complex-functions-theory-c-12- the laplace transformation II.pdf

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1、Complex Functions Theory c12 Leif Mejlbro Download free books at LeifMejlbroThaLaplaceTransformationIIComplexfunctionstheoryc-122TheLaplaceTransformationIIc-12©2011LeifMejlbro&VentusPublishingApSISBN978-87-7681-763-33TheLaplaceTransformationIIc-12ContentsContentsIntrod

2、uction51SpecialFunctions61.1TheGammaFunction61.2TheBetafunction181.3Thesineandcosineandexponentialintegrals251.4Theerrorfunction291.5TheBesselfunctions322Applications492.1Linearordinarydifferentialequations492.2Linearsystemsofordinarydifferentialequations672.3Linearpar

3、tialdifferentialequations892.4TheDiracmeasure1992.5Theztransformation1073Extensionoftheinversionformula1123.1Theinversionformulaforanalyticfunctionswithbranchcuts1123.2Theinversionformulaforfunctionswithinfinitelymanysingularities1264Appendices1464.1Trigonometricformul

4、æ1464.2Integrationoftrigonometricpolynomials1464.3TablesofsomeLaplacetransformsandFouriertransforms150Index1554TheLaplaceTransformationIIc-12IntroductionIntroductionInthisvolumewegivesomeexamplesoftheelementarypartofthetheoryoftheLaplacetransfor-mationasdescribedinVent

5、us,ComplexFunctionsTheorya-5,TheLaplaceTransformationII.Thechaptersandthesectionswillfollowthesamestructureasintheabovementionedbookonthetheory.Theexampleshavebeencollectedabout30yearsagofromsomelongforgottenbookonapplications.Itwasthenpointedoutbytheauthor,andrepeated

6、herethatoneshouldnotuncriticallyapplytheLaplacetransformationinallcases.SometimesthesimplermethodsknownfromplainCalculusmaybeeasiertoapply.LeifMejlbroMarch31,201135TheLaplaceTransformationIIc-121SpecialFunctions1SpecialFunctions1.1TheGammaFunction1Example1.1.1Compute

7、Γ−n−foreveryn∈N0.21√WeshalltakeforgrantedthatΓ=π,andalsothefunctionalequationoftheGammafunction,2Γ(z+1)=zΓ(z),fromwhich1Γ(z)=Γ(z+1)forz=0.zWegetbyasimpleiteration,1−11(−1)21Γ−n−=·Γ−(n−1)−=Γ−n−+2=···2n+12n+1n−12222√(−1)n+11(−1)n+12n+1π=Γ=n+1n−1···12(2

8、n+1)(2n−1)···3·1222√22n+1n!π=(−1)n+1.♦(2n+1)!Fast-trackyourcareerMastersinManagementStandoutfromthecrowdDesignedforgr