4.7 Properties of the Laplace Transfiorni so 75 ee For the simple sigiials we have wrisitIeicd so far, e v a h t i n g the i n t ~ g r d(4.1) Lias berii tlw shortcsl way to find thc Laplace trarisforru Wlirii tl~alitigwith cmuplicatecl signal? it i s an mlraiitagc, hottevcr to be able to use tlic pxopcrties of the and avoiding liavitig t o cvnluate the grnl clirrctly Oftcn 11 signal (e.g diflert o pcrtornr ail o p e r d h n 011 it t imc-dt twtiation) in the frecii~‘3rrc,y-doirIwin We will learn the rtiost iriiportmt propertics of the liitplncc~transform in tliis section arid foranulate thpoicxis Ihi tlicrii ‘I’hc tlicoreiris will b r rieedctl in l a k r calculalioris usiiig I hc Laplace traiisforiii, hi all v;tli~>s on the co~npltlxfi.c~lnr.ncyplane as long ab both L{j’(t)}i d L{,9(t)) exist The region of corrvt‘rgence for the cwri1)incd funclion is tlir iritcrsectioii of I lir~ rcgtoiis of convergence for the individiial tmictioris Sirigiilarities irray br ic~iovcc-l I y the addition, l i o ~ h , so in generd the conibined region o j super set of the iritri on bctwecn the indivitliial regions of convergence: [ ROC‘{cif + h q ] T: ROCC(j {RiO r / y } l (42 ) To show lincwrity, n j ( t ) -+ b y ( f ) is ixisci ted into tlw tlcxfinition of tht J,aplacc t r i t n s h m (4.1j From the linearity of the intcy,rat,ioii, (4.20) i s iinmeditLte1y ohta ivicrl