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of only sine functions or only cosine functions Recall that the Fourier series for an odd function defined on [−L, L] consists entirely of sine terms Thus we might  

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To solve a partial di erential equation typically we represent a function by a trigonometric series consisting of only sine functions or only cosine functions Recall that the Fourier series for an odd function de ned on [ L;L] consists entirely of sine terms Thus we might achieve f(x) = X1 n=1 a

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336

FourierSeries

Mathematiciansoftheeighteenthcentury, includingDaniel Bernoulliand LeonardEuler,expressed theproblemof thevibratory motionofa stretchedstring throughpartialdi ff erentialequationsthathad nosolutionsin termsof ''elementaryfunc tions.''Theirresolutionofthis di ffi cultywastointroduce infiniteseriesof sineandcosine functionsthat satisfiedtheequati ons.Inthe earlyninete enthcentury, JosephFourier, whilestudyingtheproblemof heatflow, developedacohesiv etheoryof suchseries. Consequently,theywerenamedafte rhim.Fourier seriesandFourierinteg ralsareinvest igatedinthis andthenext chapter.As youexplorethe ideas,noticethesimilaritiesand di ff erenceswiththechapters oninfiniteseries andimproper integrals.

PERIODICFUNCTIONS

Afunction fðxÞissaidto havea periodTortobe periodicwithperiodTifforall x,fðxþTÞ¼fðxÞ,

whereTisapositive constant.The leastvalue ofT>0iscalledtheleastperiod orsimply theperiodof fðxÞ.

EXAMPLE1. Thefunctionsin xhasperiods 2?;4?;6?;...;sincesinðxþ2?Þ;sinðxþ4?Þ;sinðxþ6?Þ;...all

equalsin x.However,2 ?istheleastperiodortheperiodofsinx. EXAMPLE2. Theperiodof sinnxorcosnx,wherenisapositive integer,is 2?=n.

EXAMPLE3. Theperiodof tanxis?.

EXAMPLE4. Aconstanthas anypositivenumber asperiod. Otherexamplesofperiodi cfunctionsareshownin thegraphsof Figures13-1(a),(b),and( c)below. f(x) x

Period

f(x) f(x) x x

Period

Period

(a)(b)(c)

Fig.13-1

Copyright 2002, 1963 by The McGraw-Hill Companies, Inc. Click Here for Terms of Use.

FOURIERSERIES

LetfðxÞbedefined intheintervalð?L;LÞandoutside ofthisintervalbyfðxþ2LÞ¼fðxÞ,i.e.,fðxÞ

is2L-periodic.Itis throughthis avenuethat anew functiononan infinitesetofrealnum bersis created fromtheimag eonð?L;LÞ.TheFourierseriesorFourierexpansioncorrespondingtofðxÞisgivenby a 0 2 X 1 n¼1 a n cos n?x L þb n sin n?x L

ð1Þ

wheretheFouriercoe fficientsa n andb n are a n 1 L L ?L fðxÞcos n?x L dx n¼0;1;2;... b n 1 L L ?L fðxÞsin n?x L dx 8

ð2Þ

ORTHOGONALITYCONDITIONSFORTHE SINEANDCOSINEFUNCTIONS

Noticethatthe Fourier coe

ffi cientsareinteg rals.Theseare obtainedbystartingwiththe series,(1), andemploying thefollowingproperties calledorthogonality conditions: (a) L ?L cos m?x L cos n?x L dx¼0ifm6¼nandLifm¼n (b) L ?L sin m?x L sin n?x L dx¼0ifm6¼nandLifm¼n(3) (c) L ?L sin m?x L cos n?x L dx¼0.Wheremandncanassumeany positive integervalues . Anexplanation forcallingtheseorthogonalityconditio nsisgiven onPage342. Theirapplication in determiningtheFouriercoe ffi cientsisillustratedin thefollowing pairofexamplesand thendemon- stratedindetailinProb lem13.4.

EXAMPLE1. Todeterminethe Fouriercoefficienta

0 ,integratebothsides oftheFourier series(1),i.e., L ?L fðxÞdx¼ L ?L a 0 2 dxþ L ?L X 1 n¼1 a n cos n?x L þb n sin n?x L no dx Now L ?L a 0 2 dx¼a 0 L; L ?l sin n?x L dx¼0; L ?L cos n?x L dx¼0,therefore,a 0 1 L L ?L fðxÞdx

EXAMPLE2.Todetermine a

1 ,multiplybothsides of( 1)bycos ?x L andthen integrate.Usingthe orthogonality conditions(3) a and( 3) c ,weobtaina 1 1 L L ?L fðxÞcos ?x L dx.Nowsee Problem13.4. IfL¼?,theseries( 1)andthecoe fficients(2)or(3)areparticularly simple.Thefunction inthis casehasthe period2 ?.

DIRICHLETCONDITIONS

Supposethat

(1)fðxÞisdefined exceptpossiblyata finitenumber ofpointsinð?L;LÞ (2)fðxÞisperiodic outsideð?L;LÞwithperiod 2L

CHAP.13] FOURIERSERIES337

(3)fðxÞandf 0

ðxÞarepiecewise continuousinð?L;LÞ.

Thentheseries (1)withFourier coefficientsconverges to

ðaÞfðxÞifxisapoint ofcontinuity

ðbÞ

fðxþ0Þþfðx?0Þ 2 ifxisapoint ofdiscontinuit y Herefðxþ0Þandfðx?0Þaretheright- andleft-hand limitsof fðxÞatxandrepresent lim ?!0þ fðxþ?Þand lim ?!0þ fðx??Þ,respectively.Forapro ofseeProblems 13.18through13.23. Theconditions (1),(2),and(3)impos edonfðxÞaresufficientbutnotnecessa ry,and aregenerally satisfiedinpracti ce.There areatpresentnoknown necessaryandsu ffi cientconditionsforco nvergence ofFourierseries. Itisof interestthat continuityof fðxÞdoesnotaloneensureconvergence ofaFourier series.

ODDANDEVEN FUNCTIONS

AfunctionfðxÞiscalled oddiffð?xÞ¼?fðxÞ.Thus,x 3 ;x 5 ?3x 3

þ2x;sinx;tan3xareodd

functions. 4 ;2x 6 ?4x 2

þ5;cosx;e

x þe ?x areeven functions. Thefunctions portrayedgraphicallyin Figures13-1(a)and13-1ðbÞareoddand evenrespect ively, butthatof Fig.13-1( c)isneitheroddnoreven. IntheFour ierseries correspondingtoanodd function,only sinetermscanbepresent.Inthe Fourierseries correspondingtoaneven function,onlycosineterms(andpossibl yaconstant whichwe shallconsidera cosineterm) canbepresent .

HALFRANGEFOURIE RSINE ORCOSINESERIES

AhalfrangeFourier sineor cosineseriesis aseriesin whichonly sineterms oronlycosine termsare present,respectively.When ahalfrangeseriesc orrespondingtoagiven functionis desired,the function

isgenerally definedintheinterval ð0;LÞ[whichishalf ofthe intervalð?L;LÞ,thusaccounting forthe

namehalfrange]andthenthe functionis specifiedas oddoreven, sothatit isclearly definedintheother halfofthe interval,namely, ð?L;0Þ.Insuchcase,we have a n

¼0;b

n 2 L L 0 fðxÞsin n?x L dxforhalfrangesine series b n

¼0;a

n 2 L L 0 fðxÞcos n?x L dxforhalfrangecosin eseries 8

ð4Þ

PARSEVAL'SIDENTITY

Ifa n andb n aretheFour ierco e ffi cientscorrespondingto fðxÞandiffðxÞsatisfiestheDir ichlet conditions. Then 1 L L ?L ffðxÞg 2 dx¼ a 2 0 2 X 1 n¼1 ða 2 n þb 2 n

Þ(5)

(SeeProblem 13.13.)

338FOURIERSERIES [CHAP.13

DIFFERENTIATIONANDINTEGRATIONOFFOURIE RSERIES

Di ff erentiationandintegration ofFourierseriescanbe justifiedbyusingthetheorem sonPages 271 and272,which holdfor seriesin general.It mustbeemphasi zed,however,thatthosetheorems provide su ffi cientconditionsand arenotnecessary.Thefollowingtheorem forintegration isespeciallyuseful . Theorem.TheFourierseri escorresponding tofðxÞmaybeinteg ratedterm bytermfromatox,andthe resultingserieswillcon vergeuniformlyto x a fðxÞdxprovidedthatfðxÞispiecewise continuousin ?L@x@Landbothaandxareinthis interval.

COMPLEXNOTATIONFOR FOURIERSERIES

UsingEuler's identities,

e i?

¼cos?þisin?;e

?i?

¼cos??isin?ð6Þ

wherei¼ ffiffiffiffiffiffiffi ?1 p (seeProblem 11.48,Chapter11,Page 295),theFour ierseriesforfðxÞcanbewritten as fðxÞ¼ X 1 n¼?1 c n e in?x=L

ð7Þ

where c n 1 2L L ?L fðxÞe ?in?x=L dxð8Þ Inwriting theequality(7),weare supposing thattheDirichle tconditionsaresatisfiedand further thatfðxÞiscontinuous atx.IffðxÞisdiscontinuous atx,theleftside of( 7)shouldbereplac edby

ðfðxþ0Þþfðx?0Þ

2

BOUNDARY-VALUEPROBLEMS

Boundary-valueproblemsseek todeterminesolutionsof partialdi ff erentialequationssatisfying certainprescribedconditions calledboundaryconditions.Someofthese problems canbesolved by useofFour ierseries (seeProblem13.24). EXAMPLE.Theclassicalproblem ofavibrating stringmaybe idealizedinthe followingway. SeeFig.13-2. Supposeastring istautlystre tchedbetweenpoints ð0;0ÞandðL;0Þ.Supposethetension, F,isthe sameatevery pointofthe string.The stringis madeto vibrateinthexyplanebypullin git totheparabolic positiongðxÞ¼mðLx?x 2

Þandreleasing it.(misa

numericallysmallpositiveconstant.) Itsequationwill beofthe formy¼fðx;tÞ.Theproblem ofestablishing thisequation isidealizedby( a)assumingthatthecon- stanttension, F,issolargeascomp aredto theweightwL ofthestring thatthegravi tational forcecanbe neglected, (b)thedisplacement atanypointofthe stringisso small thatthelength ofthe stringmaybe takenasLforanyof itspositions,and (c)thevibrations arepurelytransverse.

Theforceacting ona segmentPQis

w g ?x 2 y @t 2 xCHAP.13]FOURIER SERIES339

Fig.13-2

differenceintensi onsisFðsin??sin?Þ.Thisisthe forceprodu cingtheaccele rationthatacco untsfor thevibratory motion.

NowFfsin??sin?g¼F

tan?

1þtan

2 p? tan?

1þtan

2 p ?Fftan??tan?g¼F @y @x

ðxþ?x;tÞ?

@y @x

ðx;tÞ

,wherethesquaredterms inthe denominator areneglectedbecausethevibrationsare small.

Next,equatethe twoform softhe force,i.e.,

F @y @x

ðxþ?x;tÞ?

@y @x

ðx;tÞ

w g ?x 2 y @t 2 divideby?x,andthenlet ?x!0.Afterletting ?¼ ffiffiffiffiffiffi Fg w r ,theresulting equationis 2 y @t 2 2 2 y @x 2 Thishomogeneous secondpartialderivativeequati onisthe classicalequation forthevibrating string.Associate dboundaryconditionsare yð0;tÞ¼0;yðL;tÞ¼0;t>0

Theinitialco nditions are

yðx;0Þ¼mðLx?x 2 @y @t

ðx;0Þ¼0;0 Themethodof solution istoseparat evariables,i.e.,assume yðx;tÞ¼GðxÞHðtÞ

Thenuponsubstitut ing

GðxÞH

00

ðtÞ¼?

2 G 00

ðxÞHðtÞ

Separatingvariablesyields

G 00 G

¼k;

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