:I- WD J ~

lAC ... 1 ~~.J.4 1 ;> w ~ J @) t.e-+ ~rt\J::: we-Ill [+tWlf f""ev4rSCti).

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- ~ o~er ~dc;/ alAj r~GrJe Infrocil.(teJ by 11te .Ay~t +'·If-e.v-i~ op~f-; en {(' tTf.fsei-bj ~e se-Ccvtd C( b~d--"J()('~\l tdfeYl''''j Clf~flrn •

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,reed 61/1id M1, - VI qa tille -~ llAeV'e~e) t1Ae ovef&tll .f?-tlf-e.r

htts ~ t~Gtre_.,

M; 1'\; VIlli) WI & MI-. y.; IMU WI 1> ~ use S:t sk-e \tit s~ 7,20

- rt ?l 4\ lfer- J~Siql\ S1)ec. w 11 ( P ICtc~ - Ot- e.V\1 - v t /

f'~ f) i re-"", ~ +s OY\ tk~ {'r l t-e, .r; p ec-fra V//t({JIl /14.d.tJ 6uJ Ito-!- 0JIt 1'4 e phase, - -r: h i

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- The W'tw'f'afped p~"fe (5 pi ec;ew,'se (;V/&,/ . I l

blAt- CAn ittvve J'VWtP5 of- 7f; +0 accOWIodJe.. c( S13" )) -f'lr"s ;Vl H-(eo'~ J .

. t>1Nh.en. \iV\ettC fkH~ l~ Je5i'f'eol) if~ UJUt{II} )u.fhci~t tD c:b-Ijll. +cr tt M. ~

pC(."~ hC.hJ CM l

Ke-rtAJI ~ we ~"e 'Seeh ~.j- al'l AI1,.lt$~ c;-ec1-'4Il ,7,'22 CtII1\ ~e IA~ ... h:~ o.JJ \?l,.. q5 e. VJ \ -tk, U + Q f"tec-r, '" ~ ~ ~per-+-rJL ~"-~ V\\tt.(A~_ .~ GL ~'[+eA.

-? Tke ~I \'PDc~5 ser+~...., ~os q V\(NH'\~Ct.f,'ve

5~ ~-.J ~aJ\ Qv~~des 0 ~I'-vl ~ 7l,.

- ~ -:r-t CLMs :!eta'j to -tI.Aa. s'(I

~ -rk- tlMpvl~-e, '(~~se hft1J IS r~.J1 --s, -?ot",s G..rfe feet' or 0 cc uri 'VI

(OAj\Aj~ te t>~ t~v-)

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. 1?S-\- : - ...

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~ lW\pli-l"J -t~J ~ ~olc:;>.s 0;' t-{C~) Ue. lV15 ~ +~ \A V! " t c>\(dL cg. ~ :c-?tQ~ , ~

~ ~-1 ",0+ y-eyfv; c+ -k, lo (~ft

~ witty do we Celli +-'lAiS {(J.1.-\i"'i~U\vI 1,'7,2.4 1>tA.~~e)\ ?

-} -n..e. b(j i Je~ " S" 'P1:'0S~ Wz. lee) \S ~ CCiU!Ct (J ~a.ble) n;t-nO¥\~ mll\s{ey -k'AvhCv\ 1-kj kits oV\€ or MV1e. :t.:eAtVS OU+S11~. ThL LA \1\\ -t c\ c c.., Le., t

-----:0:> ~'hLe-) J~ V\£rt ~ \Ie 1\11 \., ; WItt i¥I P ~ l(J-t:.

-4 """ ~ :3 ().. co.. v JO\. ~)' s--{ "-b Ie '1M; VI ; It!A iA WI P '-' "" ~ -e... -\'V~\JISW ~V\C--\:\CV) \\1.l~) )'ucl.l. fk.,j-

~~~ (:e) -:::: Wi L~) A C.:t) )

~ wl.l\ere- P"CT;) ,~aV\ ~llpCtSS )ec-/-('~,

~ \\Q- 1A\~~C!~5 )ec~iW\ a.JJ.

~ /i.,e l1i1il'\l~1A WI j V'IVP cIelA.~ 1 7, 25~ V\A ~\ WlIA \.v\ \' '" &! s-e ~

- t1-0lAJ COVl we u~ ~~'S t\J~ a.~ ~ .. 64 S\·jY\ eVl~i~eer 1 ')

- ~ 'to ~ 'rt 't

-=:, blltev-. ~ U:~(:l;) ltJ~ IlvV-YIWtIA ICI\ or WI )t~ 'P""'~~ - f;{~cv- V\1AW\~ ~ ~Wt~~ fVtfv

fV'Oc1u.ds(J'~ '{'Ole *6(.01\5 tt~ ~~.

4ft.v: eack v~t-~~-eeJ ~O lAA..1-~{ck~. tAv,\-\ cl .. cJe, reJ1~ ~ ~ \~'cJ.e ~ c\Vdl_ ~ vW \ -teL. ~t::z,) a~ "l pV(;~C+ '

- ~~e1s ho \AJ ~

SUP\X'~ t-\L~) ~CtS q reaJ,. vt:tlue&

t-em a.t E:::;. q( ) WltIW e lol.l > i.

- T "'c fA t\ L:c ) J Des "ck ~ /} ve tI-ll>llf.t./1.I IA-\ r ~f.t! e. ~eC4l{e. tlAi I') ovt),'dt2 ~ UJ/\ t -t orc\o ...

- ~+ , H C:t:) == G L 1: ') ( I - 01. "to -I )

1- 1 - ~ -.:: G.L~) (I - 0{ ~-I) rl _ i:

r, ~[- , ~ ~ -t l :; L Gil )(~-I ~ C

-1~e 5a HAe ap?r()ti~ch wwt> W r· COlM.fIeK- lkAilAed ~S) . tAl ~ telA oc C u v

"VI- OA/ljtJ. Ja te po tvs (of h fYl] 7s fectl,

- 5IJpp5S-e. ttL.:e) ~ 0'> a ~. ~ t l: =. 0(.) W~~e (;)( €- « a",~ l tJ. \ > i. .

..... '7Ov Cl!Ve v~ffer-r 'Thrs ~ (l1s,ck ~ ubtri cfrc./t fo get a Y1et.U ~t lie! (,oJ i-tfA +~ 5a/Alle. Wl&J",;turh c;per--lruWld 1-1/ bui-~ reJlJ.C;;) p~5e; .

HC~):::' GLl-)(I- tXrl) ::2-1 ~ -:::- GC1:-)[I- 0( crt) z. - tX

, ~"" - 0( it-

=: [G,Ll-)(~-'_~K-)J [~~I::;~ J L- -r .J

A rll~ w~+k +luL ~QfAA!l 1lAU3 141 tvJa ~$ 1-/) brA.1-~ ~evo ctt 0( ~ovecJ. to -L

~

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Y'LOYl ~lft''1IV1. \ IIY'U\I/( f k~5.e (!Mite&. pw,Je or tVl.tlpUHAA-1 r'l-\u(e) )

- You CUt repe£l:f this p{p(eciurre W &c~ :rero -t~aJ- IS OI)+{'I~ ~ ULI)"rt c(' reI e ,

- T~; 5 IN If / 9 I've ~(JI). ~ VleW --PrI~ ttCfV'V1.~ ~

I, Tk 5a~e 'M~jJ1;+lAh '3fel+V't.M--1 U tjov'V 01'1~ ,;~ Nc~) )

~, M }' "1 f ~ lA U4 P h~ j ~ f

- W~y do +~,r:; 7

. /

4 h)~ appJ/C~flC1AS ,1 WlIfJ b~ lU1porfolAi 10 1Vl;V1)V0i~ & ~ rClVp JiQ.~ of +L.e Jy~~,

. "> ~ SoIMD- apfJ Ilea A"ctAJ" If /Mlnj b ~ I t~Wtpor+CI/ftr to -eV\{tlye tAd r~ . ..LJ s-telM IA ~s C?V} t I't ve,;r-e fIMj IS .

Cct vlad ~l1d S+c, £ h. ' 7.30