In Chapter we started with t i n e furic1ion5 and tlirnugli valuation of the iutegl a1 cMitiition iii (1.I ) , verified tlic Laplace transform For the reverse, thcx so-c-alletl u m m e Loplaw fran.sjorni we gave the tyiiation (4.2) without giving a jusli In t h i s chapter we %ill look at the inverse trandorIn iii depth with the c-orrrplPx fiirirtion t henry LVP restrict ourselves to Laylace transforms that oidy have poles, arid ham no sigiiificarit singularitits It will be shown that for these e.;, carrying out the iiiverse Laplace t,raiisfoirn leittls to siinplt calculations that (‘an~I)P per for iirctl without having to continually refer hack t,o function t l i e o r ~ :Tlie first step is to coinrriit tlie iiripoitarit r t w l t s from complex analysiq to i ~ i e ~ ~ r o i j r lex &(si = ( (.I + Q ( s ) e (5.1) of tlic cornplcx variable s Tlir real and imaginary parts of Q(s) axe denotrJd by Q , i s ) arid Q l ( s ) ibe the integration of such a fuiictioii it is riot siifficieiit to give two limits of intrgratirm As s may take any value in the compkx plarw, all of the vahies on t h c path between llie start and cntl points niust be given S i d r an integral is writ1 PI^ (5.2) wlicre 1V is tlic path of iritcgratioii irr the complex plane TCJ t M n r tliis path precisc>ly,the vtiliies ot s that lie on the pti,th x e given as a fiiriction of tlie real parameters I / Tlris is called a par.urrr~f?ac.~ r v e Figurc 5.1 show5 such a paiaixietric p i t h oU intrgrntioii in thc i-plane For if = v , s(v) takes thc coriipl~xvalue of thca start poirit A likewise f o r E / = ~ ( ) ) lakes thc value of the enclpoiiif [ I For V A < v < ~ € s ( v ) follows the desired path of intc~gratiori14 Thr integral (5.2) means lhnt tlie vdue of &(s)ds is; eccuiiiitlt~tedfrom all of the infinitesirrial t~icmciitt, d s on the lrngth ot the palli 1%’