FOURIER SERIES. Series expansions are a ubiquitous tool of science and engineering. The kinds of

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1 Do Bgyoko () FOURIER SERIES I. INTRODUCTION Srs psos r ubqutous too o scc d grg. Th kds o pso to utz dpd o () th proprts o th uctos to b studd d (b) th proprts or chrctrstcs o th systm udr vstgto. Powr Srs r usd th ucto udr study s cotuous d hs drvtvs o ordrs th trv [, b] whr pso s to b md. [Immdty rvw th sso o powr srs wth mphss o Tyor thorm]. Thr r two m purposs, mog othrs, or srs pso: () Appromto o ucto [ t or roud pot c bout whch pso sts] s ws ustrtd th cs o powr srs ppromtos o s d cos uctos. (b) Soutos o drt qutos. A summry ssts o powr srs pso oows: () Cotuty d drtbty t ordrs ovr th trv [, b] c [, b], ( ) ( c)! ( c) ( ) c (b) Th powr srs pso o ucto, t sts t pot c, s uqu. Hc, o c gt t usg Tyor thorm d mthmtc ducto, th bom thorm, or othr mthods (.., drtto or tgrtos usg th uorm covrgc thorm, ppcb.) (c) Th bov rrcd drtto or tgrto r ppcb to powr srs

2 Do Bgyoko () Bcus o th uorm covrgc thorm: A powr srs s uormy covrgt wth ts crc o covrgc. Powr srs psos r smpy psos o ucto trms o th st: [,,,,, ]. Not tht ths st o bss uctos s t. Gr Srs Epsos d ot b powr srs pso. Wht s commo to psos, howvr, s tht th uctos usd to pd r ry dpdt. I th cs o powr srs, o ss tht,,,, r ry dpdt [th rto o y two o thm s ot costt]. Vry ot th uctos ormg th bss sts r orthogo o to th othrs; ths s howvr ot cssry th bss uctos or powr srs pso r ot orthogo to ch othr. Comptss o th bss st s othr commo tur o pso. Th comptss o bss st s subt cocpt ps mmdty rvw th dscusso o t cocto wth orthogo uctos. Uk th cs or vctors,,, -dmsos, comptss or ucto vry ot rqurs tht thr b t umbr o bss uctos. ; & j; d ; d, j, k rspctvy costtut compt sts o bss vctors or,, d -dmsos. Ths sts r t. But [,,,,, ] s t st usd or powr srs psos. I gr, d udr spcd codtos, othr uctos c b pdd trms o compt (.., gry t) st o bss uctos tht r ry dpdt. Th spcd codto, th cs o powr srs psos, r gv by th Tyor thorm [.., cotuty d drtbty t ordrs]. Lgdr poyoms, Chbyshv poyoms, Hrmt poyoms r mps o compt (d orthoom) sts o bss uctos. S psos trms o ths sts Physcs 4 or your tt book. Th prtcur psos w dscuss t r th Fourr 4

3 Do Bgyoko () psos. Th bss uctos or ths psos r [s, s, s ], or [, cos (), cos () ] or [,,,......]. II. FOURIER EXPANSIONS: GENERALITIES From th bov dscussos, t s pprt tht o my b b to pd som uctos trms o s(), cos(), or swrd by th Drcht Thorm.. Th ky qusto s to udr wht codtos s Drcht Thorm or (Drcht codtos): th v codtos (c-c5) I (c) () s prodc wth prod T d (c) btw d () s sg vud, (c) hs t umbr o trm (mm d mm) o [, ], d (c4) t umbr o dscotuous, d (c5) ( )d s t, th th whr Fourr srs ( ) cos( ) b s( ), ( ) cos( )d d b ( ) s( ) d, sts d covrgs to () t pots whr () s cotuous; t jump (dscotuous) th Fourr srs covrgs to th mdpots. Cry, ths thorm s, or Fourr psos, wht th Tyor thorm s or powr srs psos. A pso quvt to th bov o s: ( ) C C C C... ( ) C, whr C ( ) d 5

4 Do Bgyoko () S probm o. pg 7. O c sy gt o o ths two orms rom th othr. Ps sov ths probm s prt o th homwork du o Frdy t wk. (Tody s Frdy!) Ky It s mportt to ot tht th ucto () s v,.., ( ) ( ), ( ) cos( ); th ucto s odd,.., ( ) ( ), th th ( ) b s( ). () T Th gr orm gv th Drcht thorm must b usd th ucto () s thr v or odd. Ky my prodc uctos hv prods drt rom. Th Drcht thorm s sy ppcb to ths uctos oowg smp chg o vrb. T To pd () Fourr srs, cosdr: s,cos, T T wth T. Hc, th prods o s,cos, d r qu to! T Th Drcht thorm pps s oows: ( ) cos b s whr ( ) cos( ) d 6

5 Do Bgyoko () b d ( ) s d or, trtvy, ( ) C, whr C ( ) d. III. PRACTICAL EXAMPLES Emp (s ttbook by Mry L. Bos, pg 4) () ( ), < <, < < - Th prod T ( ) ( ) s s( 5 ) s s 6 s Notc tht ths pso s ot sy put compct orm (.., ) Emp (s th ttbook o pg 9), < <, < < ( ) ths ucto, s gv, s thr v or odd 7

6 Do Bgyoko () ( ) d d d C ( ) C ( ) odd or - v or zro, d C so ( )... ( )... 5 s 5 s s Not w tht ths s ot Fourr s s srs bcus o th rst trm. Th bov mps c b usd to ustrt th ct tht th choc o th org o s, th grphc rprstto o ucto, c b utzd to hv som uctos v, odd, or wth o prty. I th grph mp two, movg th org to th pot (/,) coud d to v ucto. Homwork: Vry (or dsprov) th prty chgs suggstd bov by mkg th pproprt chgs o th org o th s d obt th Fourr psos o th rsutg uctos. Obt sprdsht mpmtto o ths psos d o th o gv mp, or vu o th prod qu to. Icud t st t trms or hrmocs. Your submsso or th mpmtto o th Fourr pso shoud cud grph o th ucto suprmposd upo tht o th rst 5 trms d tht o th sum o th rst t trms. Ps commt o th Gbbs phomo d o th vu o th sum o th hrmocs t th dscotuts o th ucto. Equy commt o th ct tht th chgs o coordt org dd ot ry ct th physcs o th probm X

7 Do Bgyoko () (though v, odd, d o-prty ucto my sm toty drt). (Ps do ot orgt Probm N., pg 7). S mp sprdshts prprd by D. Bgyoko. Thy r o most o th computrs Room.. Fourr Epso o (), or to.4;, or.5 to.4;-, or.5 to.9;, or to.5. Physcs Mthmtc Physcs 9