The Dirac Delta Function

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1、Chapter20TheDiracDeltaFunctionIdonotknowwhatIappeartotheworld;buttomyselfIseemtohavebeenonlylikeaboyplayingonaseashore,anddivertingmyselfnowandthenbyfindingasmootherpebbleoraprettiershellthanordinary,whilstthegreatoceanoftruthlayallundiscoveredbeforeme.-SirIssacNewton20.1DerivativeoftheHeavisideFunct

2、ionTheHeavisidefunctionH(x)isdefined�0forx<0�H(x)=1forx>0.ThederivativeoftheHeavisidefunctioniszeroforx�=0.Atx=0thederivativeisundefined.WewillrepresentthederivativeoftheHeavisidefunctionbytheDiracdeltafunction,δ(x).Thedeltafunctioniszeroforx�=0andinfiniteatthepointx=0.SincethederivativeofH(x)isundefine

3、d,δ(x)isnotafunctionintheconventionalsenseoftheword.Onecanderivethepropertiesofthedeltafunctionrigorously,butthetreatmentinthistextwillbealmostentirelyheuristic.TheDiracdeltafunctionisdefinedbytheproperties��∞0forx�=0�δ(x)=andδ(x)dx=1.∞forx=0�−∞Thesecondpropertycomesfromthefactthatδ(x)representsthede

4、rivativeofH(x).TheDiracdeltafunctionisconceptuallypicturedinFigure20.1.Figure20.1:TheDiracDeltaFunction.637Letf(x)beacontinuousfunctionthatvanishesatinfinity.Considertheintegral�∞f(x)δ(x)dx.−∞Weuseintegrationbypartstoevaluatetheintegral.�∞���∞∞�f(x)δ(x)dx=f(x)H(x)−f(x)H(x)dx−∞−∞−∞�∞=−f�(x)dx0=[−f(x)]

5、∞0=f(0)Weassumedthatf(x)vanishesatinfinityinordertouseintegrationbypartstoevaluatetheintegral.However,sincethedeltafunctioniszeroforx�=0,theintegrandisnonzeroonlyatx=0.Thusthebehaviorofthefunctionatinfinityshouldnotaffectthevalueoftheintegral.Thusitis�∞reasonablethatf(0)=f(x)δ(x)dxholdsforallcontinuous

6、functions.Bychangingvariables−∞andnotingthatδ(x)issymmetricwecanderiveamoregeneralformula.�∞f(0)=f(ξ)δ(ξ)dξ−∞�∞f(x)=f(ξ+x)δ(ξ)dξ−∞�∞f(x)=f(ξ)δ(ξ−x)dξ−∞�∞f(x)=f(ξ)δ(x−ξ)dξ−∞Thisformulaisveryimportantinsolvinginhomogeneousdifferentialequations.20.2TheDeltaFunctionasaLimitConsiderafunctionb(x��)definedby

7、�0for

8、x

9、>�/2b(x��)=1for

10、x

11、<�/2.�Thegraphofb(x�1/10)isshowninFigure20.2.105-11Figure20.2:Graphofb(x�1/10).TheDiracdeltafunctionδ(x)canbethoughtofasb(x��)inthelimitas�→0.Notethatthedeltafunctionsodefined