Regular singular point - Wikipedia

are classified into ordinary points, at which the equation's coefficients are analytic functions, and singular points, at which some coefficient has a ... Regularsingularpoint FromWikipedia,thefreeencyclopedia Jumptonavigation Jumptosearch Thisarticleincludesalistofreferences,relatedreadingorexternallinks,butitssourcesremainunclearbecauseitlacksinlinecitations.Pleasehelptoimprovethisarticlebyintroducingmoreprecisecitations.(June2017)(Learnhowandwhentoremovethistemplatemessage) Inmathematics,inthetheoryofordinarydifferentialequationsinthecomplexplane C {\displaystyle\mathbb{C}} ,thepointsof C {\displaystyle\mathbb{C}} areclassifiedintoordinarypoints,atwhichtheequation'scoefficientsareanalyticfunctions,andsingularpoints,atwhichsomecoefficienthasasingularity.Thenamongstsingularpoints,animportantdistinctionismadebetweenaregularsingularpoint,wherethegrowthofsolutionsisbounded(inanysmallsector)byanalgebraicfunction,andanirregularsingularpoint,wherethefullsolutionsetrequiresfunctionswithhighergrowthrates.Thisdistinctionoccurs,forexample,betweenthehypergeometricequation,withthreeregularsingularpoints,andtheBesselequationwhichisinasensealimitingcase,butwheretheanalyticpropertiesaresubstantiallydifferent. Contents 1Formaldefinitions 2Examplesforsecondorderdifferentialequations 2.1Besseldifferentialequation 2.2Legendredifferentialequation 2.3Hermitedifferentialequation 2.4Hypergeometricequation 3References Formaldefinitions[edit] Moreprecisely,consideranordinarylineardifferentialequationofn-thorder ∑ i = 0 n p i ( z ) f ( i ) ( z ) = 0 {\displaystyle\sum_{i=0}^{n}p_{i}(z)f^{(i)}(z)=0} withpi(z)meromorphicfunctions.Onecanassumethat p n ( z ) = 1. {\displaystylep_{n}(z)=1.} Ifthisisnotthecasetheequationabovehastobedividedbypn(x).Thismayintroducesingularpointstoconsider. TheequationshouldbestudiedontheRiemannspheretoincludethepointatinfinityasapossiblesingularpoint.AMöbiustransformationmaybeappliedtomove∞intothefinitepartofthecomplexplaneifrequired,seeexampleonBesseldifferentialequationbelow. ThentheFrobeniusmethodbasedontheindicialequationmaybeappliedtofindpossiblesolutionsthatarepowerseriestimescomplexpowers(z−a)rnearanygivenainthecomplexplanewhererneednotbeaninteger;thisfunctionmayexist,therefore,onlythankstoabranchcutextendingoutfroma,oronaRiemannsurfaceofsomepunctureddiscarounda.Thispresentsnodifficultyforaanordinarypoint(LazarusFuchs1866).Whenaisaregularsingularpoint,whichbydefinitionmeansthat p n − i ( z ) {\displaystylep_{n-i}(z)} hasapoleoforderatmostiata,theFrobeniusmethodalsocanbemadetoworkandprovidenindependentsolutionsneara. Otherwisethepointaisanirregularsingularity.Inthatcasethemonodromygrouprelatingsolutionsbyanalyticcontinuationhaslesstosayingeneral,andthesolutionsarehardertostudy,exceptintermsoftheirasymptoticexpansions.TheirregularityofanirregularsingularityismeasuredbythePoincarérank(Arscott(1995)harvtxterror:notarget:CITEREFArscott1995(help)). TheregularityconditionisakindofNewtonpolygoncondition,inthesensethattheallowedpolesareinaregion,whenplottedagainsti,boundedbyalineat45°totheaxes. Anordinarydifferentialequationwhoseonlysingularpoints,includingthepointatinfinity,areregularsingularpointsiscalledaFuchsianordinarydifferentialequation. Examplesforsecondorderdifferentialequations[edit] Inthiscasetheequationaboveisreducedto: f ″ ( x ) + p 1 ( x ) f ′ ( x ) + p 0 ( x ) f ( x ) = 0. {\displaystylef''(x)+p_{1}(x)f'(x)+p_{0}(x)f(x)=0.} Onedistinguishesthefollowingcases: Pointaisanordinarypointwhenfunctionsp1(x)andp0(x)areanalyticatx=a. Pointaisaregularsingularpointifp1(x)hasapoleuptoorder1atx=aandp0hasapoleoforderupto2atx=a. Otherwisepointaisanirregularsingularpoint. Wecancheckwhetherthereisanirregularsingularpointatinfinitybyusingthesubstitution w = 1 / x {\displaystylew=1/x} andtherelations: d f d x = − w 2 d f d w {\displaystyle{\frac{df}{dx}}=-w^{2}{\frac{df}{dw}}} d 2 f d x 2 = w 4 d 2 f d w 2 + 2 w 3 d f d w {\displaystyle{\frac{d^{2}f}{dx^{2}}}=w^{4}{\frac{d^{2}f}{dw^{2}}}+2w^{3}{\frac{df}{dw}}} Wecanthustransformtheequationtoanequationinw,andcheckwhathappensatw=0.If p 1 ( x ) {\displaystylep_{1}(x)} and p 2 ( x ) {\displaystylep_{2}(x)} arequotientsofpolynomials,thentherewillbeanirregularsingularpointatinfinitexunlessthepolynomialinthedenominatorof p 1 ( x ) {\displaystylep_{1}(x)} isofdegreeatleastonemorethanthedegreeofitsnumeratorandthedenominatorof p 2 ( x ) {\displaystylep_{2}(x)} isofdegreeatleasttwomorethanthedegreeofitsnumerator. Listedbelowareseveralexamplesfromordinarydifferentialequationsfrommathematicalphysicsthathavesingularpointsandknownsolutions. Besseldifferentialequation[edit] Thisisanordinarydifferentialequationofsecondorder.ItisfoundinthesolutiontoLaplace'sequationincylindricalcoordinates: x 2 d 2 f d x 2 + x d f d x + ( x 2 − α 2 ) f = 0 {\displaystylex^{2}{\frac{d^{2}f}{dx^{2}}}+x{\frac{df}{dx}}+(x^{2}-\alpha^{2})f=0} foranarbitraryrealorcomplexnumberα(theorderoftheBesselfunction).Themostcommonandimportantspecialcaseiswhereαisanintegern. Dividingthisequationbyx2gives: d 2 f d x 2 + 1 x d f d x + ( 1 − α 2 x 2 ) f = 0. {\displaystyle{\frac{d^{2}f}{dx^{2}}}+{\frac{1}{x}}{\frac{df}{dx}}+\left(1-{\frac{\alpha^{2}}{x^{2}}}\right)f=0.} Inthiscasep1(x)=1/xhasapoleoffirstorderatx=0.Whenα≠0,p0(x)=(1−α2/x2)hasapoleofsecondorderatx=0.Thusthisequationhasaregularsingularityat0. Toseewhathappenswhenx→∞onehastouseaMöbiustransformation,forexample x = 1 / w {\displaystylex=1/w} .Afterperformingthealgebra: d 2 f d w 2 + 1 w d f d w + [ 1 w 4 − α 2 w 2 ] f = 0 {\displaystyle{\frac{d^{2}f}{dw^{2}}}+{\frac{1}{w}}{\frac{df}{dw}}+\left[{\frac{1}{w^{4}}}-{\frac{\alpha^{2}}{w^{2}}}\right]f=0} Nowat w = 0 {\displaystylew=0} , p 1 ( w ) = 1 w {\displaystylep_{1}(w)={\frac{1}{w}}} hasapoleoffirstorder,but p 0 ( w ) = 1 w 4 − α 2 w 2 {\displaystylep_{0}(w)={\frac{1}{w^{4}}}-{\frac{\alpha^{2}}{w^{2}}}} hasapoleoffourthorder.Thus,thisequationhasanirregularsingularityat w = 0 {\displaystylew=0} correspondingtoxat∞. Legendredifferentialequation[edit] Thisisanordinarydifferentialequationofsecondorder.ItisfoundinthesolutionofLaplace'sequationinsphericalcoordinates: d d x [ ( 1 − x 2 ) d d x f ] + l ( l + 1 ) f = 0. {\displaystyle{\frac{d}{dx}}\left[(1-x^{2}){\frac{d}{dx}}f\right]+l(l+1)f=0.} Openingthesquarebracketgives: ( 1 − x 2 ) d 2 f d x 2 − 2 x d f d x + l ( l + 1 ) f = 0. {\displaystyle\left(1-x^{2}\right){d^{2}f\overdx^{2}}-2x{df\overdx}+l(l+1)f=0.} Anddividingby(1−x2): d 2 f d x 2 − 2 x 1 − x 2 d f d x + l ( l + 1 ) 1 − x 2 f = 0. {\displaystyle{\frac{d^{2}f}{dx^{2}}}-{\frac{2x}{1-x^{2}}}{\frac{df}{dx}}+{\frac{l(l+1)}{1-x^{2}}}f=0.} Thisdifferentialequationhasregularsingularpointsat±1and∞. Hermitedifferentialequation[edit] Oneencountersthisordinarysecondorderdifferentialequationinsolvingtheone-dimensionaltimeindependentSchrödingerequation E ψ = − ℏ 2 2 m d 2 ψ d x 2 + V ( x ) ψ {\displaystyleE\psi=-{\frac{\hbar^{2}}{2m}}{\frac{d^{2}\psi}{dx^{2}}}+V(x)\psi} foraharmonicoscillator.InthiscasethepotentialenergyV(x)is: V ( x ) = 1 2 m ω 2 x 2 . {\displaystyleV(x)={\frac{1}{2}}m\omega^{2}x^{2}.} Thisleadstothefollowingordinarysecondorderdifferentialequation: d 2 f d x 2 − 2 x d f d x + λ f = 0. {\displaystyle{\frac{d^{2}f}{dx^{2}}}-2x{\frac{df}{dx}}+\lambdaf=0.} Thisdifferentialequationhasanirregularsingularityat∞.ItssolutionsareHermitepolynomials. Hypergeometricequation[edit] Theequationmaybedefinedas z ( 1 − z ) d 2 f d z 2 + [ c − ( a + b + 1 ) z ] d f d z − a b f = 0. {\displaystylez(1-z){\frac{d^{2}f}{dz^{2}}}+\left[c-(a+b+1)z\right]{\frac{df}{dz}}-abf=0.} Dividingbothsidesbyz(1−z)gives: d 2 f d z 2 + c − ( a + b + 1 ) z z ( 1 − z ) d f d z − a b z ( 1 − z ) f = 0. {\displaystyle{\frac{d^{2}f}{dz^{2}}}+{\frac{c-(a+b+1)z}{z(1-z)}}{\frac{df}{dz}}-{\frac{ab}{z(1-z)}}f=0.} Thisdifferentialequationhasregularsingularpointsat0,1and∞.Asolutionisthehypergeometricfunction. References[edit] Coddington,EarlA.;Levinson,Norman(1955).TheoryofOrdinaryDifferentialEquations.NewYork:McGraw-Hill. E.T.Copson,AnIntroductiontotheTheoryofFunctionsofaComplexVariable(1935) Fedoryuk,M.V.(2001)[1994],"Fuchsianequation",EncyclopediaofMathematics,EMSPress A.R.ForsythTheoryofDifferentialEquationsVol.IV:OrdinaryLinearEquations(CambridgeUniversityPress,1906) ÉdouardGoursat,ACourseinMathematicalAnalysis,VolumeII,PartII:DifferentialEquationspp. 128−ff.(Ginn&co.,Boston,1917) E.L.Ince,OrdinaryDifferentialEquations,DoverPublications(1944) Il'yashenko,Yu.S.(2001)[1994],"Regularsingularpoint",EncyclopediaofMathematics,EMSPress T.M.MacRobertFunctionsofaComplexVariablep. 243(MacMillan,London,1917) Teschl,Gerald(2012).OrdinaryDifferentialEquationsandDynamicalSystems.Providence:AmericanMathematicalSociety.ISBN 978-0-8218-8328-0. E.T.WhittakerandG.N.WatsonACourseofModernAnalysispp. 188−ff.(CambridgeUniversityPress,1915) Retrievedfrom"https://en.wikipedia.org/w/index.php?title=Regular_singular_point&oldid=1101518884" Categories:OrdinarydifferentialequationsComplexanalysisHiddencategories:Articleslackingin-textcitationsfromJune2017Allarticleslackingin-textcitationsHarvandSfnno-targeterrors Navigationmenu Personaltools NotloggedinTalkContributionsCreateaccountLogin Namespaces ArticleTalk English Views ReadEditViewhistory More Search Navigation MainpageContentsCurrenteventsRandomarticleAboutWikipediaContactusDonate Contribute HelpLearntoeditCommunityportalRecentchangesUploadfile Tools WhatlinkshereRelatedchangesUploadfileSpecialpagesPermanentlinkPageinformationCitethispageWikidataitem Print/export DownloadasPDFPrintableversion Languages 한국어Italiano Editlinks