ririgrrlarity ancl to lie right ol’ {lie leftmost siiignlaritt... Rctanse 01 the properties UF 1ia;vc discimed.. should 1x2 clrar t h a t thc singularities of the Laplace tiaiisform c\it
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esal - the sum of a r ~ ~ h t - s i ~ ~ e
of c o n v ~ r ~ e ~ ~ e is a str
r i ~ ~ t - s ~ ~ e ~ and left-.sid
nsform of a bilateral sigrial c m (5ee (4.13)) b~ p r r t together
Iual rcgions of (‘0 germ’, a i t d the corxtpletcl
re>gion of wrivrrgcncr is t h i r iiitersertion ‘I’liis iriteriert ~ o i r i 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 1 h r left - b i d e d
component
In order i o fiilly imderstarirl this wv re-cxaniinc bhaiiiples 4 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 = - 2 ) 1ic.s t o tirr left of
(lie polc of tlie lrft-sicled cortipoiicnt ( 5 = 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 = - 1 ) lies to the right of the pole of Lire lcft-dw1 (‘Ol?-tpOrl(’Kl~
( s = -2) and (lie iritersection is empty s o 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 ( J f convci griicc
itself, there cariiiot be any siiigularities
E x ( t ) has a finite duration and if G ( T ( L ) ) coirverges for at least 0118)
value of s, then the r e ~ i o l ~ of c o r i v ~ r g ~ ~ i e ~ i s the entire ~~~~~~~~*
If x ( t ) has finit r tluratiorr
“c can iepreserit its I ~ p l
harid srtlc arid we clo rm
coiiceriiiiig the legion of
coritam singularities its
., whcn r ( t ) is oiily non-xrro for 4 < <: R thcu trarisform by oiily tlici first integral ori the right-
ed t o hnti auy limits All of t h v coiisicleratroiis
s a q , altliongli 5 (I) c:m
t i n integial o w r ~ ( t ) rlo ancl in that case’, t,hc L forin for ~ ( t ) therefore d