utm schiff j l laplace transformation theory and applications (1999)(245s)

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1、TheLaplaceTransform:TheoryandApplicationsJoelL.SchiffSpringerTomyparentsvItiscustomarytobegincoursesinmathematicalengineeringbyex-plainingthatthelecturerwouldnevertrusthislifetoanaeroplanewhosebehaviourdependedonpropertiesoftheLebesgueintegral.Itmight,perhaps,bejustasfoolhardytoflyin

2、anaeroplanede-signedbyanengineerwhobelievedthatcookbookapplicationoftheLaplacetransformrevealedallthatwastobeknownaboutitsstability.T.W.Korner¨FourierAnalysisCambridgeUniversityPress1988viiPrefaceTheLaplacetransformisawonderfultoolforsolvingordinaryandpartialdifferentialequationsand

3、hasenjoyedmuchsuccessinthisrealm.Withitssuccess,however,acertaincasualnesshasbeenbredconcerningitsapplication,withoutmuchregardforhypothesesandwhentheyarevalid.Evenproofsoftheoremsoftenlackrigor,anddubiousmathematicalpracticesarenotuncommonintheliteratureforstudents.Inthepresenttext

4、,Ihavetriedtobringtothesubjectacertainamountofmathematicalcorrectnessandmakeitaccessibletoun-dergraduates.Tothisend,thistextaddressesanumberofissuesthatarerarelyconsidered.Forinstance,whenweapplytheLaplacetrans-formmethodtoalinearordinarydifferentialequationwithconstantcoefficients,(

5、n)(n−1)any+an−1y+···+a0y
f(t),whyisitjustifiedtotaketheLaplacetransformofbothsidesoftheequation(TheoremA.6)?Or,inmanyproofsitisrequiredtotakethelimitinsideanintegral.Thisisalwaysfroughtwithdanger,especiallywithanimproperintegral,andnotalwaysjustified.Ihavegivencompletedetails(sometime

6、sintheAppendix)wheneverthisprocedureisrequired.ixxPrefaceFurthermore,itissometimesdesirabletotaketheLaplacetrans-formofaninfiniteseriestermbyterm.Againitisshownthatthiscannotalwaysbedone,andspecificsufficientconditionsareestablishedtojustifythisoperation.Anotherdelicateprobleminthelite

7、raturehasbeentheapplica-tionoftheLaplacetransformtotheso-calledDiracdeltafunction.Exceptfortextsonthetheoryofdistributions,traditionaltreatmentsareusuallyheuristicinnature.InthepresenttextwegiveanewandmathematicallyrigorousaccountoftheDiracdeltafunctionbasedupontheRiemann–Stieltjesi

8、ntegral.Itiselement