!lieu L { r ( t ) }e x i h fur c' < Re{s) < D As the estimates iiscd are sufiicient but not necrss;tr;y conditions for the convergence of integrals, the existence conditions for Laplace lmnsforurs derived from hc siifXicierrt but not I K y This meails that! th(i ucy$m of ~ r' i ianti U lit largcr than the stiip With the ex;myles in Sectioii 4.4, m7e saw that fiiwtiom of t,irrit> can be uniquely expressed bj7 their Laplace transforms, and likewise, the Laplace Lrmxifornis each have a uriiquc function of t iiur, if tlrr region of ccmverge~i(~ is coir into cr)nsitlcrot,icin \\'c a,r(' intcic ;tetl irl knowing wlicthcr assigning R tuncl ion of t,imc ( a Laplace traiisforrn is iiriique or does this representation of tile function lose information that, cannot be recovered by the inverse Lap1ar:c transform The lollowiug thcorcnt gives tkir ans~vcr If the following conditions for two functions of time are true: khen f ( t ) = g ( t ) rvcxywhcrr whrrc f ( t ) and y(t) izwe continuous The proof of this theorrm can be found 14,Chapt s 61 Iristead of cmmining the dttails of Ihe proof we will corisider two exaniples that clarify the conditions of the theoIerii Exaniple 4.1 The function of time f ( t ) = e ( l ) and g ( t ) = c(-t) fulfil1 bol,h conditions and For the Laplace t,rmdorru F ( s ) = G(R),hiit with e;xh diffcrenl region of convergrm'F", so that no overlapping o w i m e(t) o a S , I!ic{cs}>