Topic 23 Notes. Jeremy Orloff

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1 Topic 3 Noes Jerem Orloff 3 Fourier sie ad cosie series; calculaio ricks 3 Goals Be able o use various calculaio shorcus for compuig Fourier series: shifig ad scalig f( shifig ad scalig differeiaig ad iegraig kow series Be able o fid he sie ad cosie series for a fucio defied o he ierval [, L] 3 Udersad he disicio bewee f( defied o [, L] ad i s eve ad odd periodic eesios 3 Iroducio This opic is spli io wo subopics Firs we look a a few more calculaio ricks The commo idea i hese ricks is o use he Fourier series of oe fucio o fid he Fourier series of aoher A simple eample is if we scale a fucio, sa g( = 5f(, he he Fourier series for g( is 5 imes he Fourier series of f( Ne we ll look a fucios f( ha are ol defied o he ierval [, L] This is i preparaio for our laer sud of he wave ad hea equaios This fucio is o periodic i s o eve defied for all B eedig f( o a eve or odd periodic fucio we ca wrie he origial fucio f( as a sum of sies (sie series or a sum of cosies (cosie series 33 Calculaio shorcus Oe of our goals is o avoid compuig iegals for fidig he Fourier coefficies of a periodic fucio I his secio we ll cosider he followig calculaio shorcus for compuig Fourier series: Simplif compuaios for eve or odd periodic fucios (Alread covered i he previous opic Use kow Fourier series o compue he Fourier series for scaled ad shifed fucios 3 Use kow Fourier series o compue he Fourier series for he derivaive or iegrals of fucios Eve ad odd fucios were covered i he previous opic, so we wo go over hem agai here

2 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 33 New series from old oes: shifig ad scalig Firs, if ou scale ad shif f( he ou scale ad shif is Fourier series To avoid burdeig he saeme wih oo much oaio we sae i for period fucios You ca eed his easil o a period Suppose f( has Fourier series f( = a + a cos( + b si( Theorem (Scalig ad shifig f( The scaled ad shifed fucio g( = cf( + d has Fourier series g( = cf( + d = ca + d + ca cos( + cb si( Theorem (Scalig ad shifig i ime The fucio g( = f(c + d has g( = f(c + d = a + a cos((c + d + cb si((c + d This is o quie i sadard Fourier series form, bu i is eas o epad ou he rig fucios o pu i i sadard form The res of his subsecio will be devoed o a eeded eammple illusraig hese echiques usig ou sadard period square wave whose graph is show jus below Eample 3 (Eeded eample The graph of f( looks like his: f( = 3 3 Graph of f( = square wave We kow ha he Fourier series for f( is f( = 4 si( ( Now we will use his o fid he Fourier series for scaled ad shifed versios of f( We ll defie hese ew fucios graphicall, we could also wrie dow formulas if we waed (a f ( = (b f ( = f ( = + f( = + 4 f ( = f( = 8 si( si(

3 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 3 (c f 3 ( = f 3 ( = ( + f( = + si( Ne will look a wha happes if we scale he ime (d f 4 ( = f 4 ( = f( = 4 si( I s a lile rick o see ha f 4 ( = f( I hik abou i wo was Firs, he picure shows ha we wa f 4 ( = f(, which is give b f 4 ( = f( Secod, f 4 ( has period so is Fourier series should should have erms wih frequecies This las eample ivolves shifig he ime (e f 5 ( = / / f 5 ( = f( + / = 4 si(( + / Tha is, f 5 ( = 4 ( si(3 + 3/ si( + / + + = 4 ( cos 3 cos Differeiaio ad iegraio If f( is periodic he he Fourier series for f ( is jus he erm-b-erm derivaive of he Fourier series for f( A eample should make his clear Eample 3 Le f( be he period riagle wave from he previous opic, f( = o [ ] I s clear ha he f ( is he square wave Check had he derivaive of he Fourier series of f( is he Fourier series of f ( f( = 3 3 Graph of f( = riagle wave aswer: From he previous opic we have he Fourier series for f( is f( = 4 ( cos 3 cos 5 cos Thus, f ( = 4 ( si 3 si 5 si We kow his is he Fourier series of our 3 5 sadard square wave as claimed

4 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 4 Deca rae of Fourier series Noe ha f( has a corer ad is coefficies deca like /, while f ( has a jump ad ad is coefficies deca like / Noe also, how differeiaio chaged he power of i he deca rae Differeiaio of discoiuous fucios Term-b-erm differeiaio of Fourier series works for discoiuous fucios as log as we use he geeralized derivaive Eample 33 Le f( be our sadard period square wave Fid f ( ad he Fourier series of f ( Graph f ( aswer: Because f( has jumps we mus ake he geeralized derivaive: We kow f( = 4 f ( = 4 ( δ( + + δ( δ( + δ( + si( So, akig he erm-b-erm derivaive f ( = 4 cos( You ca check his b compuig he Fourier coefficies of f ( direcl usig he iegral formulas 8/3 8/3 8/3 8/ /3 8/3 8/3 8/3 Graph of f ( = impulse rai Eample 34 Term-b-erm iegraio Suppose ha Wha is h( = f( = + cos( + cos( f(u du? + cos(3 3 aswer: We iegrae he Fourier series erm-b-erm o ge h( = + cos(4 4 + f(u du = C + + si( + si( + si(3 3 + Noe This is a feaure of iegrals: jus because f( is periodic does mea he iegral of f( will be periodic I his case, he -erm shows ha h( is o periodic So we ca officiall sa we have a Fourier series for h( Noeheless we have a ice series for h( ha ca be used i ma compuaios Here s oe more eample of iegraio I s ver cool, bu we probabl wo ge o i i class Eample 35 For our amuseme Cosider he period discoiuous sawooh fucio f( = for < <

5 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 5 f( = 3 4 Graph of f( = discoiuous sawooh Sice f( is odd, wih period we kow ha he cosie coefficies a = For he sie coefficies i is slighl easier o do he iegral over a full period raher ha double he iegral over a half period: b = Thus f( = si( + si( + si(3 + 3 Now, le h( be he iegral of f(, specificall si( d = si u si 3u Le h( = f(u du = si u du 3 = ( cos( + cos( + cos(3 3 + = cos( a The DC erm is = This is a ifiie sum, bu we ca compue is value direcl usig he iegral formula for Fourier coefficies O [, ], h( = u du = 4 Thus, a = 4 d = 3 So a = 6 = We ve summed a ifiie series! 34 Sie ad cosie series; eve ad odd eesios 34 Defiiio of sie ad cosie series I his secio we will be cocered wih fucios f( defied o a ierval [, L] We sar b saig he heorem o how o wrie fucios as sie ad cosie series Afer ha we will use wha we kow abou Fourier series o jusif he heorem We will eed sie ad cosie series whe we sud he wave ad hea equaios Bu firs a impora semaic disicio: Fourier series are defied for periodic fucios A fucio defied ol o a ierval [, L] cao be periodic, so i does have a Fourier series The figures below show a fucio defied o he ierval [, ] ad a period fucio defied over he eire real lie

6 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 6 3 Lef: fucio defied [, ], ca be periodic Righ: periodic fucio Sie ad cosie series Wihou furher ado we sae how o wrie a fucio as a cosie or sie series ad how o compue he coefficies for he series Noe he saemes look ver much like he oes for Fourier series Cosider a fucio f( defied o he ierval [, L] f( ca be wrie as a cosie series: f( = a + a cos(/l where a = L f( also has a sie series: Impora f( = b si(/l where b = L L L f( cos(/l d f( si(/l d Sie ad cosie series are abou fucios defied o a ierval The sie ad cosie series have values for all, he agree wih f( o (, L (assumig f( is coiuous Sice f( is ol defied o (, L his is usuall wha we wa 3 Compuig a ad b ol depeds o f( o he ierval (, L 4 We will make use of sie ad cosie series whe we do he wave equaio 34 Eamples of sie ad cosie series Firs we ll give some eample compuaios We ca do his b mechaicall applig he formulas We ll gai more isigh io hese series afer we have see he proof jusifig he formulas for he coefficies Eample 36 defied o [, ] Fid he Fourier cosie ad sie series for he fucio f( = si( aswer: Cosie series L =, Usig he formula for a : a = [ si( d = cos( + = 4 B applig he formula si(a cos(b = (si(a + b + si(a b/ we ge: a = { for odd > si( cos( d = 4 for eve > (

7 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 7 Thus, f( = 4 ( cos( 3 + cos(4 5 + cos( = 4 Impora This is ol valid where f( is defied, ie o [, ] Sie series f( = si( o [, ] >, eve cos( 343 Eve ad odd periodic eesios The proof of he formulas for he sie ad cosie series coefficies urs ou o be a sraighforward applicaio of Fourier series for periodic fucios The rick is o view he fac ha f( is ol defied o [, L] as a opporui isead of a limiaio To do his we eed o defie eve ad odd periodic eesios of f( Defiiio If f( is a fucio defied o he ierval [, L] he he eve period L eesio of f( is he period L fucio { f( for L < < f e ( = f( for < < L To visualize his, we firs reflec f( i he -ais o ge a fucio defied over oe period [ L, L] We he eed his o be periodic over he eire real lie Origial fucio f( Refleced fucio o [ L, L] L L L Eve period L eesio f e ( 3L L L Makig a eve period L eesio The odd period L eesio of f( is defied similarl, wih { f( for L < < f o ( = f( for < < L L L 3L 4L 5L To visualize his, we firs reflec f( hrough he origi o ge a fucio defied over oe period [ L, L] We he eed his o be periodic over he eire real lie

8 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 8 Odd period L eesio f o ( 3L L L L L 3L 4L 5L The odd period L eesio 344 Proof of he formulas for he sie ad cosie series As we said, usig he eve ad odd period L eesios his is a sraighforward applicaio of Fourier series for periodic fucios We will give he argume for he cosie series The sie series is similar We have f( defied o [, L] ad he eve period L eesio f e ( Sice f e ( is periodic i has a Fourier series ad sice i is eve his series has ol cosie erms Tha is f e ( = a + a cos(/l Usig he smmer of eve fucios we kow a = L he ierval of iegraio we kow f e ( = f( Therefore L f e ( cos(/l d Bu o a == L This is he formula we waed o prove L f( cos(/l d Sie series You should r provig he formula for he sie series coefficies Oce more o emphasize he grammar: f( is defied for i [, L] f e ( ad f o ( are defied for all The hree fucios agree o [, L], ie f( = f e ( = f o ( for i [, L] The cosie series for f( is jus he Fourier series for f e The sie series for f( is jus he Fourier series for f o ( This is illusraed i he followig figure 3L L L L L 3L f( i orage, f e ( i ca, f o ( i purple All hree are he same for < < L

9 3 FOURIER SINE AND COSINE SERIES; CALCULATION TRICKS 9 Eample 37 Fid he sie ad cosie series for he fucio f( = defied o he ierval [, ] aswer: Sice he odd period eesio is our sadard square wave we have he sie series f( = 4 si( Sice he eve period eesio is he cosa fucio f( = we have he cosie series f( =

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