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1、Bessel'sDifferentialEquationIntheSturm・LiouvilleBoundarvValueProblem,thereisanimportsntspecialcasecalledBessel'sDifferentialEquationwhicharisesinnumerousproblems,especiallyinpolarandcylindricalcoordinates・BesseTsDifferentialEquationisdefinedas:Jy*+v1+(J一小卜
2、=0where::isanon-negativerealnumber.ThesolutionsofthisequationarecalledBesselFunctionsoforder二.Althoughtheordercanbeanyrealnumber,thescopeofthissectionislimitedtonon-negativeintegers,i.e.,*—0.1»瓷3・・・，口门
3、essspecifiedotherwise.SinceBessePsdifferentialequation
4、isasecondorderordinarydifferentialequation,twosetsandtheBesselfunctionofthesecondoffunctions,theBesselfunctionofthefirstkindkind(alsoknownastheWeberFunction),areneededtoformthegeneralsolution:However,isdivergentatK=°Theassociatedcoefficientr-isforcedtobeze
5、rotoobtainaphysicallymeaningfulresultwhenthereisnosourceorsinkat®・ImportantPropertiesBasicRelationship:TheBesselfunctionofthefirstkindoforder■-canbeexpressedasaseriesofgammafunctions.h24(2»+2)(2*4-4)TheBesselfunctionofthesecondkindoforder、canbeexpressedint
6、ermsoftheBesselfunctionofthefirstkind.砒）=珈〉卜制一老气理）（一严Jp(K)CO®(x)singleGeneratingFunction:ThegeneratingfunctionoftheBesselFunctionofthefirstkindis=f站(*«=-onRecurrenceRelation:ABesselfunctionofhigherordercanbeexpressedbyBesselfunctionsoflowerorders・刍4（龙）=兰务匕
7、）一豈」（HJhW=
8、[-6i-Lw-Ai+Iw]疳（4=煞_1何-如冏JnW=Jw-l打⑴=1（4宅匕）&()=%&)-也1("壬p%®]="
9、一1何¥『切(x)]=-x^Jji+1(RAsymptoticApproximations:Keepingthefirstfewtermsintheseriesexpansions,thebehaviorofaBesselfunctionatsmallorlargecanbecapturedandexpressedaselementaryfunctionswhic
10、haremucheasiertobeunderstoodandcalculatedthanthemoreabstractsymbols^n^andForsmall工，i.e.,fixed■-and®Forlarge-,i.e.,fixed:andX%甘CM”-扌厶（R~Orthogonality:Supposethatisthenon-negativerootofthecharacteristicequation也（尹3许=。associatedwithaphysicalproblemdefinedonth
11、eintervalofItcanbeverifiedthatByusingthisorthogonality,thecomponentofthegeneralsolutionofthephysicalproblemisHpThegeneralsolutionthusyieldswhereandisapiecewisecontinuousfunction,generallythenon-homogeneouster
