INTEGRAL TRANSFORMS

1, INTRODUCTION

li.l(r) =. ', then fte i.t.cral

I t',r''d - | t". ,1.- t\!)-o lo

is a function of 1[in fact, br Ch.pter 11, equarion 133). {e harc f(r,) -11 orf(., + l)1. SBrrins wirh a function of /, we have multiplied it bj a lunction ofl 3nd r,nund the delinire intcgral $ith respect to I, and so obtained a lunction of1. lhisanction r0) is calLcd an iDtegrat rdnsforn oil(r) Inrcsral transforms arc uscd in a

ilricry of applicltions, Io. e\ample, in solling ordinary difterenrl.l cquationsSection 3) oi p.rrial diiler€ntial equ.tions (Section 9) There are manl diFcre't linds,i intesral rr.nsforms {ith ditrerent names, dependins on shat function ofl and I wclulriph br a.d what the ranse of integntion is (the abore e!.mPle is called . ]Iellin:rnsfornt. I! this chlpter. rc shall considcr t{o inresrd ranslbrmi (Llplace andlourier) rhar are especirll! important in applications, tnd indicate sore of rheir uses.

On tie nerr rhrcc pises, you wili find a tabl€ of Llllac€ trdnsforns Io. refer€nce:hr'tng on pas€ 639. Fe shlt discus how to verifi the entries in rhe table (and find-ore fanstbrms if nccded). fhen in rhe followins sections, we consider the uses {e:r. male of Lrpl.cc transforms in sohing problens. Go dhe.d now to pase 639 ard:ltr brcli ro thc table as needcd. Note thri numbers precedcd by /, (for eramPle 11,l. , a.l5) reftr to cntries in the L.pl.ce transform t.ble

616 rYmGRAr. r'R,lNSF()R\{S ch. t5

A Short Table of Laplace Transtorms

L),:l(r)=0.r<01 v : Lnt = r\,,t: t, , , 1Lt ,tt

t,1

L3

L5

LJ

tl0

/_ll

L12

L13

Ll+

Lt5

ll

r f(*+1)-Hor _

ar r(t + l)0, + d"' lt + al"

6+r)\t+b)

lt+u)(f+b)

,!

b. + ")'!' a'

b

!+a

tll\ A

n15 IN1'RODUCTION 637

A Short Table ol Laplace Translorms (conlinued)

tr':l(i):0.r<01 1:/(rl:/lr): I

Lli

218

Lt9

L2t)

l,1l

L22

J,2l

L25

L2+

) = "11 ")l

rerrl.l.

I'.10,

JA

J\':

f(t) :

sin rr

I

t'o\ f)

a'+'\11+,n

I

t

!,,.

638 INTEGR.\ITR.INSFORMS

A Shon Table ol Laplace Translorms (conllnued)

),=l(t),t>0Lr,:/0):o,r<ol r-: z0) : r0) : j ! t"f(t) dt

L21

L28

L).9

/..10

L31 4, (if intesnbh)

t.32

L33

''r1.l+ I r\t - t)hltJ Jr: I qlt)h\t - r) lt.1,, Jo

(conlolurion ofs and l', oftcn$ritten as g * ,t see Section 5)

/-i5 Transforms of dcriratiles oll (ree Section 3):

J0-a), a>0(Sec sccrio! ,-.)

rLr:[srt u\' 1'o'uI0.

: slt a)u( a)

" ''dn

e *c(r)tc(r) nedns rk).1

C(! + d)

I "/r)

J"

I t\.{9lDI

"tr)

c(i)H(i)

{so

{dr r.

Llv') : lY t'oL6/'): t1-t tuo - )toL(r/') : !3Y !,to tto y'i, etc.

Lbt,\-?,y p' '.to t' 'to . ,t't "

!!.li2. THE LAPLACE TRANSFORMwc dcfinc Z(/), rh€ I-aplace rransform ofl(r) lalso writren a(r) since it is a f ncrionofll, br lhe equdiion

THD LAPL.{CE TR{NSFORV 6:]S

Sone books and tables deline t(/) as I rimes the ;ntegral in (2.1); you need ro ivatchror this in usins a diflcrcnr boot. Ther€ is a wide variation in notarion used;n rriring(2.1)r the Eriables r and I or r and r are ofren used instead of I and / and oiher lefiersrre used instead of t1nd r.. We shall consisrcndr use rhe norarion of a smlll lcftcr forrhc funcdon of r, lnd ihe correspondins capital lcttf for rhe iransfo.h iybich is atinction of1, for cxample/(r) and r'0) ors(r) and G(r). erc. Nore fron (2.1)rhat sincc\e intesrate fronr 0 ro .., Z(/) is rhe slhe no matter howl(r) is deiined tbr nesativcI HoFcrer, as wc shaLl see i.rer (Section 6), it is desirablc to dcfine /(r) : 0 for r < 0.

It is rery conrcnienr to h.ve a table of correspondins l(r) and -F(l) shen Fe arelsing the l-aplace transform to solve problems I-et us calculate somc of rhe entries in-hc shoft rable of Laplace transforms siven on pases 63G638. To obrain -t l in thciable, we substirutc t0) : 1 into (2.1) and find

,) 1)

Rchare assume d I > 0 to nake . !r zoo ar thc upper limit; if, is cohpiex, as irna!he. then rhe real paft oft (Rcr') mun be posirile, and rhis is the restricrion wc h.!edated in rhe tlble for ,1. For ,2. wc halc

t.1) lltpt-1,""a' R{ (, ,r'r..h l+a$e coutd conrinue in this way to obt.in the funcrion |(rr) corresponding to each l(r)

5r usins (2.1) and ev.luating the integral. Howcrcr, thcrc are some easier nethods*hlch we now itlusrr.re. First observe that rhe Laplace transform of a sun of iwonncrions h the sum of thcir L.pldce iranslbrnsi also the rr$sib.m of nO is cL(f)rhcn. is a constmr:

i. r lF\r):l I P Ptdt:Jolo|

fk)e ,'tt + I

ztl0)+s(r)l:J

:I"Lt fa)t: I"

I l0) + rk)le '' nt

t.1) sk)e 't dt : L(f) + LG),

f\t)" tt ,]t : , L\J ).

6,10 INTEGRAL TR'{\Sll ORMS

1n mlrhcmatical b.guage, we s.y rhat thc Lapl(e transform is /;.a/ (or is a lin.dt,r./arrl see chapter 3, section 7)-

\ow lct us verib rS. In (2.3), repl.ce a b] a;thcn wc halc

(2.s)f(i) : .t" : .os Lt + i sin dt,

I D+eI(r):-: - -----, Rc(, r)>0n-u t-+a-R€menberins (2.1), Fc can write (2.5) as

(2 o) / (cas J, ' , \in a/) / rLU\ a/l - ,/ tnn ,r' !'+a' ,'+a'Similarlv, r.placing , by ia in (2.3), we ser

(2.7) / ((o. ,, - , .in arl - ,' , ,o ,. R<Lp ,a) ot'+o- ,-+t-Addins (2.6) lnd (2.7), se ser Z.{i by subtractins, we set Zl.

To lerif) ,l I, start $irh Z'1, lancly

(2.8) ,n* ",, : f- " ! $s at ,1t : --\..)" !' + a'

Direrenriarc (2.8) with respeci to tbe Frdmerer u to set

'. l\ ZllI t ttt-t\i\dnd-, tl'r tar'

I )r,| , ''t,in t! at

rvhich is /-l l. \fus of findins othcr enlrics in the table are outlined in rhe problems

PFOBLEMS. SECTION 2

l. Usins dre dennnion or thc f aundnn (Chrpter_U, Section J). lerifi ,5 rnd /-a in drcLiplace rrnsroml tablc Sho* thlt ,(l/!, : vtr,r'

2. lly usinS l,:1, ye.ilr t7 drd 1-8 nr the Lrplace lnnslorn ttble

J. Using eirho ,2. or 1,3 and a,l, yerifi t9 dnd /-10.

,1. B! dill endtins rhc rfproprixte ibrnula qith respcci to d. veriii l-12.

5. Br inrcgnting rhe rppropridrc lifnul, nith resp€ct to ". verily tl96. Bt rcplacing , in t2 by r + r, and rhen bI , - ,r, and adding td subtrading the resuhs

las i. (2.6) and (2.7)], rcrily rll rnd Zl{.?. lerily /-15, l-16, Zl7. and tl6, bt combining appropdlte preceding aomlhs using (2'1)

THE I,API-AC' TRANSIiORM 6.1Ich.15

lsins (2.'1).

Irind the inve.se t.anslorns of rle tuncdons t(r) in Problems rj ro 13.

8. I +?. H,,r. Ii.. ,6,.n /.14(, + 2)'

9. -L Hnt: t.se LJ na 18.

1r . Zt I Hnn: v.u n nse Li trl L8 with complea 4

,/ 2, + l0 rnd ,, bur Zll rnd rt4 are more direcr.

,, : l,-l011. - t2 =__ ll..t it 2 p. -:a

wa) thc grlfh shitis

:4. Use ,28 to 6nd lhe I-aplace transfo.n of

C,r2)tll^:! t".:r. Use t2l dnd t4 ro find rhe inverse tmnsfo.n olrd P"i(r,r + 1).

l- Flnd rhe tunsnrnr .f

*herc r and r lre consrants.

Use z2l ro show rhrt l;' Jo(r) d = l. (see chlprer 12, Problen 1s.8.)

\ct1 +Dt+E)'15. Prove rl2 for, : Lq,,r, Difieremi,rc cquarion (2.1) {nh respect to r.16. Use l-32 and Zll ro obtain /-l l.lt. Usc /-12 and tll to obtain ,(r2 sh ,r.18. Use t.ll to derive ,21.

Iabrc cntries /-28 and L29 te kna$n as trarslatia, or rr,t,,{ theorcms. Do Problens 19 to 27

!9. ProE tle s€n*rl fornuh r29 usins (2.1).

10. Us 429 to le.iay ,6, /-1.1, ,1.1, md 218.

ll. Lsc ,29 rnd /-l I to obuin ,(t. '' sin ,t) Nhich is nor in rh€ r,blc.

ll. Obtain Z(r. '! cos rr) as in Problcn 2l

:-i. Ur rhe results *Iich rou hale obtained in Problens 21 rnd 22 to tind tlc inversetransforn ofotr + 2? r)iLr':+.1, + 5)r.

:1. sket.h on rhe sane a\es snphs of sin t, sin (r rr2), ind s'n (i - zi2), and obsdve shich

6l

l,l. Shoq that a conbination ofenFies Zl to Zl0, ,13, 21.1, rnd 218 in rle tablc, Nill gire rheinrcse transtofln .f in,v tunction ofrhe aorm

(lt+B)

3. SOLUTION OF DIFFERENTIAL EOUATIONS BYLAPLACE TRANSFORMS

\\e arc going to discuss the soludoD of linear difierential cquations viih constanlcoefficients (see Chapter 8, Sections 5 a.d 6). You aircady lnos how to solve suchcquationsj wh1, then, should Iou bother ro learn another method? Rcc.ll t]vo thing!:(l)\\hen rhe right-hand side ofa diferential equation is not zero, it is sometinres quirea lor of Fork to 6nd rhc particular solution. (2) what you find bl standlrd methods is

a general solution, containing srbirrar,r constants; you must makc a further c cularionol the rllues of rhe constaDts ,atisfyins siven initial conditions i' ordei ro solve a

particular problen. Thc Laplac€ rransform method simplifi.s both these dificultics$ihen lou use, r.blc ol Laplace trmsforms (nuch as you lvould use intesr.l rabtcs)you nray avoid somc of the rlgebra: also. rhe L.pl.cc transform merhod givcs directllthc linal soLution satisliing giten initlal conditions. Thus. although Iou raa get alongwithout using Laplsce rr.nsforms to solte ditlcrentisl equationsi in Pr.ctice the) are

\le xre going to tale the Laplace trrnslbrms of the terms in dillereniial cquarionsi t.do this we nced to lnor the transibrms of derir.tivcs l" - 4'.r//, r" = n'yidtz, etc.'lo{ind l,(}'). wc use the definilion (2.1) and intesrale by Parts, as ibllo{s

642 INTEGRAL TRANS'IORiIS ch. 15

(3.1)

13.2) L(t ):l''Llt) t)t(o) J,\0):p1t Da r'.t.

Conrinuins this proces, irc obtain the ransforms of the hishcr-order d€rirxtiles (Prob-

(quanvns \{e ,lhsrrar( rhe method b\ slme

fY t., J'o + lpY 4to + 4Y : L(t'" '^ 2'- {r * r,t

f.fL\j| | .,t). Pd. ',\'t \-t I lL ",

.Jo o Jo

: .t\0) + lL(r) = lY ro

shere ior simplicitJ ive have rvdtten r0r: v andr(0): lo. To find r(-)"'). ive thinkofl" as (1''), ud substiture r" ibr I' in (1.1) to get

L(t") : tL(J') !'\o)L\ins (3.1) as.in ro clinrinate r(,y'). $e fin3lly ha\'€

len 1 ind ,35).\c are no$ readl ro solle diflirenrial

Exafrple 1. Find the solution ol

.)r" -t t' + 4t : t'e 'z'

Fhich sldsnes rhe inirial conditions ]o - 0, r'; : 0\\e lake the L.placc trunsform of erch term in thc cqlation, using /-35 and a6 in

tb€ table of Laphce transforfts. We ger

csl

l5

: 10

solt Tlo\ or DIFffRr\-Tl \J r(l {t.o\. ts

Bur th€ initirl condirions.rc rii: 'i:0 Thus {e have

)2.l -+/-:)l - ., o- l' .tt!) (1 2)-

\o{ we w.nr J', which is rhc inverse Llplace nansibrft of v. We look in dc tablc forrhc inverse lrlnsfomr of 2,r(1 + 2)5. Br 26, s€ get

211. 2t fe 1t

- r: - ':

This is nuch siftpler thln rhe seneral solurion; Fe hav€ obtained just the solut;oniaiisfling the gilen initi.l condrtions.

f.\atuple 2. Sol'e.)" + '{1 : sin 21, subjecr Io rhc initial condirions }o : 10. r'; : 0Usins rhe labl€, we lakc rhc i.allace tr.nsform ofeach tern ofrhc cquation to get

!1Y lra lD+1Y=,Ar"rl:| .r'++Then wc substitote rhe inilial corditions and solrc for }- as follows:

(r' + a))' 101 : -1.10, 2

' t -4 u- I +r

. nally, uling lhc in!.rsc transform using Z4 and Zlt, ive hale rhe desircd solution:

.t, : l0 cos 2r + *(sin 2r - 2r cos 2,: l0 cos 2r - : sin 2r - 1r cos 2r.

fhe proc.ss of findins I' fiom v is not always as sihple rs in thesc Iso c\anPles. In:.:aons 5 and 6 we shall consider some norc adranced techniques for findins inlcrsc:rnlforns, bur risht now Fe fairLt sinrple thinss to r! if thc inrcGc:i\form we {anr is nor in the tdble. If t- contains seleral terns, wc can trJ ro find_: inverse r.nsfornrs ofthe separdre l.rmsi akernativell, w. c.n trl combiDing the,-as fi61. Suppose, lor exanplc, that re had added tog.ther the $vo fiacthn! in I in

rmplc 2 lnd so hdd r-: (1013 + 401 + 2)i(r' + +)'. Th" invcrse fansform of ftis, ::csion is not in the tdbie and obviouslt the rhing Io do is to leave the r*o fr.ctions

: combined as {e did in E!.nple 2. On the other h.nd. sometime! se sel r simplcr::cssion for I'il \e do combinc ternb. ro. e$nplc, consider

, ,+ 1,1, r , li:,'ttl, I

(, 1)(l+l)r+3 (1 + 1)(l+3)

Herc v simplified because of the canceliation of rhe factor (l l) and thc inletr.nsfbrn of lhe nn.l cxpression for v is in the table (In sener'i, v $ill not siFplrhis wa), but if ii docs, this is the simplest nerhod ol complcting the probleillo{ever, let us suppose that v contains a frirlv complic.rcd fracrion (which 'rinator.lly in thc problen,,,r becdusc we combincd sinpler fractions) and that comb

ins rcrms in l'does not simplif! it By the rechnique of partial fracrions (see a calcu

iext) we ma) b'eak thc comnLicatcd iiacrion into sevenl sinpler fraciions whosc inlerr.nsturni arc in rhc table h is unnecesarv to use a Partial lractions expansion $

all lhear denomin,rors. howevcr, since wc can iind the inlerse ransform of 'fraction $irh. qurdratic denominator from the tablc (se€ Problcm 211) We nr"v sro use pardal liacrions $hen rhe denominator is of the lbrms lnear |ifl's ttutultutit,trddtdtt. tiw! qud,lfati.

Eaamplc 3. Sohe '|'+ 4r' + I3r - 20e "-vo:l,.li:I\\ic tate rhc oansform ofedch tcrtu and solve lbr v as ibllows:

plY-t 3+lrt' 1+l3v: ?1,

L ltJ ,\ p'\l -2:t-p- +t . ',ir- '* )-,, "a'u ,'UlinF plrtial i;dcdons. ive write

t''+8p+2,' 1 Bh+Ct+1 lr++,r+ll(r + 1)(t'?+ +r + 11)

or, rftcr cledrins fi actions,

t'+8p+2"1 = (1+ B)t'1 + 111+ B + Cb + (131 +c)Sincc this is an id.ndtl, Fe equtte powcrs of, to get

A+B:1, 4A+B+C:a' 13A+C:li\Vc solve thesc cquations simultancously to find ,, = 2, /J : I , C : 1 Thc. {e h

) n+1 2 3 p 2

' ,- l' l+/-ll ,'l rr'lr: o r, 2' !o

and br' /,2, ,13. a;d r11,

J :2e' + e':'sin 3/-|':'cos3r'

64I INTEGRAL TRANSFORN1S

A set of sinuftancous ditrerenriai equations can also be sol'ed bv usins Lalrranslirns (il there is t solution;scc Kaplan, OPcrational Mcrhods. p 10) Ilere i:

!!. !!

rfiift

SOI-UTIO\ OF DIFFERE\'TIAL EQLATIO\S 6.'5

lllample .+. Solve rhe set of equdiions

)' 2J+z:0,.' | 2z:0,

;ubje.i t. rhe;nirixl cond;rions I'o: l,:o:0.$. shall call L\z): Z rnd ,(i, : )" as bcforc. \lc takc thc L.placc rnnsforn ol

.trch of drc cqr.tions ro gct

lt ]a 2Y+Z:0,lZ zt Y La :0.

\lrer substiruting the iniiial cond,tbDs rnd cotlecring rerms, s. ha!€

(! 2)Y+Z:It'Q ))/ : t\

r. sohe rhis ser of rlsebra;c equ.tions sinultaDcouslr for t' ^ntl

Z (by dny of th.:r.fiods usu.lll used for . p.ir of simuh.n.ous cquations climiDrrion, d.Icrmin.nrs..:. ) For exanple, tr€ h!] bultith the 6rsr equrion br'0 2) ind rdd rhe second to

lt., 2)'+llY: | 2 o' l': (r 2): + t

rnndt b) lool ins urr rhc inrdse trinsform of y using 1- l +. $ e set

't = rr' cos r'

: iih.l\. ynrins {i,r Z and look;ls uf thc in\crs. trrnsforn, se nrdl' (, 2)2+l'

ie couid hnd : lrom ftc first difcrcnridl cquation by slbstitutins rhc I

:=2r' r' :2i:' cos r + ir'sin r 2e" cor r : rr' sin r.

:nung lincar d;1lcrcDdli cqu.rions wirh consrrnr coetiicientr is nor tlc onlt use ol:irce trandorms. :s ro! *ill see in Secrion 9, $e m$ solve sonre liiDds of pddJ-:rntidl equations b) Lallacc rr.nsforms Also a l.ble of Ldrhce transli,.ms can bc.l tu evaluare delinitc intcsrals of de rype Jii l '7(r),lr. lor eranrple, br /-15 $ith

= I and I : 2. \'c hdrc

| ',' .o' Jr, ar'I - l\l) I)

:,rlh, rhere is morc ro thc subj.cr firn rhis. lhhoush se dr€ discus!;ns jn rhis-re. the usc of Laplicc ffansforns as. rool. Ih.I ilso c.n th] a no.e theoretical

6,16 I\'TEGR {L TR{NSFORN$ ch li

role in applicd problens. 1r is often possible to find il€sired inibrmation about a

problem directly from the Lapiace nansform of the solution $ithout eler 6nding thesolurion. Thus the use of l-apldce transforms mat lead to a better undersranding of rproblem Dr a simpler method of soludon. (Compare the use of hat.ices, for ex.nple. orthe use of loglog paper.)

PROBLEMS, SECTION 3l. Continuins the n€thod urd in deriins (l.i) and (32). ver'ar thc l-aphcc tansturnN ol

hjsheForder deriaarives of.r gilen in the t.bic (Z:15).

Bt usilg Lrplace translorms, sohc thc tullowing difercntill equrtions subieci to ihe girei inithl

2. J/ .f :2.'. J]o=l3. .1" +,1.)" + 4:!: c t', lb :0, '),; :4.1. .r," +.r' : sin I, ro = I, I| = 05. .)"' +.]' : sin t, to : 0, J; : j6. r'' 6J''+91,: rr. ro:0, ri:57. !" +!'+ 4J ='1, Io = 0. ]; : 2

8, r"+ I6i =8cos4r. ro=1;=09. ')/'+ 16r, : E cos.1r, .ro :0, .r; :8

10. .1'" +r'+'lr:61'. i/o :)l = 0

It. t +r=,1.:r,.ro:q l;=l

13. r"+r:5sinh2r, yo:0. .r;:214. t" +!': 4tr'. t,o=0. r';:l15. .r'' + 9r: cos lr, ro :0, .ri, : 6

16. .]" + 9),: cos ir, /o :2, .r;: 0

17. 1,"+5r'+5,':12, 'o:2, l;:0lli. -r' +J, = 3. ', .),0 : 1, l; : j19. ) ir' 5r=t'. l,o=l, lii=2

r" 8J, + 16r : 32t. to: ), .yi:2'+)' +5l =26dJ, lo=l, Jl;=5

5r=r0cos/. ro=2. 1;=l*,,)r-_\ l0 cos i. ro : 0, ,'; -:l.\,n) c.\ / rt - 5. r! 2

FOUIUER TR,{NSFORTIS 6t7_ !!.1!

dins th€lins of a

25. .r" +4! +5::2. "cos/. Jo:0, r;:326. !" + 2r' + I0! = 6a ' sin Jr. :r,o = 0, j,; : I

sohe the following sels ofequalions by rhe Lapl,cc rranslom nethod.2i. )t+.' l.:0 ,yo:rl,:0

28. .r, +:=2cost to:-l

29. r'+z' 2!= I ]o::o=t

10, y'+2:=t :yo=02! z:2t .o:ljl. )"+z -z=lJ lo:0, ti=l

l'+z-22:I I' zo:1, !;=l32. z + 2!:0 ro=.0:0

!' 2.:2il. r'-:' .i=cosr :yo = l

Jt+r-22:o :o:o:laluate cach oathe fotio{ins dennite inresruk br using rhe Laptrce ixnsrom hble

I I , ".,".t at ,.t1th [hi.i.(2.llLrl"1=l,r(r nn.!iu\e/iqirh, ,

,! J',.',i"na,: l"',"'a'.' 1,'" ,"'*, i'i"","64".:, I Jzt). ' ,i'

Hnt: Ve L23 ^nd

L32. See Chlprer 12, Sections 12 f. lor definition of Jo.

! FOURIER TRANSFORMS: Chapter 7, we erpanded letiodic finctioas in s€ries Df sines, cosines, and complex

r-.nentials. Physioliy, we could thint of rh€ rerms of these Fourier seri€s as rep-^-arins a set of hdmonics. In

'nusic these would be an infinire set of frequenci€s ,',

36. Jo

-

I'

". J'" r' -.rrr,{' l'" '' i" "' d,

n, J']"..'.,-',"

643 I\TI]GRAI, TRANSI'ORMS

, = l. 2, 1, ..i notice that this sctj although ininite, does not b) xnl ncus jnclud

all possible Fequ.ncics. In elecrricit,v, a Fourier series could represent. pcriodic vohage; asain {e cor d think of ihis .s nade up of xn infinne bui discretc (that is' n.contiDuous) set of !-c voliases of frequencies /.r. Similarly, in discussing lishr,Fouder series could .epresenr light consisring of a disoete ser of ivarelengrhs liin : lt 2, . , thai is, ! discrete set ol colors. Tso rehred qucstions ni8ht occur ro u

here. Firsr. is ir possiblc to repr€senr . function which is ,ot peiiodic bl somelhinan,losous to a Irouricr scries? Second, can ile som€how cxtcnd or nodifv Fourie. !eri.to coler rhe case ol a continuous spcctrum of wavelcnglhs of Ughr, or ! sound $.tcontdinins a continuous sct ol fiequencics?

llt.llr!LJlllll'Jnlnl*lJli.,'i1lllUl..Um'ilmdja"t'Ul|*{to learn that the Fou er s',!J llhai is, a gz of tcrnt is .eplac.d bI ! l'ourier i,rcszin lhe abo\e clscs. The louricr iDtcgral cd bc used to reprcscnt nonperiodic fltctionlfor eramplc ! single lolrase puhe nor relctred, or a flash of lisht. or a ,ound which i

nLr errar!.1 lhe foJ-icr ilreg rl J'o rcD'('<rr' I onrirLuL' t€r r\pt\rrLmr 'J l-(quenciesj for example ! {hole rrnge of musical tones or coiors of lighr rathcr than

Recall from Ch,tter 7, cquations (8.2) ard (E.3), lhe follosing compier Fou.i(

(+ l)l(tJ: : "..,

t'r; I I(l)s "" ',/,

Thc pcri,d of /(r) is 2l ,nd rh. frcqucncies ofthc t€rnrs in rhe scrics are ',(24 \\nu$ $J1 'o 01rdfl rle .r-c o. ( nr'runu- t(quen.i\s

Delinilion ol Fourier Translorms \lc state withour proof the lormul.s correspon(ins ro (4.1) iar . contin!ous funse of frequencies.

Conrpare (+.2) and (.{.1); s(x) corresponds to .4! d coresPonds io ,, rd J'- etrspords ro lla . This lsreet Firh our discussion of thc Phlsicrl ncanins and usc

Fourier integrah. Thc quaniiry r is t continuous analos ol the inresral-valued 'arilb,. and so rhe set of coeflicielrs ., has become , fundnrn g(a)j rhc suh o'e h

beconc an inlcsral orer d The t$o funcrions /(,) and s(a) arc called a vi or Fa,/'r/d,yi,,,r. Usually. s(d) is called rhe Fourier tianrlorn ofl(,), and/(r) is called tlinlcrsc fourier trNnsform ol s(r), but since the two intesrals difre. in forn onlv in d

sign in rhc eaponenr. it is .ather common simply to crll cither a |ourier trusl;rm

II(JURIER TRA\SFORVS 6,19

rhc.rhc. lou should chcct rhe notation ofrnr bool or irble )ou arc using. Anotherpohr on shich r.rious books d;lfer is rhc posirion of ihc ficror l,(2n) in (11)i ir sfossible to hiye it .ruldplr rhe t1,) inrcsrel ;nsread of lIe .q(() intesr.t, or ro harc rhelr.nr L\, 2xmultiplr each ofrhe inregrals.

Tne [,uti.t i,ttlrut thcat. s.ys rhat. if. tundion l(r) s]ri\lics rhc D;ichtetrndliions (Chlprcr i, sedjon 6) on ererr tinirc inre.ar, and if j:, l(r) /r js.lnitc. th.n (+.2) is corr..r. lhlr is. lfs(z) is compuied rnd subsriiure,t inro thc inregralr.l(r) [comp.re the p.occdur. of cohputins thc ..,s for a Fouricr se.jes aDd sDbsrirurrg them into rhe se.ies lirr l(rl. rhen rhe inrcsr.l slres rhe vatLre ofl(r) rnl.$herc:h!rl(iJ is contuuoxs;ar jumps ofl(t. the i c8r.l gires rhe nidro;rt ot rhc lunprssin conp,re Fouricr series, Chlprcr 7, Section 6). Tbe ti,lloxhs discuslion is nor anrrhcnrr;crl prool of rhis rhcorcm (for p.ool sec Sneddon) but is inr.nded ro hctp rouiie nrorc clcarll how Fourier intei.rlls.re rellred to Fouri.r serie\

Ir niFhi sccm rer*hablc ro rhink of rrjine to rcprcscnr r fundioD rhich ts nor.iriod;. b\ leftins rhe lcriod ( /, /) iicrcjse t., ( :., ,7_). I_€i us tr! ro do this..rfrins. sith (1.1). If wc .l "nil

:a. and ?"r1 a,,:x,1:Aa. rhenil/) = arr(2n) and (+.1) can be r€$rnren.s

-+)

-)

L. hdre chansed the dunml inreg.ation !.rtrble in." from r to, ro avoid trrerriusion.) Subsrirurins (.1.+) into (1.1). ivc hai,e

'":j.i','"n "':,-tj'

illi ,r,.,, "",.1,".'

a,r,: f '.

f(,r,*",,1,' : +I.or,, o,,

l(r):

:t*iIr'", I L.Jr

, I1.,(1,) Ar looks rarher like rhe tb.huh in catculus for Ih€ sun whose timit.11 rends ro zero, is .! inregrat. lf we ler / tend ro innDtll [ihar ir, ter t]rc perjod of

rcrd ro infin,trl, thcn Az : n,'/- 0, and rhc sum ::-,, F(d,) Aa socs oyer tor_. - .ll,l(o) ,lr; sc hale droppcd the subscripr, on l now rhar it is 3 contin-.: lariablc. $ie also lcr l rend to infinnt and r, : r in (1.6) to cct