Lecture 18: The Laplace Transform (See Sections and 14.7 in Boas)

Transcription

1 Lecure 8: The Lalace Transform (See Secons 88- and 47 n Boas) Recall ha our bg-cure goal s he analyss of he dfferenal equaon, ax bx cx F, where we emloy varous exansons for he drvng funcon F deendng on s roeres In our dscusson of he Fourer ransform we have assumed ha he funcon beng ransformed was (absoluely) negrable, Recall ha hs consran s ycally sasfed because d F (8) F s nonzero for only a fne me nerval In a wde range of neresng suaons hs s no he case and we need a mehod o rea such suaons In arcular, we wan o consder suaons where he drvng erm s no negrable, bu does urn-on a a secfc me, e, F s nonzero for only half of nfne me We can always ranslae n me so ha he funcon urns on a (or any oher fxed me ) Thus we wan o consder he case F d F,, (8) We can roceed smlarly o he Fourer ransform case bu here we assume ha he frequency (n he ransform) has a negave magnary comonen,, o render he negral n he ransform fne In fac, for he Lalace ransform we focus on he magnary ar, and defne he ransform as (noe no s here, even n my noaon) F d e F G (83) To see how he Lalace ransform works consder a few examles (see he able on age n Boas and a varey of oher exs on he subjec): Physcs 8 Lecure 8 Wner 9

2 d e Re, n n n! n d e n,re, n n a a e d e e Re a, a sn d e sn Re Im, cos d e cos Re Im, snh d e snh Re Re, cosh d e cosh Re Re (84) Noe ha n order for he negrals o exs here are consrans on boh he nal funcon and he ransform arameer As wh he Fourer ransform, he Lalace ransform s a lnear oeraon and hus he ransform of he sum of wo funcons s he sum of he ransforms of he wo funcons ndvdually, whle mullyng he funcon by a consan mulles he ransform by he same consan, cg F F G G cf, (85) If wo funcons of me are jus ranslaons (n me by ) of each oher, e, he new funcon urns on a raher han, F F (86), Physcs 8 Lecure 8 Wner 9

3 hen he Lalace ransforms are smly relaed by an exonenal facor G d e F d e F e d e F e G (87) If wo funcons of me are relaed by a (real) exonenal facor, F e F (88), hen he Lalace ransforms are smly relaed by a ranslaon (n sace) G d e F d e e F d e F G (89) If wo funcons of me are relaed by a lnear facor of me, F F, (8) hen he Lalace ransforms are smly relaed by a dervave G d e F d e F d d d d e F G d (8) The Lalace ransform of a funcon exressed as an negral roduces an exra facor of, Physcs 8 Lecure 8 3 Wner 9

4 F df G d e F d d e F e df de df df G e (8) The Lalace ransform of a dervave brngs n a facor of and magcally nroduces he usual boundary condon n me, d F F d d G d e F d F e d e F df e G F (83) Fnally consder a funcon of me defned as a convoluon of wo oher funcons of me, F d F F (84) 3 Jus as n he Fourer ransform case, he corresondng Lalace ransform s jus a roduc, Physcs 8 Lecure 8 4 Wner 9

5 G d e F de d F F 3 3 d F e de F G G (85) Thus he Lalace ransform of a convoluon (ofen wren as F F ) s he roduc of he ndvdual ransforms, jus as n he Fourer case (exce for he facor) So now consder he Lalace ransform of he usual dfferenal equaon, ax bx cx F We have x d x e x x d x d x e x x x x x, d a b cx a x x bx x d d ax bx cx a x x x b x x c x F G a b c a, G a x x bx G a b x ax (86) So now we wan o nver hs ransform (see Secon 47, age 696 n Boas, he Bromwch negral) Ofen we jus look u (afer suable manulaon) n a able as a lnear combnaon of known ransforms (or we can use Mahemaca), or we can use he exlc nverse ransform Recall ha we sared wh he Fourer ransform and hen swched o an magnary We can hnk of hs as a roaon from he real axs o he magnary axes As a resul he corresondng nverse ransform s exressed no as an negral along he real axs (as was for he Fourer Physcs 8 Lecure 8 5 Wner 9

6 ransform), bu raher along a lne arallel o he magnary axs o he rgh of any sngulares (recall he consrans above n Eq (84)), G F de G (87) Here c s larger han he real ar of he oson of any sngulary of G() n he comlex lane Noe ha he facor s back and here s a facor of due o our change of varable (o magnary ) As usual we evaluae hs negral by closng he conour (o he lef for ) wh a semcrcle a nfny and use he Resdue Theorem Noe ha for, he exonenal facor ells us o close he conour o he rgh In hs case we encrcle no sngulares (snce, by defnon, we are already o he rgh of all sngulares) and fnd ha he negral vanshes auomacally Ths s jus he desred resul as we assumed he drvng funcon vanshes for Anoher way o address Eq (86) s o nerre he frs erm n he las lne as a roduc of Lalace ransforms and use he convoluon resul Thus we can wre G c c F a a e e F (88) a x d F e e a The remanng erms n Eq (86) are jus he comlemenary soluon wh he nal condons already mosed Le us close hs dscusson wh an exlc examle Consder he dfferenal equaon x x x x, for >, = for <, wh nal condons x (n order o be conssen wh he equaon for < ) Snce he drvng force s neher erodc nor negrable, s deal for a Lalace ransform analyss Wh G (from Eq (84)) we can use Eq (86) o wre ha Physcs 8 Lecure 8 6 Wner 9

7 x G a b x ax a b c (89) To smlfy hs exresson for he nex se we can sl (aral fraconae) no ndvdual erms (oles) ha we can denfy from a able of Lalace ransforms We fnd (8) We can now fnd he nverse ransform erm-by-erm (recall ha boh he ransform and nverse ransform are lnear oeraons) usng he resuls n Eq (84) and he secal cases of Eqs (87), (89), (8), (8), (83) and (85) In arcular, we have so ha we fnd e, (8) e e x e (8) Noe ha, as execed, he nal condons (boundary condons n me) are bul no hs soluon, e, s a lnear combnaon of he smles arcular soluon,, and he comlemenary soluon, a be, wh consans fxed o mach he nal condons For racce we can also look a he nverse ransform as an exlc negral n he comlex lane We have Physcs 8 Lecure 8 7 Wner 9

8 c x x d e (83) c From he oles n Eq (8) (a and -) we have ha c (e, o he rgh of he orgn) As noed above, for we close he conour o he rgh o fnd zero For we close he conour o he lef and ck u he oles a and In order o use he Resdue Theorem of Eq (73) we mus deermne he coeffcen (e, he resdue) of he sngle oles a each on Snce we have, n fac, double oles a each on, we mus erform a Taylor seres exanson of he res of he negrand near each on o deermne he resdue Near we have, exandng boh facors n ower seres n, e, (84) where he frs facor comes from exandng he exonenal and he second facor s from exandng he denomnaor Thus near he ole he negrand s aroxmaely e (85) So we can read off he resdue a o be, he coeffcen of We recognze hs as he arcular soluon o he orgnal dfferenal equaon The corresondng analyss for he oher ole yelds e e e e e e e (86) Physcs 8 Lecure 8 8 Wner 9

9 Thus he resdue a e ogeher wh he Resdue Theorem we have s, he coeffcen of Pung hs x c e d c Resdue Resdue,, e,, (87) So, as execed, we oban he earler resul Of course, we can also fnd he resdues by usng he aral fraconaed form n Eq (8) Fnally we can use hs examle o also llusrae how we can use he convoluon resul of Eq (85) o exress he soluon of a dfferenal equaon n erms of a convoluon negral Reurnng o he algebrac resul n Eq (89), we can rewre as (beng careful ha we have a degenerae case wh and hus an exra facor of n he comlemenary soluon) x F a b x ax a b c e So fnally we exress he soluon n erms of he convoluon negral of he roduc of he nverse ransforms of he facors n he las exresson, Physcs 8 Lecure 8 9 Wner 9

10 x d e de d e e de e de e e e e e e e So agan we oban he same resul The reader s encouraged o decde whch of he above aroaches s he smles More examles of he use of he convoluon are avalable n Aendx B o hs Lecure Physcs 8 Lecure 8 Wner 9