(j!J ~~. 6~ls Iw.d~ atdf(WokiDJ' .. .
'""-..-..
Let H be a. ~ek ?lb7l-CI/Poftle1 R/~ni~1tI/,hII11/.TIe G~IS ~ ts Q S'moPM?c:f~ tJn
Hx M \ d/~(N)
swh&+
@. aOGJ)=6(g..Jr) a,J 1:x.':1..Y ' t1Ie-'~ A K 6(Xd)::' 0 f'X:feJ· __
(i;. ~ ex'd)? 0 _
~ Fixl~ !f - (l)J.~ x~rJ-, we ~ -tk-.f~ 6~ ~) ~Jf ffj t:x)"l.-fI ('+ OU)) 1l>2_
~/ -...Dy fa CX) [I-t" (fJ) 11.:<2
Tlus k+ ctSY¥-hftCIJ / ~~SJ'Clh i Ite .6~'s f~·· also ",lie,,; -fhtd A,t' G(X,~) =-Cx(j' .
FrU1>t -Me a. >11>f, -Ii ca./ khttv"fTr we Ctm f'lui -Ik:f 71 =2 rwL 71>2 , -t/.e 6'~:r fd'th.S ate. ~ ,,~
We-~ -tkf {~1j
'D?f'n-l·it~ A ~/~-fe. ~ftI t's £,aJr!10k ~!Jc.., ,.f amJ Inr!' if rIoesltlf adhtJ'f ~ !"s"f"~ 6'~~ ~.
"----.. tJ.fI~H"i.re.. · is SfM'ti -fo be 'l1b1t -f~~;"c-.
---------- .
Ex()mt,fles : If<.~ OAtd /~1
Let f? >0 be a. b(j 7UJ;n.her. Lef F{R) ::: { fG C/O(Ik.~ I f= / frr 14<: ~ /f rtrkf/tmtt/
S'Y7",~:ln t- ?
If 1t:72 , -tk.. fur. tf!J C>0 , -f/.'the exisfs a-n R0 ,r'/lc4
-Hw.+ r ~ 1< >Rofnt. /Vf(;>C .
t-. ~ f f F( R). If 11= ') • j,fm k ~ £:>0 fhe>e.. eKists R. >0 S'IlCh -f/,qi- r II:J R~R6 , tll€-Iom r.u a.. It e ~(R)r t»#cA
( l- I fJ fJ '2 < £JIk.
flqr.re re,.r, fr n" 2 ,/lie def'he. /
-h ::: rJR (JXV Such fk.I. +~R.
CJk{f) = -~)-((~ (I ~ ~~+
-~J1< f?.<f<ek.
K
.-----/ o t-~e
rAe.. tl. s+ro..JI. f/orw"d Co1>fu-fo..fl~ ,j"I-eS
].'" t l~' If>/zd-f ~1 ~R. f67 R.»o,
The 'I7lam. mu/f if +/u's .redtm Is the ft/~:J ~zqf;" of fO/rQj,~/; cAr·
Theote/'lt Lei H Je a. CC1nfIe-ie ?nanfII If His 'JJp,,
f~hDIJC ~ -/h(ffl ? rtf f~-+ f ~M~ we. Aave. 60 itj, A l-f) < f(;O
Wh-91€ A'f (I') d'(J41.Dks:le. arett Df 8~r-r). JWiia
f'trTOf 1=0 r p 6- H ~ Ief q {p;j} be +he 6reJ>n'S' Itvltc-lt~ . (o.SSLtM;(~'i +~a:f H exiSTs), Let 0' (P'J) be -Me Ort-tn.' tojktc-/u;. (J>l $" (R.;) wi -II,. -fhe Vi,..; c/, Ie! b4u"dQlt-( CPtr!; f,;"" , fJJ ie>e Ri -7 f ()o •
F07' ~ 1< R <: Ri~ lef's deno-te -I4af ri 0) = >U,f 6i (f·J)
!Jed8lJ)
i· (b) = ihf e·(p·q)t "- Jea8pl;.) J v.
Le+ f be the h4rrnAm<C fmcfc:." J1",ee! dI! lit W l.bf (I)
g"'+ISfr~ -Me ~dOlr! CPhclA+t8ns f (J) -:; S,' (J) tIL dBpJ)
f{J) =G,'(P(j) d'l 'dBp{R)
TIre 1ltQKi iItUto- rillCiple implies tfto.f
f('f) '9 6,' (~J) 1m Bp(R) \F,{j)
in faKfioJ~ , we hrw :id ..2Bi.ar ~ (7; (f)z 88p(R)
Oh.1!ae. (rMev- ha,J / ~;n~ f{J) is l,evrWt01tic / S~~es +he~'m i"1/''eS -tho. f
() - r 4f = r ~ - r ~ - J8f{I?) Iap(/) J'd$plkJ dr )'dBpJ) d yo
AIso ~ we observe +Acrl( ~'. r dG· -= ~/
./ J'dfJp{fi) dr JBpft) (
Thu.s weJed ~ !a6p<1) dt s-I
Let's lim omsit/~r ~ -fo be -lie h4lr/nl»lt'c {UhC-ftM clefneJ I71l 8f (R) \ 8,,0) SQ.-hsfhJ fie bUUn/ar/ cmJ;f,gnS
hfJ) :::. S'i O) dn oBII)
h(1) ::: ii (R) tfrL dBp{R)
AJaih 1/,e lWri 1»Umt f"'incifle. imthes fhd ~ (!J) s ftJ) 6h Jp(I<) I~(I)
-ihP.r I'wI 2. . i.e1 H £e tl R/e~1Ul s/ly'f~ 1'I<c4
.L Ik/< -t o<J
ThfJ'>, 11 /C> f QIr'(),. bD;,.c .
(;4-/t) Pro-uf of a fAelJt8.Wl of $rascamf fh>Il' f,'e /'.
The~ Let uz be t( bOZllnrleti CIJn~ cI""'aMi 7' ;,e~, Lef U be fhe f''rs+ 'D;'*t'chlef e(JenfWcc-ftQh , 1£ f
A U. = -AI l{ U >0- .
Th~ ~Cf(J U ?1Uts-t he tMIz cave '
p'r6'Of We cA~se. tn;j ~(q" Ur; >0 _ ,110 /~Q. ==0 sltc4 -Mo.. + -~ UDIS ~a~. S ((. ch a -hvnc-fwn a.!PJtr>' ex"sf~ . For eKa-tylop / ./Ale CC#n ~~ the. em vex k /1 7 +he Jrdfh of - ~~ 2(,
()m.s/d~+lre [IMU ~: = A t{ +~, U ulAA :::.0
We fll~l1Ime Mil f 'ZIf -";> U . -f/,e f'r s+ e(Jenf<V1oC/.n., We art j() i~9 +0 USe -fJ.e mfNXi}nam. [JriflC't Ie, Le+ T Je 1Ae. bijJe,rtl ~b~ Sue/, fh~+ clef ( - V~~fPJ z:) ,os ~f!Mfeq TlW..s -M~ I\S OAt %0 G H a"d '( airecft,,,,, 'Z cu-ch Ik.,-I
- ~ U.)2'{ (K.)-=O
f},ftJ f:; 6-M'f/r- i, f..if 11.)Ji (x.) >0. Lef}p: - -4 ~ 1~ +Ae (hl()ck.+~ of r' l.s
~¥ .:: LHf - I \J Cfi F" - ,,1,
By -H..e ?llai!<'~ triftc.4ple, ~'" k =: 0 , ":!t-' oS 0
A (/)" ~ 0 TIwJ.~,It!r .