1.5.Region of Corrvergerice of the Lap1uc;e Traiisform 91 ~ esal - the sum of a r ~ ~ h t - s i ~ ~ e of c o n v ~ r ~ is ea ~ str ~e r i ~ ~ t - and s ~left-.sid ~ e ~ nsform of a bilateral sigrial c m (5ee (4.13)) b~ p r r t together detl conipoiients properties we have jwt Iual rcgions of (‘0 germ’, a i t d the corxtpletcl re>gionof wrivrrgcncr is t h i r iiitersertion ‘I’liis iriteriert ~ o i ri s a lies L o the left ctf the riglitniosf ririgrrlarity ancl to (lie right ol’ {lie leftmost siiignlaritt Thc interscctiori is empty unlcss all of tlw singulnrities iu the right,-detl c o r a p o n t A i i t lie to the left of the sirigularitim in 1h r left - b i d e d component In order i o fiilly imderstarirl this wv re-cxaniinc bhaiiiples 4 and 4.5 In ICxaniylc 4.1 tlic polc of the right-sitled c - o r r i g m i w L ( A = - ) 1ic.s t o tirr left of (lie polc of tlie lrft-sicled cortipoiicnt ( = I) and thc region of coiivergelicv IS the strip Leiweri lhesc poleq Iri Exaniple 4.5, t h r pole of the right-sitled cuniporintt ( s = - ) lies to the right of the pole of Lire lcft-dw1 (‘Ol?-tpOrl(’Kl~ ( s = -2) and (lie iritersection is empty so irk t h i s case the integral (1.13) dues not (~OTIVCT gc Rctanse 01 the properties UF 1ia;vc discimed i f should 1x2 clrar t h a t thc singularities of the Laplace tiaiisform c\ithcr lic t o the left or {‘tie rty+iri of convergerice (sitigii1aritic.s of the right-sided cornpontwt) or to the riglit (singiiLLrities of Itlie left-sided coiqxmciit) In the region (Jf convci griicc itself, there cariiiot be any siiigularities E x ( t ) has a finite duration and if G ( T ( L ) )coirverges for at least value of s, then the r e ~ i o of l ~c o r i v ~ r g ~ i~s ithe e ~ entire ~~~~~~~~* 0118) If x ( t )has finit r tluratiorr , whcn r ( t )is oiily non-xrro for <