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CVT Unit 4

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217 views53 pages

CVT Unit 4

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Baba Fakruddin
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| unity Fourley Series | Fourier Series suppose taeb 2 SIV Funct «| oy 0,20) oY inoany | $a) defied 6 Gr, ay omey mfew f Can be expremed © a trigno imetyic sewy os Fo = oe + f@n Coan + baStan% ) ; | nz where ayes BA are Cours op me VArTae. Such Serves to uotinin & destred | Gorge of valugy | mown of me Fourfey sevied Ay ft ond | Be Gots do an, by ome Catled Fourier Coe.| Pheenky of $a. | Euler's Fosmulag me Fourier Series 40¥ me function fa in he mbeyvel cax se Cr2r ee ke oo a @n Coanx + bnSinn% ) 1 j where ay = air T ie tordx cet | ane | * +f 2) Copnndx j | + apon oo i 4O) sinna dx e The vdiney of a, a, bn are enon ay euler, Formulae, note! id FO a & Fourfey Sey in Me heel osxery GM Every Formulae be Comey oa +3 fo. dx ot p One J #00 cosnk -dx T 6 y wy a ee ae ae eg to be expanded, of a Fi Pagies To [ethit), pub Ca-1; the interval becontes ; the formulae (a) meduces bo ow 4.2e1r and ! “Tread. O0 im flake an = + [pe cosna' cx P ge fh sl eal at 1% bo~ te "P peaycinnn ce av “ “EX : ON S- * | CONDITIONS FOR Ex PANSIE | Condi Hons | FoURIE rnutated ceatain “Daatehlee has fom 1 ana | under + uthich , CerFain, funckons posses wall | | i ions + | | Fourie expans ‘ | | A ae pou has & valid pester server! | phe” -forteo-- ¢ | expansion ©. Ory Cancosnt + by sinew) . = . n i cyt " al PS cn Es movided + heme Qo, bn One constanks P' ia Oe ingle valued | i Po) fs well ‘Clefined, peniodic , | and finite: . ‘ ; ¢ . ; ty fon hor a fintke nuraben Of enite dtsconk- nuous in any one, period: alte roost a firike Neumben Of iy POD has ak interval of manime ahd’ natnima in the, lefini Hon: NOTE: “The above Concliions ome cu PFi nok necessomy: Some Hinpowtont Reculks: 77 °, 1 gfe casrn-ch «Lit fn aes 2, Por roxn=o ent but | mt, 2 [sine sinnt oe = fO-f0% rntn and tren-o - a, fom mzns0 3-.sinnt £0, cimanti=0. fon nez, whenez is the Sek Of all integeas- 4 Cosme C-1)2, cosant=',nez S. sin(meL}rs Cv nez 6 cos (ntl) w=o,nez Te sin. feiSE, if nis ode 0, if n ts even oumien ( j i ] 4 ; : j j 7 \ as copnm! £0,1F nis Ode Hat OE Teh Sig miseved 97 o ; | | | | | | | BE! Eapness Pos xT as foumien oF “xe mf sino cn] =e sina “7r(0)] [ro x sinnx *s even Fuockon, and’ Sinn % 0c 2 w heed Fe fasion cle > eh [(-32) “1 el: axing aes. peat - eC ed + Teco = PY pon me t2y3-- - 00) On and bn 709, We hating Fhe values Of sua neq uized fourien series for PO a the | hone cmmas Hn er ¢ FLV 2 sinon | ante rel ( <4 2 [sing sina +g Sngn ~psing + | | | ind the Pousier Senie mepaecenbing Pena of Ptr) le° POA) = Hy O06 442 sketch the gmoph iF from -4m bo 4TT tet xe B+ Sancosnn - . bysinns ——(D | 2 a Pa ur I 1 a Ll _ op [Theo Oem ae f Pea ch «Jono = “ef | land on. 12% ur one ane 2 freneosna-de = & fx corm dx ° | ° | ‘Sin a 7 - re "—cosni ' [x 20%) - | =] | our aL xsionn + sLcosnx) a + } | Fino) 2 Perf be ale a ee HJ osorsh | | | ' > ah (fh cotanm= 4, | n . (rapeseed pzeomeaee] , 2 fftbe s ‘ “t/t eas 2 -+0)- co¥6) é £ ee -Coseoe! f staid J i Substituting the valu OF 93,an and by in Ong Y 4 2 Aa ora S Easinne a JR@ KROSS S97. 2 [sinatLsina+ 43 | oben Fhe Foumen Sexties fox Plays x= aD the & | ZF | ov eaval fet] < tence Show that Sex nt eal a | iz Sa The foumiea Series OP foyex-arin [-m, ny is Im, aewre By S Concasox + bysinow) el ao ate po tle deresmire Qn and by ie wr # "fren oh = too: he eb wah faba Tuite eb fo-e Ge : x ke ged funcken ond + i / “he even funckon) PR (B)" = eet). wan 3 ond , — ooh Sine ve [o- nau) Sinoe ion Oe ox) (~ £080 JG of sos 4, = -4cosnir ne “HON Cao) [+ coon =e] RE pounzen Senies Finally, bye ae ar)sinor-ox 2 & [o- 0) (~ (:e2st)- Cina) (Si SHO) 4.62) oxy = ~200snIr eam =2C-0? a Sa etring the. values of Qe, an ond bp inQ, He g 4-2 = + Et seo? ee “pecwPat nD) ay FS pyc Atty S [ut sana ¢ 2 SD" ssn] net n = a cosx az tu | ~~ 082% 5 cosan | cost ge Sead rnc oie [es inn Sina% 4 singe 3 Deduction: 120 fe 9 lence the to co). ae Jn Point GP continuity of fev, Fourier sexier op f¢x) at A=0 connayt fyrng HO TO), HE gee pattiog Zz » Punctions Having points of Discortinutys 15 04 x4IT gxptand — Few= fi a 5 OT KAA 27 the ftesval (o,217) 6 @ Founien senies The ourien senies op pea) fo ts given os o pad= pad + 20m corm t ar Sinn —AOD ere Qo - Efron de = ° [Jee Ot eo] Pe aT, On et J tenecn® dx = HL casnx dus fe cron w? i v 22 cu = Lessin psy L fcosnt-Ou = a (0-0) = af ( 7 ) a, =o oe [he and bye ‘i 7 VAP . " s fens nnach = it [f-som ch fousnena COSNe | = fina ei ceane ® el ni 2 —Lfe> ee ee ee 2 2, uthen 9 if odd z ir and bn inenesrt yg Subostituting the values Of 2070 inne Lsingn rhs Leno} fo tg ee F Asinne = Let LBS sin tel A 166 Eran wery, ; Mig Substituting the values of a,, a, and b, in (1), we Bet ’ sin ne I a- tangs 4 += | inde a sin dx +. Le) 4 ta oO a ym fe Graph offlg mx inf de 4°) Af Example 6 : Obtain the Fourier series for f(r) = x ~ x7 in the interya) a show that 4) (or) wz 7 eTDe e Solution: The Fourier series of f(x) = x~ x? in (-n,} is given by eae! a Bey (a, cos nx +b, sin nx) Using Fuler’s formulae, we determine a, and by, eto. 2 * (; xis odd function and x? is even function) ~0-2y(=2 mem) a ay (=e) bys (n40) (% cosnn =(-1)") goutor Sao it Finlly By = f(e~2?)sinme ae cones Hest) n ned ned cose) sons] 7 7m ) [sts _ sindx , sin3x _ nee 2 aes ~Q) Deduction :x=0 is a point of continuity of f(x). Hence the Fourier series of f(x) at x=0 converges to f(0). Putting x= 0 in (2), we get ove oo yor m7 : 7 Example 7 : Expand /(x)= (5) 0 sin nx = 0] 2acosne sinha )cosmn, = 24cosnmsinhan [ean sf na? +n?) (-1)"2asinh ax i bem FF c0smn=(-1)"] e a | Use the formula J e* sinbxde = —— (asinbx- beoshx) | L +h } 170 Substituting the values of $ sinhar yl (A) 2asinh ar ony ax S| xa +H) _2sinhan [( 1 _ 2a t a ncosmm (e (=ncostt)~ a ; xa +") which is the required Fourier series. Deduction : Putting x=0 and a= 1 in (2), we get 1 = 2sinhe * k a oF sinh ftsghaghe a 1 = (e which is the required result. (Notice that x = 0 is a point of continuity of fx) =e), * (-asinax =!" cS. “| t " wv wet) _ (este x(a tnt) acos 3x “Fea +My and b, it (1), we get (A)"2nsinhar aa x(a’ +n) Engineering Mathonay, co (0- -rcost) AQ) Example 10: Obtain the Fourier series for the function ix) = x sin x, 0 sinnx 22 2 Lai oon or xoosx=nmeosx—5sinx-2 si 5 Fa in me ann SL pra mmanan RRA VC) #7 rd the pe mesent the | “Sy ourxien Sexies to we given by fea) fr fon “T<* | 1f6 l Hginooring M i 208 sing 1 jaf sins n| cot) ‘J n| n "” ” LBA Gog ne 2 (ly form = h 23 fe n ‘ith Nn, "ai, \, mon 0 Xt { Lf Le o4 Finally b,, : J soysinnrde = fo.sinm dx = fx 2 sin nx de men 1 0 a 2 = {x7 sin nx dv t 0 1 | 2f ~cosmx | ' als (- st asl using, Bemoullis Ry . Mtl ee od) cos nn-+-%(cosnm-l) ln ae =e" 3 [c yt ‘| Substituting the values of a’s and b’s in (1), we gel pe (-l)” o : 10) 2EF canes S| eat 4 Ao 1 |x ae mur a nel ry Frouriem Semes fom Eveo ond Oda Functions - I fe finehon foo ay even in (i, ) tently fowicy SEXVES Cxpantéon Contatns only Cosine ford 7 * © . $ anCarny ney fw = % where Oo a A= a [sovennads | ° metion Fe va on ond seas yhy Fourtey ifa (mn) Then gine cexig Cxpan tion Coc oing term : ua [ fe - & bnSinnx ny only w Sia where bn >= 7 J fessinns dx ese teeceern rea os a Fouriey Sexi in ene tT = expand FO > x ® ginte TED ena, a8 $m , £6) fy odd the veveguired Fousiey SERS Pex = SP by Shame a Pn — there be 2 | fensinnn dn o inna % o(~ sien + bx ( Geen ae ng & (sive 9 J" s = of <3 cosmn + ot : ae : 2 a on conn | 2 reoonn far £1) ra Atle gt : Hone SF a el)! a= &. ne NEED Ham od @ we oo day b Sinna ae eee XL Express Faye % as @ founien senies in (ona ' oo) Since PEN ~*~ OO \ < - fox ts on odd punchor in Emm | 1 tn tle Fouaier Senter €XPorsion, the Cocin ence ' : : absent and only sine teams one Preig fears ane oe hE SF bosinnx —+O et w > wthene jb 2 2 inng-dy af sm) - = 2 fas =| y 2 enh + (+ Sinnreo) GubsHtule the value of ba in O, He get 2S cue, te DS EE sion ye ‘ I a =. [sin-sinc sing —Lsin yas - sol 2 3 Am kthich fs Fhe mequined Founien Seniec. Obtain the Founies senies fon the 2unchen Peay Juin -1 <2<7 and deduce that seeks me \e Olt i Since #2) = bal=%= It) = BOY) , thene fore Po=1x} a ) fs an even Punchon. tence the Founren Series talitl Consist OP cosine teams only: 2 Bas Itt= Qe + Sancesne 30 nm “= Klhease, Qo = 2 wv wT = fewdn == Jind = = de ° 22 eM T zh And, - 7 22 w seal = fe C4). carn ch = [lai-cosnn ce a To = vel conn «dy * sinina ) [-sesony™ 7 ne © 2 =! TL AT Ai] > 2 fer-1] : Tht STOLE His even hy ik nis odg = — HO me the Values in), we gee Cos, CO3 [Dede pihen ¥2 Os [nfs lol=® [path wzo- in Cy), We have ioe be] weu ftriehe Eo ae gt te ate i ee st & — % rpc cosh ar, expand fC) 06 0 Pour er genies in CTU) Ot lek Fa)= coshon = te 2a then pleay]e EEX 2 coshaxe POY z fs PCW) 11S} On ey even Punckon oe vee pene 22 + San come mers) nel ps 2 Oo= 7+ hay-dx ° Sree 2 ec ard ease 2_ (sinhor - sinha) ao to or “Thea ale = 2 (sinhor -0) = 3_ sinha | otr on | ancl 1. ard, One = if Po conn dn | Te r 7 Uf peo, 2°) cosnn dy =z hoax - Cosma de az] (Pee Jeasnrdn ar, fom oe & [eteoer: od + festeosns dx] r] elf S coccomensnnlfe (S (occnn ncn i om ts (acosniro) =! carafe eae acornia)- caro? Cae a’n aha BAe a OTe 1), Same Faqm) [¥ cot f+ f- 97% red] acy? arian) 4j CeO G97) = zat)? [= - au) (nad + 2a¢-1Psinhat 7 (n>40%) Subst tu bing Fhe value of Qo ond an M@,u har s 2A SIDhAT cee. or hey ina = 20 cinhan fe + Ss cPeos ne oe za Ban This is the nrequined Founies Seni es. | Hair Range Fourier Sentess “The Sine Sevig The Hatf vonge Sine Sevvey of £m tn HW) 1% Qivun by o fm = © bn Sinnx nr) wT j Fovsinnx dx 0 where bn = = U. The Casime Sevies The waif -onge Gesine Serry ef FO fa (1) 4 given by fos An Coan col at 7 J fordx | ° 7 \ anit = J $ercernx dx | 2 a which is the required Fourier series. ; Example 8 ; Find the Fourier series to represent the function Ji) aS eel NTU Peedi. s) Solution : Since sin x is an odd function, ay Let f(x) = 2b, sin nx, where Ea fe Lp b= 2] sinx sinnx dx == [2sin xsin nx de nto 7 = + Fcost =n)x ~ cos(1-+n)x] de x0 - feats. stems] (n#1) t I-n 1+ I-n l+n = 0(n#1) Ifm= 1, then {oe are tn n#1) _ 2p wo b= sf sin’ x de = + [id cos2xjax 1 , om) =1 + f@) =, sin x = sin x Example 9 ; Find the Fourier series to represent the function AX) =|sinx|,-n cy 2 sin ne =2 [xis in sus.) Example 2 : Find the half range sine series for f(x) = x(n —), in0 mA 35. 228 Engineering Mat, a ggsinse— +] [snags - x as ne series expansion of ) is given by (ii) Cosine Series. The Cost wd (:@sC foe) E32], le w2( a2} ow( 2) 2) o( 22 a) Lol “Rin C2) 2) nr 2n 2 pea | 0 = 2 [i xcosnrdr + Joe-scosm | a ca When n is odd, a, = ie, a =a; =a, 2 nt =| 005 i and a, -(: cas = is even 0) Putting »=2,4,6,8,10,...in (3), we get ay «2 (2888=2)_2. 5, ® Ed z 2 ( 2c0s2i a =2 (2emeze=2 ™ 4 229 | 2 (2cos2n-2)_ EL ee 2cos4n—2 as = 2 ge SO 2 Tt -2 (2esse=2 =2(-4\) 4 0 1?) ao?) Fe ggbsttting the values of ay and a, in (2), we get 2 1 fee vtec gone rm] nm 2(1 1 aaa Foret sreonsr+ A cosors, ) liter: ; since a, =0 when n is odd, we have n=2,4,6,0 ete 2 A nos —1- (2 ose Ue Gh eee (-17"] cos 2nx (Replacing n with 2n) n=l t sl n eI SICp" = =24— PY [2 (A)" 2) cos2n =. 2b lO) Nos, 4 as 4 a w m nl (3) e284 1 okie = eee 500s 2nx 7 | cos 2nx 2 4A BBM 4 yt 33 222 J cos2x+ + cos6x +5 cosl0x +. | 4 «le 3 3 the halt - mange Cosine and cine cesies ‘by fa lve gunckion f= «in the Monge o Fi coor) a5 [en] | Tw oF | et ore toe | » for 0 odd- | ; t | subastg there - Volues In (1 We ge | ge LS Aycan | \ nets" | | es) fon) yam 4 [ cots be SE ] j SL a joey PlOIO ars eo in ol, we get £ Zz 5 4 4 4 A p $ 7 “The Sine Sem The olf Onge “given by | F | aye & there br = | * | | | Tag basinn® sine semies expansion eg a Lv POO Ie —7@ a Jpensonreb FT bre Speer sxe a [= (eset ces > 2h ar cosomr substituting the value oF nl Scud 5 2 r 2 bo ina, Kle Je : ian Le Soo" ae o ale yrs z eanies Por PON) 29K incom, \ fousiien cosine cenies is given by Oe + F An conn ne ue a [am sis ts | Obtain halt mange cosine Henee short that GO The halk mange | fOO= Sine na Hese, land On: 2 Jee de 7 Efenst Qn = w, 2 [ simeosor-olx a eA ° at 3 ' = Lf feo tern tain ten)x] he 7 “ell [[7= costlenye eosenn |" 1-9 fens ces CI+n aT care ~amaiiad 1h f eo w = (-cosx), aanW arf vsirncosn-ok ° [: asinasse = sinfare) sine cos 1-0) a TA 5] Ceo cesar] putting xo in 1, 2 — f-coset teeny? a(n!) ] wemy! P], nel pals (1D | Tr ina)tne) is even (on) odd one poxding as 9 va cere ch = 2 fSina-dx " ar = 2 ° 1 fe cosex]” Le eee = LL Ccos2m-Coso, of | = Ceosan-coso) os Qs O3 2082 77° are 4 and 0° Fa ojeue cl ee ae cra (ut) Cat) ginte 2 S rv 2D, — © D=1) 6 kod inte 2 — 402% _ ooche __Y cose r case tetas) Tos) 1-9 (oH) COs2% - COHX — Conen rs le Be S17 which is mequixed semies for sin TE is easy to ge that ay can he ttmitten . OS Gare 2-4 S Cost on te 7® 4 tue obtain ho Which ig the mequined aesutl Sine Senties. Eipond pexy- cosy 9 eae 7 halk nange let pons 5 bp sinnx lO. az wv _[cessinnr de 2 ie aL rt 2 ban 2 = ‘due & = | Frwsino de ; ate we f2sonxeos ad = bff sinentn sing ide > EE r CL fe cor cnet Coscn-1)% Tony Hem ntl eis ‘ a fe cost — cosCn-T + L +2] ote nel aoe tT a eo 4] cnet) ntt n-t nt me Per 5 eyo, 1 + nel nm ne om da tit bata |! t_\} coe) «£ [foreni(aa tas] . a ee Cal ae nel ' 2 bye then IS odd and net = Un then n ts ever arte) TE Wie, then WY, a ‘i ee Ze “feosusinnche = 1 fsinza-ch Tv o . ° Sie © . +/ al = x1 Ccasen -coso) = = or Seas (ane mill 0 ° Ths bpeo when nis odd = HO then nis even artn=1) Sub, the values Of bs In(i), ule He mye FH cine iz 2 on _gpnx x (ney aes Sa a2 MG -- + r Runs atin 9 coms AS 2N cima [vm is even, seplace n by AYN oe [os 32 3 4 =— 24 va i es aT 13 Feoran th sno + | Me In an “Aréthrary Wferval’. o> a he wu 7 fi enaion Aer Hoey. Foo e Ty ogivua by Pe ana courte Serre exp | Sete me CF at - A eo eee ConA B.S. Sn Sin nit rm. . AD » ef2h “| in (0,24) courier S007 of Feu pees | pe Conse fom@ PEO’ | fay Con MTA alot | AOS. deacge ty ere Te ot 7 / LeysinnTx de di baat ae J Feo seine dx Anke. > 5 Feuprers PU os & fourier geries foE J, Q. , Since 4-H a LE W=PD therefore 4X) 76 ar Be, function: Hence. the ‘fourier, gevies of 4ex) fo ELD ig ghion by Pea) = Oo. 2 £0, cos MX _sO) whove ee Hine [Pence Y= Hh fata -H(F fy. a at On = % flops tm MIA dha. = 2 cos TA E L =a? tot cst “(ag | fet mh i e per \* Be | - | Bain | 2 cos DUA rt Ts] eee: or , i | Goce the Hast and fast Lorms vanish a bots | Upper and ower. Itmits - Oy = x 3 ee) deosnr EWA s™ ne Ma | a Rubgtttubing, - these values Ph (1 te ety 2 = Dt og niet nt Ret pa Lae Peps oa 3 oT Tei n> a L oP aly ) zie [estat ~ ose tert) cst} ), we get find the tourter gextos of 4o)- T= wy fy 02h eo | tleve Length Of intewvol ale ¥ fet jo) fp Z [oyccsmm 4! Lby8n i t] ie : | = ey E faocosrnra +n 8% MMA, Since Le) | 2 ; | “thon 0 Feb fe ae on oi ort Sta cos 2 nwa Sine oo. + fet cosnTin dtd 2 1 * =L fiweosnna de | ‘ ° pet Sona) _ 2 2 2 [eo ) cof-zsami \ o = t] — cog ant. | ; [38a “aa cosgrn) = stelle de i pi wt ido CL) a {tran omade 6° eo fa en (eset ms = Sogn “ales a) ai (ra aja = ustltng the wause - als andes to W> pe Ih) = ea Tet 2 oe Rona: ; whfch te the. creqtriod “Pourter, gore. Kapand “fea =e ogo. -fourter certo, i the fnterialC,) lek He Bey Zo cos OT x 25, iq deve Let note San Z ances eZinto mx-30 “hen sunt 4 fend. fee ce Ly) ° ee ere -a! = 9(se € “£a8iobh}. : : and One § (ie COS Ma « fe “amend C esate). “Seat te) SEE l |= 1 Wee (ecostsit tong to nit) = =e (=cosnt sei ni iy san belen” ret DP ote ee ate (e- elie “em” ED" asioh : HS The Finally bp = iste) So: re -fPemn (# Ie )- . (tor “oe oo tt | : nog CSiorm— seas) e(sionn= ree) “aa = Le, frmeosnir(end q ; byte aos piled ) ~ anit aot gin,’ bo : | fubgitantag the. values ©f a, Anand by §ntt),tue f Lea) = aioh) 42 RT etaby ae Pe B cose Serco gyn het Oe. & ot =Sfohy [w92 Sot scosmma +82 SF orm] War 7 net el(o~ arrest) ~ e@- aireeson} Find the fourier geates to epee 4d= 2a, when —2 Sree ( Ad = 1% § an even tunckibo tove L=g det Yas) = 4 2 arncos-*t | 2 80+ Hayes M1 _5.@ . La not 2 ay ic ; 902 4 Shenae See jae gen & x) = a 73 4 [deo co NE on dt = J ‘oi oes om dx (7 ed so 2) “feos HUF ome (O2 *) “f oft Ae (Say Ce) “eee sfosrmr +0 a wg} | Op ob, cosnTt dhs / i. pn . ganged the “values -Of a ae Ay ints) ,we get we ABs 20 tno ; sm ie} ce 43 co tl ye) Bae B- 1 pie ; a ets ; amt_| cosy Bigs a+b, feos - Lome 78 WR 3 oe cos 3 es ei tag, the pourtar gertes Por PLY) = = Ur Win 02H 22d. land hence decluce 1 sheet heap =z, the fourier “govtes for $i) sala 42 fo the fotenal (mal) te given by ‘ fo) -% + 2% cos (ME) + 2 yp 80(SH) 0 bee oh dts fee eae wi? e “es if nt® ae 2% ed 1 (at) = as 1 feo cos (trh)aa = st = cos MH Pr | ee one unt oP oy we AAS =I re 3 “ihe tate-Dgtn( + tfal- -sa)eas De (tap, E 3 B el i> Is = fhe - Je sath Pe soften HE) mffean 32) 8 ame det a gif Et “fi _Y).7 tf) “ey ya fe) or PEI go = yh de lees a Subset tubing he volurs ef 9,2, And by tl) ibe get RO = My | 2 al eos(ON) : i bel He i | fey ghia gp) ong. Eo (ot nr) Poduckton : aie Busing 420 in@) , we obtaty ul oO. onal He 2 yx Gh 2 Te ee Sie | Pind the fourcter gortes expansion fer 46) if | tea) fae MH -adaco G | by te o4ner | Here bag : ‘ deb dev) = Be + Boge mB gin M4 mm (: “fe 3)-30 Then a, wf “ififiapeg : -t[ear(2}] ae ard nt $ Seo) Oo a ~Ctde2) a (fast dn fa cog Mae] an * L 92) ob fa) + |) @ | 2a a = a Re ny a -A5 eo) sv O = =0 oshen n ts even | eB, when n Ss odd. ew ‘rally b= > 4 fi Fea) ste MY dat | ou 2| fash ni anf see by at afd + al Sin OA an a : aa by =a. ! 4 Boson tihséin ott] 3 a =) = = wk 4 rain io a | staking the values 15h: An Q hee § : 2 £t 4 gy L-a. A, cos SA 2 ape neh 135? - pie Stes 244 gaample 3‘ Find the Fourier series to represent 1 — x? in the interval -1 roe 2 =4 [ine jnas fo] : 2 -l 2 ola [rereos re (-1=2) 2 “le cota cate cos = a =f £ fata =k j cos a, since cos oi is even function _ (nme)) sin } 2k nm _ 12 )| = —-sin— “hig m2 2h = 0 when nis even Ez k mm =2@=0, 7 : 2) since sin“ is odd Sostitng the values of a’s and b’s in (2), we get _ 1 ge3tt, 1 Se 17, S@ £428 [on gg, 388 | 2 — < cos EE 2 FD TSO OOS Dt which is she ‘required Fourier series, ¢ Halt Reinge,,.. » Feeniex ses arg. - Range Cosine SevieS : 20 jos Fore Gey Senne: esi va an f ) ae - Viet | te Ceantix 4 eee where: ba = 2 — Pind the. tal}-range! Sine: “peites ‘of 40> 1 to {o,/) The. touster gine gevtes of tt) | o (6/4) ig qhen by ‘foyer = 22 by tts see. b= Shay enamt a aft STOMA hy. Cg acon =A 2. wel ces t) = 3. CEO ~ bal? s when n fs even TOO "Bey when nts add thence the T senjitied fourster seviog fs 4ea) Sapte &in Pt Come) = 2 pant) heey t= A(ate Tag ato STE 4b SPOT ef the h COR ine Series ay oot on in the tonal (0/2) i ae | 2 gortes efes? | os Tal 1-2 4 Loess (7) | 50: ! | |pen A= Affe +8 ngs a, afin ye a(t} roe 2 iL ee 0 “fh aos LU mn an = at cos mde ee “eee so l i pr): “ye Jo? eto : Fal] fio. ntT=0) Fl cit ay Cs fo A : | a te 0 “even Qy = lw tp nts sod... a! ugg, te “che values ofa 1 oie AL eng ON hee a Se -) 0 y) we get 8. Fok . ‘gir top Ak ‘cng B44 OS ihe a om i ee fia) = Z bot OT 6" Aces Then by = Pf 4) &o - fat da ppt ee dL z) OTN hes at CSF) 2 afin O18 iy) pe Pe af yt 8 gubgtttating (9 to&) we a qe See or Laer 7 fel wee Mee ae Nod the 2 cosine sextes® or #2) » 2184) fy) 04% Za cea Races an, of, the herfer + we 1m # ter et ‘The voqpst ved ovis 88 of the form ila) -.% + 2, C68 ia’C "tex. —¥ O whdite. a - 4 fs cos AA. “Afra e os ATH om "CG J=2) © nT d : , fe 2) cos" ran oe : eg oat ae & et daing foteqration by parts |” a ah F ayo > =a nr oe ~ galt ‘] ie when nis even Oye 0, when nf odd and ao 3tei- dts af, | fay ‘the. vanes eat) we got ile xs tt) I gre 2-16, fee ee 4. eos wi oS aT, 83 o. ie uv 1 coS a’ i T+ Se puto () we gee ptr ae} ta 3 2d (csr einen tse casaT +, art boson. -) ale a 2d te = ee bier - ot Sgegthey). ye vee rm “ a db = or 4 se pte fied the fouster extos tor 4). Sata tn 0223 and wud <4) rs the fourfer sine gertes he itd= at tn the. interval, (od $ qhen, by Hu) = = b,ein( 2) ce ye HE. bp. 5 fowtee = 9 foe» CR), my cas aH af e) ae (2): pa 21) ow | oe z a “ape 3 etd -cos(e! Bi nile stig Bes) fe emia Bese laa i} by = eeeash even 1 bien] eae spents odd. ————— ntti SSE SOT Te Hen i) i - 1 HE 2g {at fh. sees +8 C8) Tthis the “ty trol fourfer: gine series. Prgevalls ttheoven): Foomula! L Puan! a J Festan at [ahz 2 ‘cabsee)] a we Knows the Hourrtes gerier of-fC\)In the . fnterval C4, £) tS P Fei) =90 4 2 (agpes HE t bolo mt) _»@ voheve. : ; ee Og= pone, { ee Efe a > 6 bo=} ae ded a Mi dx a F Mutbyng @& by tx) we ge | Gea: So4e0-+ Zeotidos Tt g 2 fe)" STI 39 lu egrating 6 a oe fe debs Prot 12 ger peat date Se a 2 teas mT ay if Sb. “flow om pi 30 “4 (2) ( becoines fi eae 1 OF 4 Slds+ soso so “Thi FS the parsevalls (théoreny:dorrnuda. gy oe (atelf rage coblne gents): fee x 4d a Zap) yaort Ce Range Sine ere): 5 ya) a= [2 5] ind tbe comple fourior goxtes of PCa) =o tf a L& zm and fea)= PoGem) .. ee complex form 4 fone series “F ae welt giao bY» ft) = Zoe! ae ae fe atte ope? Me af’ ety, i soyh, tox “xf tite oft a sgh lt” ys “x [ee ame a) aT | (er- amen(ee “Zeosn isin ! “ate a(t) = COSNT = ey” oe, ee ttto Chin (etodlieta) ond, ee T_ gstah (m) oO EI)? ae) atx — ie “Ths the complen -fourier certes of 400 és a gtohtt So tag e™ (stom 024) fad the complen fourier perios Of e tunekion Rta) ok, he the compl a ouster serfes of tex) ool ts Gin) by ta. 2 Ch em ->@ ; neo shove Cy =k Steal dup NeOvk bt2-- where a, = Lf Tether da” pale oly Ls eater)» (Sans eel: "6 Ay (i+ Po) Jae - Tw : iT lrio) grat “ain (2 ui ool fe "eames "(cant tstom)] antes) Ha eat ene Le r)) artad) : Ahr = Stal (1-fcosnit = “eBtet) sob ran) STC) “Thus ; the. Bates orm of the ovate Serres OL fOD oR Rae + 4s) = Stohr. Serge im. ‘ a

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