Piecewise-Defined Functions and Periodic Functions

Transcription

1 28 Piecewie-Defined Funcion and Periodic Funcion A he ar of our udy of he Laplace ranform, i wa claimed ha he Laplace ranform i paricularly ueful when dealing wih nonhomogeneou equaion in which he forcing funcion are no coninuou Thu far, however, we ve done preciou lile wih any diconinuou funcion oher han ep funcion Le u now recify he iuaion by looking a he or of diconinuou funcion (and, more generally, piecewie-defined funcion) ha ofen arie in applicaion, and develop ool and kill for dealing wih hee funcion We will alo ake a brief look a ranform of periodic funcion oher han ine and coine A you will ee, many of hee funcion are, hemelve, piecewie defined And finally, we will ue ome of he maerial we ve recenly developed o re-examine he iue of reonance in ma/pring yem 28 Piecewie-Defined Funcion Piecewie-Defined Funcion, Defined When we alk abou a diconinuou funcion f in he conex of Laplace ranform, we uually mean f i a piecewie coninuou funcion ha i no coninuou on he inerval (, ) Such a funcion will have jump diconinuiie a iolaed poin in hi inerval Compuaionally, however, he real iue i ofen no o much wheher here i a nonzero jump in he graph of f a a poin, bu wheher he formula for compuing f () i he ame on eiher ide of So we really hould be looking a he more general cla of piecewie-defined funcion ha, a wor, have jump diconinuiie Ju wha i a piecewie-defined funcion? I i any funcion given by differen formula on differen inerval For example, if < f () = if < < 2 if 2 < if and g() = if < < 2 if 2 are wo relaively imple piecewie-defined funcion The fir (keched in figure 28a) i diconinuou becaue i ha nonrivial jump a = and = 2 However, he econd 555

2 556 Piecewie-Defined and Periodic Funcion Figure 28: 2 T 2 (a) (b) The graph of wo piecewie-defined funcion T funcion (keched in figure 28b) i coninuou becaue goe from o a goe from o 2 There are no jump in he graph of g By he way, we may occaionally refer o he or of li ued above o define f () and g() a condiional e of formula or e of condiional formula for f and g, imply becaue hee are e of formula wih condiion aing when each formula i o be ued Do noe ha, in he above formula e for f, we did no pecify he value of f () when = or = 2 Thi wa becaue f ha jump diconinuiie a hee poin and, a we agreed in chaper 24 (ee page 496), we are no concerned wih he precie value of a funcion a i diconinuiie On he oher hand, uing he formula e given above for g, you can eaily verify ha lim g() = = lim g() and lim g() = = lim g() ; o here i no a rue jump in g a hee poin Tha i why we wen ahead and pecified ha g() = and g(2) = In he fuure, le u agree ha, even if he value of a paricular funcion f or g i no explicily pecified a a paricular poin, a long a he lef- and righ-hand limi of he funcion a are defined and equal, hen he funcion i defined a and i equal o hoe limi Tha i, we ll aume f ( ) = lim f () = lim f () + whenever lim f () = lim + f () Thi will implify noaion a lile, and may keep u from worrying abou iue of coninuiy when hoe iue are no imporan Sep Funcion, Again Mo people would probably conider he ep funcion o be he imple piecewie-defined funcion Thee include he baic ep funcion, if < ep() = if (keched in figure 282a), a well a he ep funcion a a poin α, ep α () = ep( α) = if < α if α <

3 Piecewie-Defined Funcion 557 T α T (a) (b) Figure 282: The graph of (a) he baic ep funcion ep() and (b) a hifed ep funcion ep α () wih α > (keched in figure 282b) We will be dealing wih oher piecewie-defined funcion, bu, even wih hee oher funcion, we will find ep funcion ueful Sep funcion can be ued a wiche urning on and off he differen formula in our piecewie-defined funcion In hi regard, le u quickly oberve wha we ge when we muliply he ep funcion by any funcion/formula g() : } g() if < α if < α g() ep α () = = g() if α < g() if α < Here, he ep funcion a α wiche on g() a = α For example, and 2 ep 3 () = in( 4) ep 4 () = if < 3 2 if 3 < if < 4 in( 4) if 4 < Thi fac will be epecially ueful when applying Laplace ranform in problem involving piecewie-defined funcion, and we will find ourelve epecially inereed in cae where he formula being muliplied by ep α () decribe a funcion ha i alo ranlaed by α (a in in( 4) ep 4 () ) The Laplace ranform of ep α () wa compued in chaper 24 If you don recall how o compue hi ranform, i would be worh your while o go back o review ha dicuion I i alo worhwhile for u o look a a differenial equaion involving a ep funcion! Example 28: Conider finding he oluion o y + y = ep 3 wih y() = and y () = Taking he Laplace ranform of boh ide: L [ y + y ] = L [ ep 3 ] L [ y ] + L[y] = e 3 [ 2 Y() y() y () ] + Y() = e 3

4 558 Piecewie-Defined and Periodic Funcion [ 2 + ] Y() = e 3 Thu, Y() = [ y() = L e 3] ( 2 + ) ( 2 + ) e 3 Here, we have he invere ranform of an exponenial muliplied by a funcion whoe invere ranform can eaily be compued uing, ay, parial fracion Thi would be a good poin o paue and dicu, in general, wha can be done in uch iuaion 282 The Tranlaion Along he T-Axi Ideniy The Ideniy A illuraed in he above example, we may ofen find ourelve wih L [ e α F() ] where α i ome poiive number and F() i ome funcion whoe invere Laplace ranform, f = L [F], i eiher known or can be found wih relaive eae Remember, hi mean Conequenly, e α F() = e α F() = L[ f ()] = f ()e d = f ()e d f ()e α e d = f ()e (+α) d Uing he change of variable τ = + α (hu, = τ α ), and being careful wih he limi of inegraion, we ee ha e α F() = = = f ()e (+α) d = τ=α f (τ α)e τ dτ (28) Thi la inegral i almo, bu no quie, he inegral for he Laplace ranform of f (τ α) (uing τ inead of a he ymbol for he variable of inegraion) And he reaon i i no i ha hi inegral limi ar a α inead of Bu ha i where he limi would ar if he funcion being ranformed were for τ < α Thi, along wih obervaion made a page or o ago, ugge viewing hi inegral a he ranform of if < α f ( α) ep α () = f ( α) if α The obervan reader will noe ha y can be found direcly uing convoluion However, beginner may find he compuaion of he needed convoluion, in() ep 3 (), a lile ricky The approach being developed here reduce he need for uch convoluion, and can be applied when convoluion canno be ued Sill, convoluion wih piecewie-defined funcion can be ueful, and will be dicued in ecion 284

5 The Tranlaion Along he T-Axi Ideniy 559 Afer all, τ=α f (τ α)e τ dτ = = = = =α α = α = = f ( α)e d f ( α) e d + f ( α) ep α ()e d + f ( α) ep α ()e d = L [ f ( α) ep α ] =α Combining he above compuaion wih equaion e (28) hen give u e α F() = = τ=α f ( α) e d =α f ( α) ep α ()e d f (τ α)e τ dτ = = L [ f ( α) ep α () ] Cuing ou he middle, we ge our econd ranlaion ideniy: Theorem 28 (ranlaion along he T axi) Le F() = L[ f ()] where f i any Laplace ranformable funcion Then, for any poiive conan α, Equivalenly, L [ f ( α) ep α () ] = e α F() (282a) L [ e α F() ] = f ( α) ep α () (282b) Compuing Invere Tranform The Baic Compuaion Compuing invere ranform uing he ranlaion along he T axi ideniy i uually raighforward! Example 282: Conider finding he invere Laplace ranform of Applying he ideniy, we have e L [ e ] = L [ ] e 2 } 2 + } = L [ e 2 F() ] = f ( 2) ep 2 () F() Here he invere ranform of F i eaily read off he able: f () = L [F()] = L [ 2 + ] = in()

6 56 Piecewie-Defined and Periodic Funcion π 2 + 2π T Figure 283: The graph of in( 2) ep( 2) So, for any X, f (X) = in(x) Uing hi wih X = 2 in he above invere ranform compuaion hen yield L [ e ] = f ( 2) ep 2 () = in( 2) ep 2 () Keep in mind ha in( 2) ep 2 () = if < 2 in( 2) if 2 < The graph of hi funcion i keched in figure 283 Oberve ha, a illuraed in figure 283, he graph of L [ e α F() ] = f ( α) ep α () i alway zero for < α, and i he graph hifed by α of f () on [, ) on α Remembering hi can implify graphing hee ype of funcion Decribing Piecewie-Defined Funcion Ariing From Invere Tranform Le u ar wih a imple, bu illuraive, example! Example 283: Conider compuing he invere Laplace ranform of F() = 2 e 2 e 2 Going o he able, we ee ha G() = 2 g() = Uing hi, along wih lineariy and he econd ranlaion ideniy, we ge f () = L [F()] = L [ 2 e 2 e 2 ] = L [ 2 e ] L [ 2 e 2 ] = ( ) ep () ( 2) ep 2 ()

7 The Tranlaion Along he T-Axi Ideniy 56 Noe ha he ep funcion ell u ha ignifican change occur in f () a he poin = and = 2 While he above i a valid anwer, i i no a paricularly convenien anwer I would be much eaier o graph and ee wha f really i if we go furher and compleely compue f () on he inerval having = and = 2 a endpoin: For <, hen f () = ( ) ep () }} For < < 2, hen f () = ( ) ep () }} For 2 <, hen f () = ( ) ep () }} ( 2) ep 2 () }} ( 2) ep 2 () }} ( 2) ep 2 () }} = = = ( ) = = ( ) ( 2) = Thu, if < f () = if < < 2 if 2 < (Thi i he funcion keched in figure 28b on page 555) A ju illuraed, piecewie-defined funcion naurally arie when compuing invere Laplace ranform uing he econd ranlaion ideniy Typically, ue of hi ideniy lead o an expreion of he form f () = g () + g () ep α () + g 2 () ep α2 () + g 3 () ep α3 () + (283) where f i he funcion of inere, he g k () are variou formula, and he α k are poiive conan Thi expreion i a valid formula for f, and he ep funcion ell u ha ignifican change occur in f () a he poin = α, = α 2, = α 3, Sill, o ge a beer picure of he funcion f (), we will wan o obain he formula for f () over each of he inerval bounded by he α k Auming we were reaonably inelligen and indexed he α k o ha we would have For < α, f () = g () + g () ep α () }} < α < α 2 < α 3 <, + g 2 () ep α2 () }} = g () = g () + g 3 () ep α3 () }} +

8 562 Piecewie-Defined and Periodic Funcion For α < < α 2, f () = g () + g () ep α () }} For α 2 < < α 3, + g 2 () ep α2 () }} + g 3 () ep α3 () }} = g () + g () = g () + g () f () = g () + g () ep α () }} And o on + g 2 () ep α2 () }} + g 3 () ep α3 () }} + + = g () + g () + g 2 () + + = g () + g () + g 2 () Thu, he funcion f decribed by formula (283), above, i alo given by he condiional e of formula f () if < α f () if α < < α 2 f () = f 2 () if α 2 < < α 3 where f () = g (), f () = g () + g (), f 2 () = g () + g () + g 2 (), Compuing Tranform wih he Ideniy The ranlaion along he T axi ideniy i alo helpful in compuing he ranform of piecewiedefined funcion Here, hough, he compuaion ypically require lile more care We ll deal wih fairly imple cae here, and develop hi opic furher in he nex ecion! Example 284: Conider finding L[g()] where g() = if < 3 2 if 3 < Remember, hi funcion can alo be wrien a g() = 2 ep 3 () Plugging hi ino he ranform and applying our new ranlaion ideniy give L[g()] = L [ 2 ep 3 () ] = L [ f ( 3) ep 3 () ] = e 3 F()

9 Recangle Funcion 563 where f ( 3) = 2 Bu we need he formula for f (), no f ( 3), o compue F() To find ha ha formula, le X = 3 (hence, = X + 3 ) in he formula for f ( 3) Thi give f (X) = (X + 3) 2 Thu, and F() = L[ f ()] f () = ( + 3) 2 = , = L [ ] = L [ 2] + 6L[] + 9L[] = Plugging hi back ino he above formula for L[g()] give u L[g()] = e 3 F() = e 3 [ ] 283 Recangle Funcion and Tranform of More Complicaed Piecewie-Defined Funcion Recangle Funcion Recangle funcion are ligh generalizaion of ep funcion Given any inerval (α, β), he recangle funcion on (α,β), denoed rec (α,β), i he funcion given by if < α rec (α,β) () = if α < < β if β < The graph of rec (α,β) wih < α < β < ha been keched in figure 284 You can ee why i i called a recangle funcion i graph look raher recangular, a lea when α and β are finie If α = or β =, he correponding recangle funcion implify o and rec (,β) () = rec (α, ) () = if < β if β < if < α if α < And if boh a = and b =, hen we have rec (, ) () = for all

10 564 Piecewie-Defined and Periodic Funcion α β T Figure 284: Graph of he recangle funcion rec (α,β) () wih < α < β < All of hee recangle funcion can be wrien a imple linear combinaion of and ep funcion a α and/or β, wih, again, he ep funcion acing a wiche wiching he recangle funcion on (from o a α ), and wiching i off (from back o a β ) In paricular, we clearly have rec (, ) () = and rec (α, ) () = ep α () Somewha more imporanly (for u), we hould oberve ha, for < α < β <, } } if < β if < β ep β () = = = rec (,β) (), if β < if β < and if < α ep α () ep β () = if α < < β if β < if < a = if α < < β = rec (α,β) () if β < In ummary, for < α < β <, and rec (α,β) () = ep α () ep β (), rec (,β) () = ep β () (284a) (284b) rec (α, ) () = ep α () (284c) Thee formula allow u o quickly compue he Laplace ranform of recangle funcion uing he known ranform of and he ep funcion! Example 285: L [ rec (3,4) () ] = L [ rec (3,4) () ] = L [ ep 3 () ep 4 () ] = L [ ep 3 () ] L [ ep 4 () ] = e 3 e 4

11 Recangle Funcion 565 Tranforming More General Piecewie-Defined Funcion To help u deal wih more general piecewie-defined funcion, le u make he imple obervaion ha g() if < a if < a g() rec (a,b) () = g() if a < < b = g() if a < < b, g() if b < if b < and } g() if < b g() if < b g() rec (,b) () = = g() if b < if b < So funcion of he form if < a f () = g() if a < < b if b < can be rewrien, repecively, a and h() = g() if < b if b < f () = g() rec (a,b) () and h() = g() rec (,b) () More generally, i hould now be clear ha anyhing of he form g () if < α g () if α < < α 2 f () = g 2 () if α 2 < < α 3 can be rewrien a (285a) f () = g () rec (,α )() + g () rec (α,α 2 )() + g 2 () rec (α2,α 3 )() + (285b) The econd form (wih he recangle funcion) i a bi more concie han he condiional e of formula ued in form (285a), and i generally preferred by ypeeer Of coure, here i a more imporan advanage of form (285b): Auming f i piecewie coninuou and of exponenial order, i Laplace ranform can now be aken by expreing he recangle funcion in formula (285b) a he linear combinaion of and ep funcion given in equaion e (284), and hen uing lineariy and wha we learned in he previou ecion abou aking ranform of funcion muliplied by ep funcion! Example 286: Conider finding F() = L[ f ()] when From he above, we ee ha f () = 2 if < 3 if 3 < f () = 2 rec (,3) () = 2 [ ep 3 () ] = 2 2 ep 3 ()

12 566 Piecewie-Defined and Periodic Funcion So F() = L[ f ()] = L [ 2 2 ep 3 () ] = L [ 2] L [ 2 ep 3 () ] The Laplace ranform of 2 i in he able, while he ranform of 2 ep 3 () ju happened o have been compued in example 284 a few page ago Uing hee ranform, he above formula for F become F() = 2 [ 2 3 e ] 2! Example 287: Conider finding F() = L[ f ()] when if < 2 f () = e 3 if 2 < < 4 if 4 < From he above, we ee ha Thu, f () = e 3 rec (2,4) () = e 3 [ ep 2 () ep 4 () ] = e 3 ep 2 () e 3 ep 4 () L[ f ()] = L [ e 3 ep 2 () ] L [ e 3 ep 4 () ] (286) Boh of he ranform on he righ ide of our la equaion are eaily compued via he ranlaion ideniy developed in hi chaper For he fir, we have where L [ e 3 ep 2 () ] = L [ g( 2) ep 2 () ] = e 2 G() g( 2) = e 3 Leing X = 2 (o = X + 2 ), he la expreion become g(x) = e 3(X+2) = e 3X+6 = e 6 e 3X So and g() = e 6 e 3 G() = L[g()] = L [ e 6 e 3] = e 6 L [ e 3] = e 6 3 Thi, along wih he fir equaion in hi paragraph, give u L [ e 3 ep 2 () ] = e 2 G() = e 2 e 6 3 = e 2( 3) 3 The ranform of e 3 ep 4 () can be compued in he ame manner, yielding L [ e 3 ep 4 () ] = e 4( 3) 3

13 Recangle Funcion 567 (The deail of hi compuaion are lef o you) Finally, combining he formula ju obained for he ranform of e 3 ep 2 () and e 3 ep 4 () wih equaion (286), we have L[ f ()] = L [ e 3 ep 2 () ] L [ e 3 ep 4 () ] = e 2( 3) 3 e 4( 3) 3! Example 288: Le find he Laplace ranform F() of 2 if < f () = e 3 if < < 3 2 if 3 < To apply he Laplace ranform, we fir conver he above o an equivalen expreion involving ep funcion: f () = 2 rec (,) () + e 3 rec (,3) () + 2 rec (3, ) () = 2 [ ep () ] + e 3 [ ep () ep 3 () ] + 2 ep 3 () = 2 2 ep () + e 3 ep () e 3 ep 3 () + 2 ep 3 () Uing he able and mehod already dicued earlier in hi chaper (a in example 287 and 284), we dicover ha L[2] = 2, L [ 2 ep () ] = 2e, and L [ e 3 ep () ] = e ( 3) 3, L [ e 3 ep 3 () ] = e 3( 3) 3 L [ 2 ep 3 () ] = e 3 [ ] Combining he above and uing he lineariy of he Laplace ranform, we obain F() = L[ f ()] = L [ 2 2 ep () + e 3 ep () e 3 ep 3 () + 2 ep 3 () ] = 2 2e + e ( 3) 3 e 3( 3) 3 + e 3 [ ]

14 568 Piecewie-Defined and Periodic Funcion 284 Convoluion wih Piecewie-Defined Funcion Take anoher look a example 28 on page 557 A noed in he foonoe, we could have by-paed much of he Laplace ranform compuaion by imply oberving ha y() = in() ep 3 () and compuing ha convoluion Bu in he foonoe, i wa claimed ha compuing uch convoluion can be a lile ricky Well, o be hone, i no all ha ricky I more an iue of careful bookkeeping When compuing a convoluion h f in which f i piecewie defined, you need o realize ha he reuling convoluion will alo be piecewie defined, wih (a you will ee in he example) he formula for h f changing a he ame poin where he formula for f change Hence, you hould compue h f eparaely over he differen inerval bounded by hee poin Moreover, in compuing he correponding inegral, you will alo need o accoun for he piecewiedefined naure of f, and break up he inegral appropriaely To implify all hi, i i rongly recommended ha you compue he convoluion h f uing he inegral formula h f () = (and no wih he inegrand h(x) f ( x) ) One or wo example hould clarify maer h( x) f (x) dx! Example 289: Le compue in() ep 3 () Since ep 3 i piecewie defined, we will, a uggeed, ue he inegral formula So, in() ep 3 () = in( x) ep 3 (x) dx Fir, we compue he inegral auming < 3 Thi one i eay: in() ep 3 () = in( x) ep 3 (x) dx = }} = ince x<<3 in( x) dx = in() ep 3 () = if < 3 (287) On he oher hand, if 3 <, hen he inerval of inegraion include x = 3, he poin a which he value of ep 3 (x) radically change from o Thu, we mu break up our inegral a he poin x = 3 in compuing h f : in() ep 3 () = = = 3 3 in( x) ep 3 (x) dx in( x) ep 3 (x) dx + }} = ince x<3 in( x) dx in( x) ep 3 (x) dx }} = ince 3<x in( x) dx

15 Convoluion wih Piecewie-Defined Funcion 569 = + co( ) co( 3) = co( 3) Thu, in() ep 3 () = co( 3) if 3 < (288) Combining our wo reul (formula (287) and (287)), we have he complee e of condiional formula for our convoluion, in() ep 3 () = if < 3 co( 3) if 3 < Glance back a he above example and oberve ha, immediaely afer he compuaion of in() ep 3 () for each differen cae ( < 3 and 3 < ), he reuling formula for he convoluion wa rewrien along wih he value aumed for (formula (287) and (287), repecively) Do he ame in your own compuaion! Alway rewrie any derived formula for your convoluion along wih he value aumed for And wrie hi omeplace afe where you can eaily find i Thi i par of he bookkeeping, and help enure ha you do no loe par of your work when you compoe he full e of condiional formula for he convoluion One more example hould be quie enough! Example 28: Le compue e 3 f () where if < 2 f () = 2 if 2 < < 4 if 4 < For hi convoluion, we can do a lile pre-compuing o implify laer ep: e 3 f () = = e 3( x) f (x) dx e 3+3x f (x) dx = e 3 e 3x f (x) dx Now, if < 2, e 3 f () = e 3 e 3x f (x) dx = e }} 3 e 3x x dx = x ince x < < 2 Thi inegral i eaily compued uing inegraion by par, yielding e 3 f () = e 3 [ 3 e3 9 e3 + 9 ] = 9 [ 3 + e 3 ] Thu, e 3 f () = 9 [ 3 + e 3 ] (289)

16 57 Piecewie-Defined and Periodic Funcion On he oher hand, when 2 < < 4, e 3 f () = e 3 e 3x f (x) dx = e 3 [ 2 e 3x f (x) dx + }} = x ince x < 2 [ 2 = e 3 e 3x x dx + = = e 3 [ 9 = = ( 5e 6 + ) [ e 6 ] e 3 2 e 3x f (x) }} = 2 ] dx ince 2 < x < < 4 ] e 3x 2 dx ( e 3 e 6)] Thu, e 3 f () = [ e 6 ] e 3 for 2 < < 4 (28) Finally, when 6 <, e 3 f () = e 3 e 3x f (x) dx = e 3 [ 2 e 3x f (x) dx + }} = x ince x < 2 [ 2 = e 3 e 3x x dx + = = e 3 [ 9 ( 5e 6 + ) = = [ 6e 2 + e 6] e e 3x f (x) }} = 2 dx + ince 2 < x < 4 e 3x 2 dx + ( e 2 e 6) ] ] e 3x dx e 3x f (x) }} = ince 4 < x ] dx Thu, e 3 f () = 9 [ 6e 2 + e 6] e 3 for 4 < (28) Puing i all ogeher, equaion (289), (28) and (28) give u [ ] 3 + e 3 if < 2 9 e 3 2 f () = 3 + [ ] e 6 e 3 if 2 < < 4 9 [ 6e 2 + e 6] e 3 if 4 < 9

17 Periodic Funcion 57 Figure 285: T (a) (b) Two periodic funcion: (a) a baic aw funcion, and (b) a baic quare wave funcion T 285 Periodic Funcion Baic Ofen, a funcion of inere f i periodic wih period p for ome poiive value p Thi mean ha he graph of he funcion remain unchanged when hifed o he lef or righ by p Thi i equivalen o aying f ( + p) = f () for all (282) You are well-acquained wih everal periodic funcion he rigonomeric funcion, for example In paricular, he baic ine and coine funcion in() and co() are periodic wih period p = 2π Bu oher periodic funcion, uch a he aw funcion keched in figure 285a and he quare-wave funcion keched in figure 285b can arie in applicaion Sricly peaking, a ruly periodic funcion i defined on he enire real line, (, ) For our purpoe, hough, i will uffice o have f periodic on (, ) wih period p Thi imply mean ha f i ha par of a periodic funcion along he poiive T axi Wha f () i for < i irrelevan Accordingly, for funcion periodic on (, ), we modify requiremen (282) o f ( + p) = f () for all > (283) In wha follow, however, i will uually be irrelevan a o wheher a given funcion i ruly periodic or merely periodic on (, ), In eiher cae, we will refer o he funcion a periodic, and pecify wheher i i defined on all of (, ) or ju (, ) only if neceary A convenien way o decribe a periodic funcion f wih period p i by f () = f () if < < p f ( + p) in general The f () i he formula for f over he bae period inerval (, p) The econd line i imply elling u ha he funcion i periodic and ha equaion (282) or (283) hold and can be ued o compue he funcion a poin ouide of he bae period inerval (The value of f () a = and inegral muliple of p are deermined or ignored following he convenion for piecewie-defined funcion dicued in ecion 28)

18 572 Piecewie-Defined and Periodic Funcion! Example 28: Le aw() denoe he baic aw funcion keched in figure 285a I clearly ha period p =, ha jump diconinuiie a ineger value of, and i given on (, ) by if < < aw() = aw( + ) in general In hi cae, he formula for compuing aw(τ) when < τ < i aw (τ) = τ So, for example, aw ( 3 /4 ) = 3 /4 On he oher hand, o compue aw(τ) when τ > (and no an ineger), we mu ue aw( + ) = aw() repeaedly unil we finally reach in a value in he bae period inerval (, ) For example, ( ) ( ) ( ) 8 5 aw = aw = aw 3 ( ) ( ) 2 = aw = aw = Ofen, he formula for he funcion over he bae period inerval i, ielf, piecewie defined! Example 282: Le qwave() denoe he quare-wave funcion in figure 285b Thi funcion ha period p = 2, and, over i bae period inerval (, 2), i given by qwave() = if < < if < < 2 So, qwave() = if < < if < < 2 qwave( 2) in general Before dicuing Laplace ranform of periodic funcion, le make a couple of obervaion concerning a funcion f which i periodic wih period p over (, ) We won prove hem Inead, you hould hink abou why hee aemen are obviouly rue If f i piecewie coninuou over (, p), hen f i piecewie coninuou over (, ) 2 If f i piecewie coninuou over (, p), hen f i of exponenial order = Tranform of Periodic Funcion Suppoe we wan o find he Laplace ranform F() = L[ f ()] = f ()e d

19 Periodic Funcion 573 when f i piecewie coninuou and periodic wih period p Becaue f () aifie f () = f ( + p) for >, we hould expec o (poibly) implify our compuaion by pariioning he inegral of he ranform ino inegral over ubinerval of lengh p, F() = = p + f ()e d f ()e d + 4p For breviy, le rewrie hi a 3p F() = 2 p p f ()e d + k= (k+)p kp f ()e d + 5p 4p 3p 2 p f ()e d f ()e d + f ()e d (284) Now conider uing he ubiuion τ = kp in he k h erm of hi ummaion Then = τ + kp, e = e (τ+kp) = e kp e τ, and, by he periodiciy of f, So, (k+)p =kp f (τ + p) = f (τ) f (τ + 2p) = f ([τ + p] + p) = f (τ + p) = f (τ) f (τ + 3p) = f ([τ + 2p] + p) = f (τ + 2p) = f (τ) f (τ + kp) = = f (τ) f ()e d = = (k+)p kp τ=kp kp p f (τ + kp)e (τ+kp) d f (τ)e kp e τ d = e kp p f (τ)e τ d Noe ha he la inegral doe no depend on k Conequenly, combining he la reul wih equaion (284), we have [ p ] p F() = e kp f (τ)e τ d = e kp f (τ)e τ dτ k= Here we have an incredible roke of luck, a lea if you recall wha a geomeric erie i and how o compue i um Auming you do recall hi, we have e kp = k= [ ] e p k k= k= = e p (285)

20 574 Piecewie-Defined and Periodic Funcion We alo have hi if you do no recall abou geomeric erie, bu will you cerainly wan o go o he addendum on page 576 o ee how we ge hi equaion Wheher or no you recall abou geomeric erie, equaion (285) combined wih he la formula for F (along wih he obervaion made earlier regarding piecewie coninuiy and periodic funcion) give u he following heorem Theorem 282 Le f be a piecewie coninuou and periodic funcion wih period p Then i Laplace ranform F i given by F() = F () e p for > where F () = p f ()e d There are a lea wo alernaive way of decribing F in he above heorem Fir of all, if f i given by f () if < < p f () =, f ( + p) in general hen, of coure, F () = p f ()e d Alo, uing he fac ha p f ()e d = f () rec (,p) () e d, we ee ha or, equivalenly, ha F () = L [ f () rec (,p) () ] F () = L [ f () rec (,p) () ] Wheher any of alernaive decripion of F () i ueful may depend on wha ranform you have already compued! Example 283: Le find he Laplace ranform of he aw funcion from example 28 and keched in figure 285a, if < < aw() = aw( + ) in general Here, p =, and he la heorem ell u ha L[aw()] = F () e = F () e for >

21 Periodic Funcion 575 where (uing each of he formula dicued for F ) F () = = aw()e d e d (286a) (286b) = L [ rec (,) () ] (286c) Had he auhor been ufficienly clever, L [ rec (,) () ] would have already been compued in a previou example, and we could wrie ou he final reul uing formula (286c) Bu he wan, o le ju compue F () uing formula (286b) and inegraion by par: F () = e d = e = ( ) e d = e + 2 [ e e ] = 2 [ e e ] Hence, L[aw()] = F () e = e e 2 e = [ ] e 2 e = 2 e e Thi i our ranform If you wih, you can apply a lile algebra and implify i o L[aw()] = 2 e, hough you may prefer o keep he formula wih e p in he denominaor o remind you ha hi ranform came from a periodic funcion wih period p Ju for fun, le go even furher uing he fac ha e e = e 2e/2 e 2e /2 = 2 2e /2 = e /2 e /2 2 e /2 inh(/2) Thu, he above formula for he Laplace ranform of he aw funcion can alo be wrien a L[aw()] = 2 2 e /2 inh(/2) Thi i ignifican only in ha i demonrae why hyperbolic rigonomeric funcion are omeime found in able of ranform

22 576 Piecewie-Defined and Periodic Funcion Addendum: Verifying Equaion (285) Equaion (285) give a formula for adding up k= e kp auming p and are poiive value To derive ha formula, we ar wih he N h parial um of he erie, S N = N k= e kp p = e}} = + e p + e 2 p + e 3p + + e N p Muliplying hi by e p, we ge e p S N = e [ p + e p + e 2 p + e 3p N + + e p] = e p + e 2 p + e 3p + + e N p + e (N+)p The imilariy beween S N and e p S N naurally lead u o compue heir difference, [ e p ] S N = S N e p S N = + e p + e 2 p + e 3p N + + e p} e p + e 2 p + e 3p + + e N p + e (N+)p} = e (N+)p Dividing hrough by e p hen yield The above formula for S N aumed here, hen and hu, S N = e (N+)p e p hold for any choice of p and, bu if p > and >, a lim N e (N+)p =, k= e kp = lim N N k= e kp = lim N S N e = lim (N+)p N e p = e p = e p, confirming equaion (285)

23 An Expanded Table of Ideniie 577 Table 28: Commonly Ued Ideniie (Verion 2) In he following, F() = L[ f ()] h() H() = L[h()] Rericion f () f ()e d e α f () F( α) α i real f ( α) ep α () F() e α α > d f d F() f () d 2 f d 2 2 F() f () f () d n f n F() n f () n 2 f () d n n 3 f () f (n ) () n =, 2, 3, f () n f () d F d ( ) n dn F d n n =, 2, 3, f (τ) dτ f () F() F(σ) dσ f g() f i periodic wih period p F()G() p f ()e d e p 286 An Expanded Table of Ideniie For reference, le u wrie ou a new able of Laplace ranform ideniie conaining he ideniie lied in our fir able of Laplace ranform ideniie, able 25 on page 54, along wih ome of he more imporan ideniie derived afer making ha able Our new able i able 28

24 578 Piecewie-Defined and Periodic Funcion m f y() Y Figure 286: A ma/pring yem wih ma m and an ouide force f acing on he ma 287 Duhamel Principle and Reonance The Problem Now i a good ime o re-examine ome of hoe forced ma/pring yem originally dicued in chaper 7 and 22, and diagramed in figure 286 Recall ha hi yem i modeled by m d2 y d 2 + γ dy d + κy = f where y = y() i he poiion of he ma a ime (wih y = being he equilibrium poiion of he ma when f = ), m i he ma of he objec aached o he pring, κ i he pring conan, γ i he damping conan, and f = f () i he um of all force acing on he pring oher han he damping fricion and he pring reacion o being reched and compreed ( f wa called F oher in chaper 7 and F in chaper 22) Remember, alo, ha m and κ are poiive conan Our main inere will be in he phenomenon of reonance in an undamped yem Accordingly, we will aume γ =, and reric our aenion o olving m d2 y d 2 + κy = f (287) Ulimaely, we will furher reric our aenion o cae in which f i periodic Bu le wai on ha, and derive ome baic formula wihou auming hi periodiciy Soluion Uing Arbirary f The General Soluion A you know quie well by now, he general oluion o our differenial equaion, equaion (287), i y() = y p () + y h () where y h i he general oluion o he correponding homogeneou differenial equaion, and y p i any paricular oluion o he given nonhomogeneou differenial equaion The formula for y h i already known In chaper 7, we found ha y h () = c co(ω ) + c 2 in(ω ) where ω = κ m

25 Duhamel Principle and Reonance 579 Recall ha ω i he naural angular frequency of he ma/pring yem, and i relaed o he yem naural frequency ν and naural period p via ν = ω 2π and p = ν = 2π ω For fuure ue, noe ha y h i a periodic funcion wih period p ; hence y h ( + p ) y h () = for all Tha leave finding a paricular oluion y p Le ake hi o be he oluion o he iniialvalue problem m d2 y d 2 + κy = f wih y() = and y () = Thi i eaily found by eiher applying he Laplace ranform and uing he convoluion ideniy in aking he invere ranform, or by appealing direcly o Duhamel principle Eiher way, we ge where Since ω = κ /m, y p () = h f () = [ ] h(τ) = L τ m 2 + κ h(τ) = [ 2] τ m L 2 + (ω ) Thu, he above inegral formula for y p can be wrien a y p () = ω m h( x) f (x) dx = [ τ m L 2 + κ / m] = ω m in(ω τ) in(ω [ x]) f (x) dx (288) The Difference Formula and Fir Theorem For our udie, we will wan o ee how any oluion y varie over a cycle (ie, a increae by p ) Thi variance in y over a cycle i given by he difference y( + p ) y(), and will be epecially meaningful when he forcing funcion i periodic wih period p For now, le conider he difference y( + p ) y() auming y = y p + y h i any oluion o our differenial equaion Of coure, he y h erm i irrelevan becaue of i periodiciy, y( + p ) y() = [ y p ( + p ) + y h ( + p ) ] [ y p () + y h () ] Now, uing formula (288) for y p, we ee ha y p ( + p ) = ω m = ω m +p +p = y p ( + p ) y p () + y h ( + p ) y h () }} in(ω [( + p ) x]) f (x) dx in(ω [ x] + ω p }} ) f (x) dx 2π

26 58 Piecewie-Defined and Periodic Funcion = +p ω m = ω m in(ω [ x]) f (x) dx in(ω [ x]) f (x) dx + +p in(ω [ x]) f (x) dx ω m Bu he fir inegral in he la line i imply he inegral formula for y p () given in equaion (288) So he above reduce o y( + p ) = y() + +p in(ω [ x]) f (x) dx (289) ω m To furher reduce our difference formula, le u ue a well-known rigonomeric ideniy: +p in(ω [ x]) f (x) dx = = +p +p = in(ω ) in(ω ω x) f (x) dx [in(ω ) co(ω x) co(ω ) in(ω x)] f (x) dx +p +p co(ω x) f (x) dx co(ω ) in(ω x) f (x) dx Combining hi reul wih he la equaion for y( + p ) and recalling he previou reul derived in hi ecion hen yield: Theorem 283 Le m and κ be poiive conan, and le f be any piecewie coninuou funcion of exponenial order Then, he general oluion o m d2 y d 2 + κy = f i y() = y p () + c co(ω ) + c 2 in(ω ) where Moreover, ω = κ m and y p () = in(ω [ x]) f (x) dx ω m y( + p ) y() = ω m [I S() in(ω ) + I C () co(ω)] for where I S () = +p +p co(ω x) f (x) dx and I C () = in(ω x) f (x) dx

27 Duhamel Principle and Reonance 58 p p p a a + p T Figure 287: Illuraion for lemma 284 Reonance from Periodic Forcing Funcion A Ueful Fac Take a look a figure 287 I how he graph of ome periodic funcion g wih period p, and wih wo region of widh p greyed in in wo hade of grey The darker grey region i beween he graph and he T axi wih < < p The ligher grey region i beween he graph and he T axi wih a < < a + p for ome real number a Noe he imilariy in he hape of he region In paricular, noe how he piece of he ligher grey region can be rearranged o perfecly mach he darker grey region Conequenly, he area in each of hee wo region, boh above and below he T axi, are he ame Add o hi he relaionhip beween inegral and area, and you ge he ueful fac aed in he nex lemma Lemma 284 Le g be a periodic, piecewie coninuou funcion wih period p Then, for any, +p g(x) dx = p g(x) dx If you wih, you can rigorouly prove hi lemma uing ome baic heory from elemenary calculu? Exercie 28: Prove lemma 284 A good ar would be o how ha d d +p g(x) dx = Reonance Now conider he formula for I S () and I C () from heorem 283, I S () = +p co(ω x) f (x) dx and I C () = +p in(ω x) f (x) dx If f i alo periodic wih period p, hen he produc in hee inegral are alo periodic, each wih period p Lemma 284 hen ell u ha I S () = +p co(ω x) f (x) dx = p co(ω x) f (x) dx

28 582 Piecewie-Defined and Periodic Funcion and +p p I C () = in(ω x) f (x) dx = in(ω x) f (x) dx Thu, if f i periodic wih period p, he difference formula in heorem 283 reduce o where I S and I C I S = p y( + p ) y() = ω m [I S in(ω ) + I C co(ω)] are he conan co(ω x) f (x) dx and I C = p in(ω x) f (x) dx Uing a lile more rigonomery (ee he derivaion of formula (78b) on page 363), we can reduce hi o he even more convenien form given in he he nex heorem Theorem 285 (reonance in undamped yem) Le m and p be poiive conan, and f a periodic piecewie coninuou funcion Aume furher ha f ha period p, he naural period of he ma/pring yem modeled by Tha i, Alo le I S = p m d2 y d 2 + κy = f period of f = p = 2π ω wih ω = κ m p co(ω x) f (x) dx and I C = in(ω x) f (x) dx Then, for any oluion y o he above differenial equaion, and any >, y( + p ) y() = A co(ω φ) (282) where A = (I S ) 2 + (I C ) 2 ω m and wih φ being he conan aifying φ < 2π, co(φ) = I C (IS ) 2 + (I C ) 2 and in(φ) = I S (IS ) 2 + (I C ) 2 To ee wha all hi implie, aume f, y, ec are a in he heorem, and look a wha he difference formula ell u abou y( n ) when τ i any fixed value in [, p ), and n = τ + np for n =, 2, 3, The value of y(τ) can be compued uing he inegral formula for y p in heorem 283 To compue each y( n ), however, i i eaier o ue hi compued value for y(τ) along wih difference formula (282) and he fac ha, for any ineger k, co(ω k φ) = co(ω [τ + kp ] φ) = co(ω τ φ + k ω p }} ) = co(ω τ φ) 2π

29 Duhamel Principle and Reonance 583 Doing o, we ge and o on In general, y( ) = y(τ + p ) = y(τ) + A co(ω τ φ), y( 2 ) = y(τ + 2p ) = y(τ + p ) + A co(ω [τ + p ] φ) = [y(τ) + A co(ω φ)] + A co(ω τ φ) = y(τ) + 2A co(ω τ φ), y( 3 ) = y(τ + 3p ) = y(τ + 2p ) + A co(ω [τ + 2p ] φ) = [y(τ) + 2A co(ω φ)] + A co(ω τ φ) = y(τ) + 3A co(ω τ φ), y( n ) = y(τ) + n A co(ω τ φ) (282) Clearly, if A = and ω τ φ i neiher π / 2 or 3π / 2, hen y( n ) ± a n Thi i clearly runaway reonance Thu, i i he A in difference formula (282) ha deermine if we have runaway reonance If A =, he oluion conain an ocillaing erm wih a eadily increaing ampliude On he oher hand, if A =, hen he oluion y i periodic, and doe no blow up By he way, for graphing purpoe i may be convenien o ue he periodiciy of he coine erm and rewrie equaion (282) a y( n ) = y(τ) + n A co(ω n φ) Replacing n wih, and recalling wha n and τ repreen, we ee ha hi i he ame a aying y() = y(τ) + n A co(ω φ) (2822) where n i he large ineger uch ha np and τ = np! Example 284: Le u ue he heorem in hi ecion o analyze he repone of an undamped ma/pring yem wih naural period p = o a force f given by he baic aw funcion keched in figure 288a, f () = aw() = The correponding naural angular frequency i if < < aw( ) if < ω = 2π p = 2π The acual value of he ma m and pring conan κ are irrelevan provided hey aify 2π = ω = κ m

30 584 Piecewie-Defined and Periodic Funcion Y T T (a) (b) Figure 288: (a) A baic aw funcion, and (b) he correponding repone of an undamped ma/pring yem wih naural period over 6 cycle Alo, ince he oluion o he correponding homogeneou differenial equaion wa prey much irrelevan in he dicuion leading o our la heorem, le aume our oluion aifie y() = and y () =, o ha he oluion formula decribed in heorem 283 become y() = y p () = 2πm in(ω [ x]) aw(x) dx In paricular, if x <, hen aw(x) = x and we can complee our compuaion of y() uing inegraion by par: Thi implifie o and y() = 2πm in(2π[ x]) x dx = [ x 2πm 2π co(2π[ x]) = 2πm x= [ co(2π[ ]) co(2π[ ]) 2π 2π ] co(2π[ x]) dx 2π + in(2π[ ]) in(2π[ ]) (2π) 2 (2π) 2 y() = 8π 3 [2π in(2π)] when < (2823) m In a imilar manner, we find ha I C I S = p co(ω x) f (x) dx = p = in(ω x) f (x) dx = co(2π x) x dx = = in(2π x) x dx = = 2π Thu, A = (I S ) 2 + (I C ) 2 = 2πm 4π 2 m Since A =, we have reonance There i an ocillaory erm whoe ampliude eadily increae a increae ]

31 Addiional Exercie 585 To acually graph our oluion, we ill need o find he phae, φ, which (according o our la heorem) i he value in [, 2π) uch ha co(φ) = I C (IS ) 2 + (I C ) 2 = and in(φ) = I S (IS ) 2 + (I C ) 2 = Clearly φ = So le Then, employing formula (2822) (derived ju before hi example), y() = y(τ) + n A co(ω φ) = 8π 3 [2πτ in(2πτ)] + n m 4π 2 m co(2π) where (ince p = ) n i he large ineger wih n and τ = n Thi i he funcion graphed in figure 288b Addiional Exercie 282 Uing he fir ranlaion ideniy or one of he differeniaion ideniie, compue each of he following: a L [ e 4 ep 6 () ] b L [ ep 6 () ] 283 Compue (uing he ranlaion along he T axi ideniy) and hen graph he invere ranform of he following funcion: a d e 4 3 b π 2 + π 2 e 2 e e c e 4 ( 5) 3 f 284 Finih olving he differenial equaion in example 28 π 3 / 2e ( + 2)e 5 ( + 2) Compue and hen graph he invere ranform of he following funcion (expre your anwer a condiional e of formula): a d e 2 b π ( + e ) 2 + π 2 e e + e 3 ( + 4)e 2 8e c f e 2 e 2 2e 4 + e Find and graph he oluion o each of he following iniial-value problem: a y = ep 3 () wih y() = b y = ep 3 () wih y() = 4 c y = ep 2 () wih y() = and y () = d y = ep 2 () wih y() = 4 and y () = 6 e y + 9y = ep () wih y() = and y () = 2

32 586 Piecewie-Defined and Periodic Funcion 287 Compue he Laplace ranform of he following funcion uing he ranlaion along he T axi ideniy (Trigonomeric ideniie may alo be ueful for ome of hee) a f () = if < 6 e 4 if 6 < b g() = if < 4 4 if 4 < c ep 6 () d e 3 ep 2 () e 2 ep 6 () f in(2( )) ep () g in(2) ep π/2 () h in(2) ep π/4 () i in(2) ep π/6 () 288 For each of he following choice of f : i Graph he given funcion over he poiive T axi ii Rewrie he funcion in erm of appropriae recangle funcion, and hen rewrie ha in erm of appropriae ep funcion iii Then find he Laplace ranform F() = L[ f ()] a f () = c f () = e f () = g f () = i f () = e 4 2 if < 6 if 6 < 2 if < 3 2e 4( 3) if 3 < if < 3 9 if 3 < if < 2 if 2 < < 3 4 if 3 < if ( ) 2 if < < 3 4 if 3 b f () = d f () = f f () = h f () = j f () = 2 2 if < 2 if 2 < in(π) if < if < if < 2 3 if 2 < < 4 if 4 < if < in(π) if < < 2 if 2 < if 2 4 if 2 < < 4 if The infinie air funcion, air(), can be decribed in erm of recangle funcion by air() = (n + ) rec (n,n+) () n= = rec (,) () + 2 rec (,2) () + 3 rec (2,3) () + 4 rec (3,4) () +

33 Addiional Exercie 587 Uing hi: a Skech he graph of air() over he poiive T axi, and rewrie he formula for air() in erm of ep funcion b Auming he lineariy of he Laplace ranform hold for infinie um a well a finie um, find an infinie um formula for L[air()] c Recall he formula for he um of a geomeric erie, X n = n= X when X < Uing hi, implify he infinie um formula for L[air()] which you (we hope) obained in he previou par of hi exercie 28 Find and graph he oluion o each of he following iniial-value problem: a y = rec (,3) () wih y() = b y = rec (,3) () wih y() = and y () = c y + 9y = rec (,3) () wih y() = and y () = 28 Compue each of he following convoluion: a 2 ep 3 () b ep 4 () c co() rec (,π) () d 2e 2 rec (,3) () e e 2 [ e 5 rec (,3) () ] f in() [ in() rec (2π,3π) () ] g f () where f () = if < 4 2 if 4 < if < 2π h in() f () where f () = co() if 2π < < 3π if 3π < 282 Each funcion lied below i a lea periodic on (, ) Skech graph of each, and hen find i Laplace ranform uing he mehod developed in ecion 285 e 2 if < < 3 a f () = f ( 3) if > 3 b f () = qwave() (from example 282) if < < c f () = if < < 2 f ( 2) if > 2 d f () = 2 2 if < 2 f ( 2) if 2 < (ee exercie 288 a)

34 588 Piecewie-Defined and Periodic Funcion T (a) (b) 2 T Figure 289: (a) The quare wave funcion for exercie 283 a, and (b) he recified ine funcion for exercie 283 b e f () = f f () = in() if < < 2 4 if 2 < < 4 f ( 4) if > 4 (ee exercie 288 i) 283 In each of he following exercie, you are given he naural period p and a forcing funcion f for an undamped ma/pring yem modeled by m d2 y d 2 + κy = f Analyze he correponding reonance occurring in each yem In paricular, le y be any oluion o he modeling differenial equaion and: i Compue he difference y( + p ) y() ii iii Compue he formula for y() auming y() = and y () = (Expre your anwer uing τ and n where n i he large ineger uch ha np and τ = np ) Uing he formula ju compued for par ii along wih your favorie compuer mah package, kech he graph of y over everal cycle (For convenience, aume m i a uni ma) a p = 2 and f i baic quarewave keched in figure 289a Tha i, if < < f () = if < < 2 f ( + 2) in general b p = and f () = in(π), he recified ine funcion keched in figure 289b c p = and f () = in(4π)