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1、August7,201221:04c06Sheetnumber1Pagenumber309cyanblackCHAPTER6TheLaplaceTransformManypracticalengineeringproblemsinvolvemechanicalorelectricalsystemsactedonbydiscontinuousorimpulsiveforcingterms.ForsuchproblemsthemethodsdescribedinChapter3areoftenratherawkwardtouse.Anothermethodthatisespe
2、-ciallywellsuitedtotheseproblems,althoughusefulmuchmoregenerally,isbasedontheLaplacetransform.Inthischapterwedescribehowthisimportantmethodworks,emphasizingproblemstypicalofthosethatariseinengineeringapplications.6.1DefinitionoftheLaplaceTransformImproperIntegrals.SincetheLaplacetransformi
3、nvolvesanintegralfromzerotoinfin-ity,aknowledgeofimproperintegralsofthistypeisnecessarytoappreciatethesubsequentdevelopmentofthepropertiesofthetransform.Weprovideabriefreviewofsuchimproperintegralshere.Ifyouarealreadyfamiliarwithimproperintegrals,youmaywishtoskipoverthisreview.Ontheotherha
4、nd,ifimproperintegralsarenewtoyou,thenyoushouldprobablyconsultacalculusbook,whereyouwillfindmanymoredetailsandexamples.Animproperintegraloveranunboundedintervalisdefinedasalimitofintegralsoverfiniteintervals;thus∞Af(t)dt=limf(t)dt,(1)aA→∞awhereAisapositiverealnumber.IftheintegralfromatoAex
5、istsforeachA>a,andifthelimitasA→∞exists,thentheimproperintegralissaidtoconvergetothatlimitingvalue.Otherwisetheintegralissaidtodiverge,ortofailtoexist.Thefollowingexamplesillustratebothpossibilities.309August7,201221:04c06Sheetnumber2Pagenumber310cyanblack310Chapter6.TheLaplaceTransformLe
6、tf(t)=ect,t≥0,wherecisarealnonzeroconstant.ThenEXAMPLEA∞Aect1ectdt=limectdt=lim0A→∞0A→∞c01cA=lim(e−1).A→∞cItfollowsthattheimproperintegralconvergestothevalue−1/cifc<0anddivergesifc>0.Ifc=0,theintegrandf(t)istheconstantfunctionwithvalue1.InthiscaseAlim1dt=lim(A−0)=∞,A→∞0A→∞sotheint
7、egralagaindiverges.Letf(t)=1/t,t≥1.ThenEXAMPLE∞Adtdt2=lim=limlnA.1tA→∞1tA→∞SincelimlnA=∞,theimproperintegraldiverges.A→∞Letf(t)=t−p,t≥1,wherepisarealconstantandp=1;thecasep=1wasconsideredinEXAMPLEExample2.Then3∞A1−p−p1−ptdt=limtdt=lim(A−1).1A→∞1A→∞1−p∞AsA→∞,A1
