5 Cornplcx Anal) sis m t i the Inversc Laplace Transform 102 we obt,dil;i K =; C = X'e (*anexpert that fur a second-order denominator with rcal cvelliclrihs arid ccimplcx conjup,aiCA p i k , that tlic col rcbpoiiding f wirt ion of t imr i\ romp(mtl ol ternis WP Ilricrefore waiit t o bring sccond term of t,kio partial fraction espanuion (5.49) into a form tEiat corresp t o siiie ~ r i t cminc l function5 of tinie '2lliis can be done by coinpletiiig t,lie squ Bs + c 92 I s t 10 + 52 + s + 40 -6(s 2) '-6 ( -+ ) L 36 + (5.52) Piittirig this into (5.49) givcs the rcprrse~itatiorr (5.53) F(s) = (5 + A - -+- B1 l)( 4-2)s s 3- s + _I + -.L;1' ( s -r ay (5.55) For simple poles the coefiiicienis c m be calcula~etl(5.39): A = Lirn [ I - ' ( s ) ( s 8- -1 + I)] =- I (5.56) (5.57) again Substit uling s = -1 gives llie exact value ol A becmise the othcr two lorins becoiiic zero This prtrccdure also ivorlrs for Nil so F ( s )(s + ) = A-( ss -I+2I) d + BI(S + ) ,132 (5.58)