化工应用数学ppt课件.ppt

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1、化工應用數學授課教師：郭修伯Lecture6FunctionsanddefiniteintegralsVectorsChapter5FunctionsanddefiniteintegralsTherearemanyfunctionsarisinginengineeringwhichcannotbeintegratedanalyticallyintermsofelementaryfunctions.Thevaluesofmanyintegralshavebeentabulated,muchnumericalworkcanbeavoidedifthe

2、integraltobeevaluatedcanbealteredtoaformthatistabulated.Ref.pp.153Wearegoingtostudysomeofthesespecialfunctions…..SpecialfunctionsFunctionsDetermineafunctionalrelationshipbetweentwoormorevariablesWehavestudiedmanyelementaryfunctionssuchaspolynomials,powers,logarithms,exponenti

3、als,trigonometricandhyperbolicfunctions.FourkindsofBesselfunctionsareusefulforexpressingthesolutionsofaparticularclassofdifferentialequations.Legendrepolynomialsaresolutionsofagroupofdifferentialequations.Learnsomemorenow….TheerrorfunctionItoccursinthetheoryofprobability,dist

4、ributionofresidencetimes,conductionofheat,anddiffusionmatter:0xzerfxz:dummyvariableProofinnextslidexandyaretwoindependentCartesiancoordinatesinpolarcoordinatesErrorbetweenthevolumedeterminedbyx-yandr-Thevolumeofhasabaseareawhichislessthan1/2R2andamaximumheightofe-R2Moreabou

5、terrorfunctionDifferentiationoftheerrorfunction:Integrationoftheerrorfunction:Theaboveequationistabulatedunderthesymbol“ierfx”with(Therefore,ierf0=0)Anotherrelatedfunctionisthecomplementaryerrorfunction“erfcx”Thegammafunctionforpositivevaluesofn.tisadummyvariablesincethevalue

6、ofthedefiniteintegralisindependentoft.(N.B.,ifniszerooranegativeinteger,thegammafunctionbecomesinfinite.)repeatThegammafunctionisthusageneralizedfactorial,forpositiveintegervaluesofn,thegammafunctioncanbereplacedbyafactorial.(Fig.5.3pp.147)MoreaboutthegammafunctionEvaluateCha

7、pter7VectoranalysisIthasbeenshownthatacomplexnumberconsistedofarealpartandanimaginarypart.Onesymbolwasusedtorepresentacombinationoftwoothersymbols.Itismuchquickertomanipulateasinglesymbolthanthecorrespondingelementaryoperationsontheseparatevariables.Thisistheoriginalideaofvec

8、tor.Anynumberofvariablescanbegroupedintoasinglesymbolintwoways:(1)Ma