'79 4.7 Properties of the Laplace Tranbfurnr i t is the sitim' as the dcrivative of ~ ( twlicre ) I ( f ) r m be tlifcrciitiated (t # (1) a n d it is not dt.firred at t =- This h C J k in the defined regioii is iiot a problcm, as wt- h a v ~alrcady see11 in 011 4.6.2 that tlc\iaiicc~sat isolakd Imints not, infliiorice the ViililP of the LA I1 diould also be notrtl that a t t = the d u e u ( ) is e q t ~ to l ilie right-sickcl limit o-t ~ ( tat) i This vitliie is also kriown as x ( i 1: i) fi > u(0)= lint r(0 i- 6) - ,c(O+) d -0 (4.33) We h a w now obtained the> cliflertmtiation throrciti ot the (1)iliXteral) Laplace t ransform for iigIit-t-,idetlsigiials: C{ I " ( f ) ) = S,U(S) ~ ZjO+) ('1 34) It expiebses t h r h t c t i o n ~ " ( t-)which like (4.32) iepmimts ?lie dcrivtiitive of J ( t ) ~cLnshmof x ( t ) arid thr 2(0 t ) t ifre inilialh oiily tielirretl f o ~f > tlic ~ ~ n ~ l n ~Lwplnce eral oiily for positivc t i m e f < x),is oitcii used is not clistingiiishabie from the bilateral Laplace transform, For iiniliatc>rai signal uiir differentiation theoierri (3.34) i t i s often given 111 the a i ~ dtherefore hits tl x: fOK 121' I LJ{ mid tlic value of ~ ( slio~ild ) \hp iinderstood I,o hr t h e light-hand side liiirit (4.33) Tlw iiitrgratioii theorrrrr of tlic tiriilatcral Laplate trailstorm m;\trhes the integratiori tlicoiem for thc iinilatrrsl LC>p1~i(-(-' tr msform (4.27)