~~~~~~~ oft If an analytic fwiction f (.) ii kiiowri along a closcrl path I T - aiicl F ( s ) i s arrnlyiic* enclosed region (Fig ti.5), tlw Caudi> inityy,jI (5.16) can be rxspd to calculat,~F ( s ) (vc~rywhcwin 1he rnclowl region T l i ~polr i n (5.16) IS niovod t o location oI interest It is thcwforc rompletdy su-fficicnt t o know F ( s ) or1 tlir bortic.1 of a n analytic rcbgion It is likcwse siifficieiit to ltnovv F ( s) along a line parallel to I ~ ini;tgiriaiy P axis (5 28) to h c able to cnlculate every valiie to tlrr riglit of thr line Dccau\c it is ililrercLntiAAe a11 aiialytic fimction has a strong inner structurti 1x1 het, F ( ) can tvem hc audyticallv conti*ruetl outside of closed path W a n d a when F ( s ) is only kriowri along d the Caiichy intrgral, tlw reviem '\;OK that w e haw ft c o n d i d d "A?>will now us(' the result t o tlrrive the l(Jri1111lil for the invase Laplace iraristorrri arid give siiriple wtys to prrlorni it In llris rt3sp , tlicl Cancliy iirtt.gral i s impor twnt tor two reasons: L~ B T2ir sirripltfic*atioiiof the pittli of irit rgrat ion cwrisidcr ed in sec.kiuii 5.4.2 l ( w h directly t o tlie formiila for the inverse Lnplace traiisfoim ion 5.4.1) allows a simplified ( alciilation of the irtver-x transhrrn (wit Inoiit complex integration) for systcms with single 01 multiple polcis E To derive the iiivcrsr Laplam Ir a n s h i m wc start with tlir (:auchy integral in the foirti of (5.28) The condition (5.23) i b always fulfilled by a rational fiaation