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1、TomyparentsvItiscustomarytobegincoursesinmathematicalengineeringbyex-plainingthatthelecturerwouldnevertrusthislifetoanaeroplanewhosebehaviourdependedonpropertiesoftheLebesgueintegral.Itmight,perhaps,bejustasfoolhardytoflyinanaeroplanede-signedbyanengineerwhob
2、elievedthatcookbookapplicationoftheLaplacetransformrevealedallthatwastobeknownaboutitsstability.T.W.Korner¨FourierAnalysisCambridgeUniversityPress1988viiPrefaceTheLaplacetransformisawonderfultoolforsolvingordinaryandpartialdifferentialequationsandhasenjoyedm
3、uchsuccessinthisrealm.Withitssuccess,however,acertaincasualnesshasbeenbredconcerningitsapplication,withoutmuchregardforhypothesesandwhentheyarevalid.Evenproofsoftheoremsoftenlackrigor,anddubiousmathematicalpracticesarenotuncommonintheliteratureforstudents.In
4、thepresenttext,Ihavetriedtobringtothesubjectacertainamountofmathematicalcorrectnessandmakeitaccessibletoun-dergraduates.Tothisend,thistextaddressesanumberofissuesthatarerarelyconsidered.Forinstance,whenweapplytheLaplacetrans-formmethodtoalinearordinarydiffer
5、entialequationwithconstantcoefficients,(n)(n−1)any+an−1y+···+a0yf(t),whyisitjustifiedtotaketheLaplacetransformofbothsidesoftheequation(TheoremA.6)?Or,inmanyproofsitisrequiredtotakethelimitinsideanintegral.Thisisalwaysfroughtwithdanger,especiallywithanimproper
6、integral,andnotalwaysjustified.Ihavegivencompletedetails(sometimesintheAppendix)wheneverthisprocedureisrequired.ixxPrefaceFurthermore,itissometimesdesirabletotaketheLaplacetrans-formofaninfiniteseriestermbyterm.Againitisshownthatthiscannotalwaysbedone,andspeci
7、ficsufficientconditionsareestablishedtojustifythisoperation.Anotherdelicateproblemintheliteraturehasbeentheapplica-tionoftheLaplacetransformtotheso-calledDiracdeltafunction.Exceptfortextsonthetheoryofdistributions,traditionaltreatmentsareusuallyheuristicinnatu
8、re.InthepresenttextwegiveanewandmathematicallyrigorousaccountoftheDiracdeltafunctionbasedupontheRiemann–Stieltjesintegral.Itiselementaryinscopeandentirelysuitedtothislevelofexposition.Oneofthehi
