Chapter5.page.20-40.fall.2011.pdf - Jt. SER\GS £w : :tot....

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Jt. SER\GS $0\..liT\ON -HE.lHOO OF FRO S~NIUS t::ot' £us:: 0 3L-?..o £w :: w"(t) + pl-t.)"''(.~') +<fC'tr)W(i:) =o :tot. is tl'~~vl t.o o.vu~ . .t~.t}e. the. ~4lucs.iiot) ~o¥ :sinqulctk' )'o·,.,ts (. tc. 'De d.e.~rne.O"'). 1nis hi!~p.s -to clca..ssi~v ~C\uo...iions a..ncl to ehoos~ po~.,i.s c.Doui. whic\.t we.. Oro··~cu-'1 pci•nt C\'equla..vo W>o\n'l) _ o..n11 point. wne~e. pt~) o..t~d '6ti~) ho.ve. ne. S\nqv\cn;,t.\i!s (do .,ot ~ ±.o ') ; o.t. Ye~ula.Y )'oi¥')1:.5, i:, p'£1-) cu.,cl ~1(!.) e.x,·st \n c.. ne\~'n'ooY"hood. o.v-ouV'ld ~. OY' i~ p'CI.) a.nd~'C~) 00 not ex'1s-\. ln (4¥)~\qbbo.-hBGCl a.round 'i.o ~o = s,'V')qu\cU-po'1nt: ha.s a.\-worst ! ' b e,.ha.vioV' a.. ~--i:o ho.! 6-i:. \.&.lord. ca.. I ~e..h~VIov--(~-to) t pli.) _L. ~ i: "(i,') ___:;. (~·:£o) G ."t C~ 0 hY.,·,t ~ ~-f.o) p(:t)1 ~ oO i: -lio "~a~. r (i-~)~,t-!/J--.:, ce a-.t:."-c:J: ~: &.o Clot =l :.-eo.
(SUMMARY OF ,;ERoES sm.unoN t<JtfHD 0 j ( Froben'nJ s m~d:l-lod) (a) ,, ac; -e--,.;?,0(.b) I I 1-<e. (c) j{: no -requlax-S~V"l~ula."' p~·~ts el(ist 1 let -l;-6.::: an~ pe>lnt u..:bere. ne..lt.he't" pO:) nor ~t-i) d11Je~e-.1\) k+A. a. S.;bsttute, w<.-c) = L d;.,(:Z:.-ro) 1\:::.o , ... 4. ~t'to'pl~ e'1va..:hoV) lolj . ';)'nl ( ~-t;() to set <'•d 0~ $•()&~\CU.:t·tte.s ~~ pc.U 0(' <6 < ?::) . t.e. 0 ( ~ -Ci::t~) ·, !=--2o IS o-rd; n ClJ/'j po;"'t c~-~o)' ·.~ o(, ~I o-r ~~I ( :t--t:c;) 2. ·, \-I <0 ~tiL S. Set -a.ncl-Scpf>o5e m:o (p o..r.d 1:, ha.ve no .:>·"'flc..l~d.r,t.<es c>...t ~0). Ther-. -the. powell' o~ (1;-~o) Is ('l-.:U.)I<+M-01. : 0.0(K+o)(tt+o-l) =o ( ao ts undel:.c.lf"<m;,..ed) \i.(K-IJ =0 * k'.::O or 1<=1 t.e . )(,:0 ) J.<2..-=.l Se.t coef:.h'cle'()t o~ l"oelo(t lcx.v-~e(" r,>owel(' o~ (i:-r~) ::::: 0 Th1:S should couple a.~ AVId-a;,.., or ll.>._ and. a.)\_+2.. ()..hGiues c;\.l"e.C.OCSioV'I ('e\o...tton h>c-#le. aii.~S Obia;.-, '('eCt>./'Slon '('e\a .. :'liO\{") ~or a.h. 1s. (Aimcst C>\010 0~ a), j:.o'('e~h h:. IS uVloeievm,'ned) 7. w ... ite. 5olu+to,""'. ·. Flt) = ~. ,\_ tl< I ciJ z a./\, (~-:CoJ -+ C.z. z::_ I ')~+K~ a ,t (~-=to A=o >t::o one a..ll. 1fl be IJ"f\de-t.e-c~tned O..ol\0 CO.. C) be 1·1\~o<'too'fc..te.d ~nto c, I w'•ll be tm d e-bcv-ytllt'i e~ v..nd. 'oe lo ,, Ce. one Cl>.. C()l(") S. II=-k', = kz.. we ~et prol:;s Dl'le ;ncl-epe.rodcVIt ~olv+l_,;n J ·,r-k',-1-<2. *<':I) a, 3. w_e set t~o Ind. ~olo's, ·~ 1"(,- t<z. ~ o , 1. « .3 mo~t c..n e ck. i-u ~ee •f two ~ef"te!.S rure. iio-)d ... pe>?d~Vlt-.
GENE~AL RESU\.'TS ~ w l.C) ::: A SERlE5 SOLUTIONS \0 etw =0 K the l<Lrqer c...,'•\1 '1t'eld a. :>olutlo'J; -+:.h.r. $eco~d '< Ma.'f or Mel" ~o't. "ield o-solution which is \;~C!Clr'l'1 i~depe_,de-nt o~ t.he. .S.olu'\.loh obt.CA.ined ~rowa -lhe IG.r4er k. r. Expo.nsion o.bout e•= o...-cqutQN' ~,";.,qulA-v-point is t'Yllle~ more. l~kel'1 t.e. 't;e\d i:.wo inclepeV\d.eV'lt Serie.s ~olutiaV\5, (~~e. C<l.Se.$ E (11\'l:ol) o.nd !II' (l'th='2) J
~. :r~ $<:Y\ts :!lalut•on Cjiues onl~ one lthe.c...\"l~ lncl ~~~ e.nd,nt ih::al'-l:li 'S!!J ~6 $w"'o ~ S.(Cona t'nCl.l( be c;,b t.a.in ed 5e ~~ w1. t :c) :: w, (-i;) exp ( ~ plt) Jt) J i-' 0. b (W,<i)),_ 'fou ecu, sho~ t\-lo.:l w~<.c) so..:t:.ls ~.·tts tw-z. =o '~ £w,.::.o ~-7. =oO L
.SPECIAL FuNcnONS: THEIR. P~oPE~TIE:S ANO PfJ:J:Eftt!l-/TlAL-eGuATlONS (C.l ;/w = w" + pl't) t.u' + ~(~)W ::o (3) tht. ~olvtion is uau4.ll'f w ('!) = . smq '"''CUt' pcmrt:.s : ~a.d e. Q.~"t. "" V"eq\Jla.Ar .. , "'fu\Q.r ~c)u,t 1 Zo W+~ ( ~-ro) ~ i cp'ue& Q. soluttoVJ ' w i (. t1.
A COENEftAI.. FoRM FOR £w:oo ~is complex !.:.!:!'] I 1\"' •114.'(1l·b)(~·C.) + ~~'<b-c)(b-o.') u'<t-O..)(C•b) :iL w ~. c:.. w .,. \\. -1:-(\. 2-b + ~ -c -:J(-i-'""o."":'x-~---b)-<-~--c.-) Co""ment.s.: pC~).: p(t) II ,, CC. +col' +{!,t~11'li+1S1 =I) [ ·-el-«.' + i--o.. he..$ 0.. 'V'equ\o..'"' " " s.inq ula..-=1 ~ " t) ~(t)=t··-·] Q.~ ~:.a.

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