complex-functions-theory-c-11- the laplace transformation I.pdf

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1、Complex Functions Theory c11 Leif Mejlbro Download free books at LeifMejlbroTheLaplaceTransformationIComplexFunctionsTheoryc-112TheLaplaceTransformationIc-11©2010LeifMejlbro&VentusPublishingApSISBN978-87-7681-753-43TheLaplaceTransformationIContentContentIntroducti

2、on51TheLaplacetransformation41.1Nullsetsandnullfunctions;theLebesgueintegral61.2TheLaplacetransformation101.3ThecomplexinversionformulaI511.4Convolutions761.4.1Convolutions,general761.4.2Convolutionequations801.4.3Besselfunctions851.5Linearordinarydifferentialequa

3、tions901.5.1Lineardifferentialequationofconstantcoefficients901.5.2Lineardifferentialequationsofsimplepolynomialcoefficients1021.5.3Linearequationsofconstantcoefficientsanddiscontinuousrighthandside1081.6Thetwo-sidedLaplacetransformation1121.7TheFouriertransformat

4、ion1141.8TheLaplacetransformofafunctionviaadifferentialequation1202Appendices1222.1Trigonometricformulæ1222.2Integrationoftrigonometricpolynomials1222.3TablesofsomeLaplacetransformsandFouriertransforms126Index1314TheLaplaceTransformationIIntroductionIntroductionIn

5、thisvolumewegivesomeexamplesoftheelementarypartofthetheoryoftheLaplacetransfor-mationasdescribedinVentus,ComplexFunctionsTheorya-4,TheLaplaceTransformationI.Thechaptersandthesectionswillfollowthesamestructureasintheabovementionedbookonthetheory.LeifMejlbroFebruary

6、18,201135TheLaplaceTransformationI1TheLaplacetransformation1TheLaplacetransformation1.1Nullsetsandnullfunctions;theLebesgueintegral12Example1.1.1LetA0=[0,1],andletA1:=0,∪,1denotetheclosedset,whichisobtained33byremovingtheopenintervalinthemiddle.Thenlet

7、1236789A2:=0,∪,∪,∪,32323232323232betheset,whichisobtainedbyremovingalltheopenintervalsinthemiddleineachofthetwoclosedsubintervalsofA1.SketchA1andA2.ThendefinethesetsAnbyinduction,followingthesamepatternasdescribedabove,alwaysremoving+∞theopenintervalinthemiddleofe

8、achsubinterval.LetA=.n=01)ProvethatA=∅.2)ProvethatAisanullset.3)ProvethatAcontainsanon-countablenumberofpoints.Figure1:ThesetsA1andA2.EachAconsistsof2n