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1、TheLaplaceTransform:TheoryandApplicationsJoelL.SchiffSpringerTomyparentsvItiscustomarytobegincoursesinmathematicalengineeringbyex-plainingthatthelecturerwouldnevertrusthislifetoanaeroplanewhosebehaviourdependedonpropertiesoftheLebesgueintegral.Itmight,perhaps,bejustasfoolha
2、rdytoflyinanaeroplanede-signedbyanengineerwhobelievedthatcookbookapplicationoftheLaplacetransformrevealedallthatwastobeknownaboutitsstability.T.W.Korner¨FourierAnalysisCambridgeUniversityPress1988viiPrefaceTheLaplacetransformisawonderfultoolforsolvingordinaryandpartialdiffer
3、entialequationsandhasenjoyedmuchsuccessinthisrealm.Withitssuccess,however,acertaincasualnesshasbeenbredconcerningitsapplication,withoutmuchregardforhypothesesandwhentheyarevalid.Evenproofsoftheoremsoftenlackrigor,anddubiousmathematicalpracticesarenotuncommonintheliteraturef
4、orstudents.Inthepresenttext,Ihavetriedtobringtothesubjectacertainamountofmathematicalcorrectnessandmakeitaccessibletoun-dergraduates.Tothisend,thistextaddressesanumberofissuesthatarerarelyconsidered.Forinstance,whenweapplytheLaplacetrans-formmethodtoalinearordinarydifferent
5、ialequationwithconstantcoefficients,(n)(n−1)any+an−1y+···+a0yf(t),whyisitjustifiedtotaketheLaplacetransformofbothsidesoftheequation(TheoremA.6)?Or,inmanyproofsitisrequiredtotakethelimitinsideanintegral.Thisisalwaysfroughtwithdanger,especiallywithanimproperintegral,andnotalwa
6、ysjustified.Ihavegivencompletedetails(sometimesintheAppendix)wheneverthisprocedureisrequired.ixxPrefaceFurthermore,itissometimesdesirabletotaketheLaplacetrans-formofaninfiniteseriestermbyterm.Againitisshownthatthiscannotalwaysbedone,andspecificsufficientconditionsareestablished
7、tojustifythisoperation.Anotherdelicateproblemintheliteraturehasbeentheapplica-tionoftheLaplacetransformtotheso-calledDiracdeltafunction.Exceptfortextsonthetheoryofdistributions,traditionaltreatmentsareusuallyheuristicinnature.Inthepresenttextwegiveanewandmathematicallyrigor
8、ousaccountoftheDiracdeltafunctionbasedupontheRiemann–Stieltjesintegral.Itiselement
