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1、BesselfunctionFromWikipedia,thefreeencyclopediaInmathematics,Besselfunctions,firstdefinedbythemathematicianDanielBernoulliandgeneralizedbyFriedrichBessel,arecanonicalsolutionsy(x)ofBessel'sdifferentialequation:foranarbitraryrealorcomplexnumberα(theorderoftheBessel
2、function).Themostcommonandimportantspecialcaseiswhereαisanintegern.Althoughαand−αproducethesamedifferentialequation,itisconventionaltodefinedifferentBesselfunctionsforthesetwoorders(e.g.,sothattheBesselfunctionsaremostlysmoothfunctionsofα).Besselfunctionsarealsokn
3、ownascylinderfunctionsorcylindricalharmonicsbecausetheyarefoundinthesolutiontoLaplace'sequationincylindricalcoordinates.Contents¢1ApplicationsofBesselfunction¢2Definitions¢2.1Besselfunctionsofthefirstkind:Jα¢2.1.1Bessel'sintegrals¢2.1.2Relationtohypergeometricseri
4、es¢2.1.3RelationtoLaguerrepolynomials¢2.2Besselfunctionsofthesecondkind:Yα(1)(2)¢2.3Hankelfunctions:H,Hαα¢2.4ModifiedBesselfunctions:I,Kαα¢2.5SphericalBesselfunctions:j,ynn¢2.5.1Generatingfunction¢2.5.2Differentialrelations¢2.6SphericalHankelfunctions:hn¢2.7Riccat
5、i–Besselfunctions:S,C,ξ,ζnnnn¢3Asymptoticforms¢4Properties¢5ConnectionwithFourierTransform¢6Multiplicationtheorem¢7Bourget'shypothesis¢8DerivativesofJ,Y,I,H,K¢8.1p-1dependency¢8.2p+1dependency¢8.3Otherrelationships¢9Selectedidentities¢10Seealso¢11Notes¢12Reference
6、s¢13ExternallinksApplicationsofBesselfunctionBessel'sequationariseswhenfindingseparablesolutionstoLaplace'sequationandtheHelmholtzequationincylindricalorsphericalcoordinates.Besselfunctionsarethereforeespeciallyimportantformanyproblemsofwavepropagationandstaticpot
7、entials.Insolvingproblemsincylindricalcoordinatesystems,oneobtainsBesselfunctionsofintegerorder(α=n);insphericalproblems,oneobtainshalf-integerorders(α=n+½).Forexample:¢Electromagneticwavesinacylindricalwaveguide¢Heatconductioninacylindricalobject¢Modesofvibration
8、ofathincircular(orannular)artificialmembrane(suchasadrumorothermembranophone)¢Diffusionproblemsonalattice¢SolutionstotheradialSchrödingerequation(insphe
