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1、TheLaplaceTransform:TheoryandApplicationsJoelL.SchiffSpringerTomyparentsvItiscustomarytobegincoursesinmathematicalengineeringbyex-plainingthatthelecturerwouldnevertrusthislifetoanaeroplanewhosebehaviourdependedonpropertiesoftheLebesgueintegral.Itmight,perhaps,bejustasfoolhardy
2、toflyinanaeroplanede-signedbyanengineerwhobelievedthatcookbookapplicationoftheLaplacetransformrevealedallthatwastobeknownaboutitsstability.T.W.Korner¨FourierAnalysisCambridgeUniversityPress1988viiPrefaceTheLaplacetransformisawonderfultoolforsolvingordinaryandpartialdifferential
3、equationsandhasenjoyedmuchsuccessinthisrealm.Withitssuccess,however,acertaincasualnesshasbeenbredconcerningitsapplication,withoutmuchregardforhypothesesandwhentheyarevalid.Evenproofsoftheoremsoftenlackrigor,anddubiousmathematicalpracticesarenotuncommonintheliteratureforstudent
4、s.Inthepresenttext,Ihavetriedtobringtothesubjectacertainamountofmathematicalcorrectnessandmakeitaccessibletoun-dergraduates.Tothisend,thistextaddressesanumberofissuesthatarerarelyconsidered.Forinstance,whenweapplytheLaplacetrans-formmethodtoalinearordinarydifferentialequationw
5、ithconstantcoefficients,(n)(n−1)any+an−1y+···+a0yf(t),whyisitjustifiedtotaketheLaplacetransformofbothsidesoftheequation(TheoremA.6)?Or,inmanyproofsitisrequiredtotakethelimitinsideanintegral.Thisisalwaysfroughtwithdanger,especiallywithanimproperintegral,andnotalwaysjustified.Ihav
6、egivencompletedetails(sometimesintheAppendix)wheneverthisprocedureisrequired.ixxPrefaceFurthermore,itissometimesdesirabletotaketheLaplacetrans-formofaninfiniteseriestermbyterm.Againitisshownthatthiscannotalwaysbedone,andspecificsufficientconditionsareestablishedtojustifythisopera
7、tion.Anotherdelicateproblemintheliteraturehasbeentheapplica-tionoftheLaplacetransformtotheso-calledDiracdeltafunction.Exceptfortextsonthetheoryofdistributions,traditionaltreatmentsareusuallyheuristicinnature.InthepresenttextwegiveanewandmathematicallyrigorousaccountoftheDiracd
8、eltafunctionbasedupontheRiemann–Stieltjesintegral.Itiselement
