2. The Laplace Transform

Transcription

1 Th aac Tranorm Inroucion Th aac ranorm i a unamna an vry uu oo or uying many nginring robm To in h aac ranorm w conir a comx variab σ, whr σ i h ra ar an i h imaginary ar or ix vau o σ an w viw a a oin in h comx an b a uncion o im in, W orm a rouc on h inrva [ ) an ingra i rom zro o ininiy Thu, whr in h owr imi o ingraion man -, i h han i imi o Hnc, i incu h uni imu a h origin, i i incu in h inrva o ingraion Th uncion i ca h aac ranorm o h im uncion an i ca h comx rquncy Thu, w hav a ranormaion rom imomain ino rquncy omain Th oraion o h aac ranorm i ab ( ) Th oraion o going bac o h im omain uncion ranorm i no by rom i aac

2 ( ) Throughou h who cour w wi u ur-ca r or h aac ranorm an owr-ca r or h im uncion No a uncion hav a aac ranorm or xam grow o a a ha hr i no vau o or which h ingra i ini I can b hown ha any uncion or which hr ar oiiv conan M > an c > uch ha ha h aac ranorm Now w conir numrica xam Exam a c M or a >, whr h owr coicin a i a ra conan Subiuing ino w obain R σ > a a a ( a) ( a), hn ( a) ( σa) a bcau ( σa) a Hnc, w may wri a a ( ) Sinc h ingra xi ony i σ > a, w ay ha h rgion o convrgnc o h aac ingra, in h conir ca, i a ar o h comx an crib by inquaiy R > a Howvr, h ruing xrion ma n or a in h comx an, xc or a Conquny, w conir o b w in by quaion or a a

3 Exam u conir h uni uncion, in in Scion, crib by h quaion (4) ra bow: u or < or > u () A o o u i hown in ig u ig Uni uncion To comu h aac ranorm o h uni uncion w vaua h ingra ( u ) u u aum R σ >, hn a an ( u ) (4) ho Thu, h righ ha o h -an ( R ( ) > ) rmin h rgion o convrgnc Howvr, h uncion (4) ma n or a in h comx an, xc or ; conquny, w conir o b w in by quaion (4) or a Obrv ha h aac ingra i in ony bwn an Th bhavior o h uncion or < ha no ac on i ranorm or inanc h

4 4 uncion or a, an u hav h am aac ranorm Simiary, a h hr uncion hown in ig hav h am aac ranorm bcau a h uncion ar inica or (,) a ( a ) a ( b ) a u a > a > ( c ) a a > ig uncion having h am aac ranorm Baic rori o h aac ranorm Uniqun rory Th aac ranorm i a on-o-on raion bwn h im uncion in on [, ) an i aac ranorm xanaion I wo uncion an hn h quaion Thi rory n an hav h am aac ranorm ( ) ( )

5 ho I man ha h uncion can b irn ony in oin o iconinuiy rom h nginring oin o viw, an can b conir o b h am uncion or inanc, h oowing uncion: i < i i > i i i < > hav h am aac ranorm rom h nginring oin o viw boh uncion can b conir a h am uncion ahough, ricy aing, h uncion ar iinc Thu, h uniqun rory a ha or a givn im uncion hr xi a uniqu aac ranorm o hi uncion an convry, or a givn uncion hr xi a uniqu im uncion, in h abov crib n, bing h invr aac ranorm o Thi rory aow u o ranorm a im-omain robm ino a rquncyomain robm an hn go bac o h im-omain ouion inariy rory ) I ( an ar wo uncion having h aac ranorm an, rcivy, an c an c ar arbirary conan, ra or comx, hn Proo ( c ) c c c () Th roo i ba on h inariy rory o h ingra c ( c c ) ( c c ) c c c

6 6 Exam W u hi rory o in h aac ranorm o co By Eur ormua w wri co Hnc, w obain co Simiary w can in in Ao aii inariy rory (6) Dirniaion ru (7) whr man - Proo Th ingra can b ingra by ar by ing

7 7 v v u u W u h ormua b a b a b a u v uv v u an a a uiciny arg, o ha R im im σ Hnc, ho To in h con rivaiv w ay h ru wic [ ] iwi, w can riv an aroria ormua or h n-h rivaiv Th irniaion ru nab u o ov inar irnia quaion by man o h aac ranorm a i wi b iura via an xam Exam 4 u conir h irnia quaion

8 8 in x x x W ay h aac ranorm o boh i o h quaion an u h inariy rory ( X x X x x in in ) To rmin h aac ranorm o w can u Eur ormua in in Subiuing h abov xrion an h iniia coniion x, w hav X W ov hi quaion or X X A w now in co Hnc, w obain X x in co

9 9 No u Sinc δ ( (9)), w can ay h irniaion ru o in h aac ranorm o h uni imu: u ( δ ) ( u ) u 4 Ingraion ru Proo Th ingraion ru i a oow u conir h uncion an comu i im rivaiv g τ τ (8) g ( τ ) τ an g ( τ ) τ Aying h irniaion ru, w hav or quivany g G G

10 6 Exam W comu h aac ranorm o h uncion obain rom h uni uncion by h ucciv ingraion r > or or u( τ ) τ u Now w rmin h aac ranorm o h uncion ingraion ru r uing h ( r ) ( u ) u τ ( u ) τ Thu, ( u ) (9) Nx, w conir h uncion r r( τ ) τ τ u( τ ) τ u an comu i aac ranorm u r τ ( r ) τ Hnc, w obain Coninuing hi way w can how ha ( u ) ( ) n n n! ( u ) ( ) n

11 6 Shiing rory u conir a uncion an muiy i by h uni uncion A a ru, w obain h rouc u which i inica o or oiiv an i qua o zro or ngaiv Nx, w hi h rouc u by om oiiv conan h Th hi ( or ay ) uncion i ( h) u( h) ig how an xam uncion, a w a u an ( h) u( - h) ( a ) ( b ) u ( c ) (-h) u(-h) h ig Craing h uncion ( h) u( - h) h aac ranorm o b ( h) u( - h) can b oun a oow Th aac ranorm o ( ( h) u( h) ) ( h) u( h) h x, hn

12 6 ( x h) ( ( h) u( h) ) ( x) u( x) h h x h ( x) x x Thu, w hav h ( ( h) u( h) ) Exam 6 W comu h aac ranorm o h rcanguar u uncion hown in ig4 a h ig 4 Rcanguar u uncion Th u uncion can b xr in rm o h uni uncion u u( h) () Hnc, uing h hiing rory w hav h h ( ) u u( h) 6 Convouion horm W conir a cia ca o h convouion (4) whr h uncion an ar qua o zro or <, ca h uniara convouion

13 6 Th convouion horm a ha (4) an whr an ar h aac ranorm o rcivy Proo W u h aac ranorm o an ubiu Hnc, w hav ( τ ) ( τ ) τ τ ( τ ) τ W inrchang h orr o ingraion () τ ( τ ) τ, () Now w x τ o ha an x A a ru, w obain ( x τ ) x τ τ τ x x x τ τ

14 64 or x <, an τ < w can rac h owr imi o Sinc ( x) ingraion qua o (-τ) o h inrna ingra by τ τ x x x Sinc h inrna ingra in (6) i, w hav which n h roo No τ τ τ Sinc an or < w in Thu, ( τ ) ( τ ) τ ( τ ) ( τ ) τ ( τ ) ( τ ) ( τ ) ( τ ) τ τ (6) τ ho ( τ ) ( τ ) τ (7) Paria racion xanion In hi Scion w conir h robm how o go rom h aac ranorm bac o h im uncion, or, in ohr wor, rom h rquncy omain o h im omain A vry uu chniqu in hi i i h aria racion xanion u conir a raiona uncion

15 6 n whr n an ar oynomia wih ra coicin W aum ha n an ar corim, i any common acor ha bn canc ou, an h gr o n i mar han h gr o A ir, w conir a ca whr ha im zro,,, m Th zro o h oynomia ar ca h o o Thn h uncion can b wrin a oow n α n ( )( ) ( ) m (8) Wihou o o gnraiy w can aum α I i a w nown ru o agbra ha can b cii a m whr h numbr i ca h riu o a h o I i ay o ha in h nighborhoo o any o bow u To obain w muiy by (- ) an quaion ho In gnra, w obain ( ) ( m ) (9) Thn a rm in (9), xc, n o zro Thu, h im ( ) ( ),, m im Exam 7 u conir h uncion

16 66 Th o o ar: -, - Th aria racion xanion o i whr an can b comu uing im im Now w can in u conir a raiona uncion which ha i oub o a an im o a,,, m Thn, w hav m n I i w nown rom agbra ha in uch a ca h aria racion xanion ha h orm m To rmin w muiy by an Uing w hav

17 67 im To in w roc a oow W muiy by m () an irnia boh i o () wih rc o m Hnc, w hav im (4) Th coicin, m ar comu a bor Uing h aria racion xanion w rmin m m u conir h ir rm on h righ han i an ay h raionhi (9) ra bow an h raionhi

18 68 ( ) ( g ) g G( ) Hnc, w obain () ( ) Uing (), w hav m (6) Aicaion o h aria racion xanion o comuing h invr aac ranorm wi b iura via xam Exam 8 u conir h raiona uncion ( )( ) To in h o o hi uncion w ov h oynomia quaion ( ) ( ) an orm h aria racion xanion whr: (7)

19 69 im im ( ) ( ) ( )( )( ) ( )( ) ( ) im 4 im 4 ( ) ( ) ( ) im W h abov coicin ino (7) an rmin h invr aac ranorm 4 ( ) 4 ( ) No ha or any h con an h hir rm orm a air o comx conuga numbr Conquny, hir um i oub ra ar Hnc, w hav Exam 9 4 ( ) R co( 4 ) W in h invr aac ranorm o h uncion ( )( ) uncion ha on oub o - an on im o - Th aria racion xanion i whr: ( )

20 7 im im im im im Hnc, w may wri i