588 Chopter 9 Fourier Series Methodsf[ FroblemsIn Problems I through 10, sketch the graph of the function fdefined for all t by the given formula, and determine whetherit is periodic. If so, find its smallest period.If f (t) is a function of period 2n, it isreadily verified (problem 30) thatf (t) dt (28)for all a. That is, the integral of f (t) over one interval of length 2z is equal to itsintegral over any other such interval. In case /(r) is given explicitly on the interval[0,2n1rather than on [-2, nl, itmay be more convenient to compute its Fouriercoefficients asf ", r at d, : l"'*'"t n2ta,:! I turcosntdrir Jo(29a)(zeb)andh-.f (t) sinnt dt.: I,^f (r) : cos2trt.ftt/(r):sln-.f (t) : cot2t tf (t) : sinhzrJ (t) : cos2 3rIn Problems I I through 26, the values of a period2tt functionf (t) in one full period are given. Sketch several periods of itsgraph andfnd its Fourier series.ll. f (t):1 , -tr ! / ( z-..+3, *r <t < 0:12. f rr,: l_;. ;.;a;t3. .f (t) :0, *r <t t01, 0<tat14. f (t) :3, -n<t301'-2, 0<ttt21. f (t): t2 , -ir ! t A r22. f(t):t2,0!t<2tI^23. t-rt,:lu- -'.{r{o:" [t'. }at<n24. f(t) : lsin/1, -t ! t ! r25. f(t) : cos22t, -n ! t { n(^26. r-rr, :lu' -zstso:" lsint, 0<r{r(-27. Yerify Eq. (9). (Suggestion: Use the trigonometric iden-titycosA cosa : I [cos(A + B) + cos(A - B)].)28. Verify Eq. (10).29. Yerify Eq. (11).30. Let f (t) be a piecewise continuous function with periodP. (a) Supposethat0 { a < P.Substitute u : t - P bshow thatfa+P faI taldt: I fu)dt.J P JOConclude thatfo-P fpI f"tdr: I Ittldt'Ja Jo(b) Given A, choose n so that A : nP * a witL0 { a < P. Then substitute n : t -nP to show that1.?19.15.16.17.18.l (/) : sin 3tirf(/) : cos al"2f (t) : tantJ'(t): cosh3l/(/) : lsin/lf ttt --T<t<JT0<t<2t-r{t{n+t, -n <t{0;-t, 0<ttt7T +t, -r t t <0;0, }tttr0, -ntt<*7r/2;l, -r/2ttar12;0, r/2<t{r,)4.6.8.10.l,(rt-JoIrtt-Ir l,l"lnte. f (t):20. f (t) :rA+PI31. Multiply each side in Eq. (13) by sinnt and then inregra:.term by term to derive Eq. (17).f (t)dr : I,'*' f (t)dt : fn' f {,)a,594Chopter 9 Fourier Series MethodsHence the Fourier series of / isf (t):+. +i g=g - i i sinnrt 15)5 r('-n__t ,- nand Theorem 1 assures us that this series converges to f(t) for all r. IWe can draw some interesting consequences from the Fourier series in 65).If we substitute / : 0 on each side, we find that4A*lf (o):2:;*;D;On solving for the series, we obtain the lovely summation$ I _,, I , I I 7r2la-r+2r+y*++12.13.L4.15.(16 rthat was discovered by Euler. If we substitute t : 1in Eq. (15), we getwhich yieldsIkl P'oblemsIn Problems 1 through 14, the values of a periodic functionf(t) in one full period are given; at each discontinuity thevalue of f(t) is that given by the average value condition in(13). Sketch the graph of f andfind its Fourier series.Inl. f(r):l-t' -3<r<o;[ 2. 0<r<3(^z. ftn:10' -5<t<o:11, 0<r<5If we add the series in Eqs. (16) and (17) and then divide by 2, the ,,even" termscancel and the result isf(-tl'.t,_t_t I Ik--;--:t- 2r+ *- 4r+"'=t2I2.llln2rf_I__L__l_-325272g1\:rLn2n odd '-(ll t(18 'De,l'at1E-19.5. f(t):t, -2n <t <2n6. f(t):t,0<t<37. f (t): l/1, -1 <t <l[0. o<r<l;I8. /(r): | 1. | <t <2;I[0. 2<r <39. f(t):t2, -7<.t<lI^ro. fu):19' -2<t<0"- [r'. 0 <t <21r. ftr):cosa. -l<r<l2tl.II^r. frrl :l 2' -2tr<t<o:" [-1. 0<t<2tr4. f(t):t, -2<t <2'(pelcedsns 3uo1 se)Jequnu puoqeul u€ eq ol pe,tord su.4d selJes oqnJ-esJo,\ulorD Jo rrrns eql ]€ql 6161 Ipun ]ou s€1( 1l 'peepul 'relngeJurs seunluec o.4\1 tsel eql JeAo ssecsns lnotpl^\ peul e^eq,{ueur rog Jlesrnof ro; eure; leer8 uI^\ IILt no,{-(e) ued uttuns s.Jolng ol Jelrruls ,z;o eldrllnu IEuorlBJ e se 'ecuels-ur JoJ-sJaqurnu JellIIueJ Jo suuel ur selJes eqnc-esJei.ulsrql Jo rrrns eql Sulsserdxo ur peeccns no[' 11 :Ttnuray'ureldxg alnJssecrns tduepe rno,{ s1 '(e) ued Jo seuesJerrnod eql q I Jo onle^ eleudordde uu 8urln1t1sqns ,{q,t?,tE,tZ,.'-r--r-f-flIIIssues eql elenp^e o1 fiwaily (c) ',(ltnuquocstpqcea le enp^ eqt Surlectpur '/ go qder8 oql qrle{s pueZE tL eS e€'.-l_----1_--tI,TIIIuoDeruurns eql e^uep 01 (e)gedyo serres eql osll (q) 'z > I > )t- JI 6J : (l)/ q1t,r,r/ uorlcun; u7 poued eqt Jo serles reunod eql puIC (8)(t)r+"g(t)r,d,(I-) + ... - (t)tg(t),d *(t)zg(t)d - (t)t1(t)d : 1p11)s(t)a Iletp sged,{q uorler8elur Peleeder,{q .roqg 'u eat?ap;o prurou.{1od e * (l)d 1eq1 esoddn5(r > 1> )t-) 7l : '" , a7.zlt-zx JUSOJT_,(t-)*luz> I > 0)7u: " <'02,u soJ ;rCL'rzZIfu+IB*Z+ tx9-.ttsuorl?{r[uns eqlernpep'(e) ued ul seues reunod eI11 urord (q) '.&1nurtuoc-srp qcse te onlu^ eqt Suqectpur'/ go qder8 eql qclels pue'sa*tatuoc sauas q)Da qrlq&r ot uot|cunl uTpouad aqt qdofipuo '17 q?notLlt gt sualqord m palsq sauas raunol aqi a^uaQsa.IJas s(zIuqIo'I ocnpepol I Jo enl?^ elerrdordde ue elnl4sqns (q) '.(lmu4uocstpqreo te enle^ eqt Suqecrpur '/ 1o qder8 eql qcle{s pu€u +u .l"r"tt*Z-t:(1)Jleql .^Aotls'7,> t > 0tolt : (t).{ qtlar g poued yo uollsunJ e sr / teql esoddn5 @) 'tt'(e) uud ur seues rounod oql tuog(St) 'bg ur uolte{rnuns seues oql ernpe(I (q) ',{ltnuquoc-sry qcee te enle,\ eqt Sunecrpur '/ 3o qdur8 aql qcle{s pu?7Vt> 1> Y-) I(vz> t > o)r:u DDO ,u -x, zu ryzr*r"r: ",rl-l* T - rnrs"t'\ 7 -u>:--------------- ('6r,Jlluts ,1r( I-) FI=U7U__L: - ( .gIl-u ltlurssV ,L 9,8!III>LDDO I96 nu}a\'/TS.UZL 'vu -(1tt 11-1 r-ll-L I+r\r / @puP06 vu ,-\'/TSn-r@Y:Q){I,uuu(L -\ z-**"(x -'J I tsocpT - lul.s lvz - Jso) ztTI +lur.s EiV+ I soJ nl- : lP ,ur, ,, fJ + luIsVZ * l soc tvz - tuls 217,1 -tsox tlu+ luls ?l : lp ls6 tl Icu\4- l.(ir9l+t/ @,u\- c' I (st+L:vtzuZ/*-- vrgl'uZ> t > 0roJ1uq1 moqg (e) 'v7lBql puelpr! .4Aoqs ol ZZ uelqord Jo elnllrroJ ler8elur aqt ,i1ddy 'g7'slerurou.(1od 5o sluolcUJeoo reunog 8ur-lnduroc ur InJesn sr elnuuoJ sql '(l)30(,-e) : G)19o^rtulrreprtue peleroll qry oql selouep (l)19 ereqmleq] ,&\or{s'I > 1 > gJIl : (t)l pue0 > I > l-JI0 : (r)"/teql qcns 7 poued Jo uorlcunJ e sr / teqt esoddn5 (E) '9I'(e) ued ur seuos rerrnod eq] urorJ (tt) pue(gt) 'sbg ur suort?utuns souos eql ocnpe( (q) '.fimu4uoc-srp gceo le enp^. eqt ?uBecrpur '/ 3o qder8 oql qcle{s pueu-7ur-t.(utr_- (V+_:u)J,4urs H\ /tlsoJ r-i zuvleql .{\oqs'ltz> I >0roJzl: (t)! tDIt^ u1,poued;o uorlcunJ e sr / tuqt esoddn5 (u) 'Sf)r7> t>o 'luttl\ : \r){ .vr''0> t> )rZ- '0f.)l>J>0 'tilultl:(/)/.€l:o>J>I- 'olI>i>0"//urs:(t\{'Zl969 eJUAoleAuO3 puo sol.les lounol lDjouec z'6606 Chopter 9In Problems I through 10, a function f (t) defined on an inter-val 0 < t < L is given. Find the Fourier cosine and sine seriesof f and sketch the graphs of the two extensions of f to whichthese