1 1 b RBParisWeexcludethislastcasefromourasymptoticconsiderationsThefuncisentireinandisconsequentlycompletelydescribedinThebehaviourofforlargeand ID: 484745
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AppliedMathematicalSciences,Vol.7,2013,no.133,6601-6609HIKARILtd,www.m-hikari.comhttp://dx.doi.org/10.12988/ams.2013.310559ExponentiallySmallExpansionsoftheConßuentHypergeometricFunctionsR.B.ParisSchoolofEngineering,ComputingandAppliedMathematicsUniversityofAbertayDundee,DundeeDD11HG,UKCopyright =1+ +1) b( R.B.ParisWeexcludethislastcasefromourasymptoticconsiderations.Thefunc-)isentireinandisconsequentlycompletelydescribedinThebehaviourof)forlargeandxedparametersisexponen-tiallylargeinRe(0andalgebraicincharacterinRe(0.Thewell-knownasymptoticexpansionof)forisgivenby[4,p.328] (b)1F1(a;b;z)E(z(1 2+z3 wherethroughout0denotesanarbitrarilysmallquantity.Heretheformalexponentialandalgebraicasymptoticseries)and)aredenedby):=):= (1+ wherethecoecientsaregivenby Theexponentialexpansion)isdominantasinRe(0andbecomesoscillatoryontheanti-Stokeslinesarg ,whereitisofcomparablemagnitudetothealgebraicexpansion.InRe(0,theexponentialexpansionissubdominantwiththebehaviourof)thencontrolledby).ThenegativerealaxisargisaStokesline)ismaximallysubdominant.When)isoptimallytruncatedat,ornear,itsleastterm,theexponentialexpansionundergoesasmoothbutrapidtransitionintheneighbourhoodofarg;see[4,p.67]and[6,Chapter6].Itisclear,however,that(1.1)cannotaccountcorrectlyfortheexponentiallysmallexpansiononarg,sinceitpredictstheexponentiallysmallbehaviour(|.Whenisnon-integerwithreal,thisisacomplex-valuedcontributionwhereas)isreal.Theinterestinexponentiallypreciseasymptoticsduringthepasttwodec-adeshasshownthatretentionofexponentiallysmallterms,previouslyne-glectedinasymptotics,isessentialforahigh-precisiondescription.AnearlyexamplethatillustratedtheadvantageofretainingexponentiallysmalltermsintheasymptoticexpansionofacertainintegralwasgiveninOlverswell-knownbook[1,p.76].AlthoughsuchtermsarenegligibleinthePoincar´sense,theirinclusioncansignicantlyimprovethenumericalaccuracy.Ouraiminthisnoteistoobtaintheexponentiallysmallexpansionas-sociatedwith),andalsothatofthesecondKummerfunction R.B.Paris.Theexpansion(2.2)wasrstobtainedbyOlver[2,3]whobasedhisanalysisonaLaplaceintegralrepresentationfora,b,z).AdierentderivationusingaMellin-Barnesintegralrepresentationwasgivenin[5];seealso[6,p.241etseq.2.1Theexpansionwhenisnon-integerTheanalogousexpansionfor)forcanbeobtainedbymakinguseoftheidentity[4,p.323](1+ iba,b,ziba,b,zeThen,with0,substitutionoftheexpansion(2.2)intotheaboveidentityyields )(1+ iba,b,xeiba,b,xe (1+ ):=iiaSincethetruncationindexischosentobeoptimal,theindexappearingin(2.1)satises.Theasymptoticexpansionof)for,when||,hasbeendiscussedindetailbyOlverin[2].Byexpressing)intermsofaLaplaceintegral,whichisassociatedwithasaddlepointandasimplepolebecomingcoincidentonarg,Olver[2,5]establishedthat 2i 2N1k=0(1 2)kg2k(j)(1 wherethecoecients)resultfromtheexpansion 1d dw=1 w+k=0gk(j)wk,1 Thebranchof)ischosensuchthat1as1andtheparameterisspeciedby1)(2.7) R.B.ParisTheorem1isnon-integerwehavetheexpansion (b)1F1(a;b;x)xa(a) (1+ 2sin ,whereistheoptimaltruncationindexofthealgebraicexpan-sionsatisfyingisapositiveintegerandthecoecientsaredenedin(1.3)and(2.9)respectively.2.2Theexpansionwhenisaninteger,itisseenfrom(1.2)thatthealgebraicexpan-0andthecoecientsintheexponentialexpansionvanishforjn.FromKummerstransformationthehypergeometricfunctionontheright-handsideisapolynomialin.Uponsomestraightforwardrearrangementwendtheniteexpo-nentialseriesgivenby =(1validforallnite,wherethecoecientsfollowfromthein(1.3).,thealgebraicexpansionconsistsoftermsandsocannotbeoptimallytruncated;theparameterin(2.1)(with)isthereforeniteinthislimit.Fromtheequationimmediatelyabove(2.8)itisseenthatiinthiscase.Consequently,wethenobtaintheresultgiveninthefollowingtheorem.Theorem2Letdenotepositiveintegers.Forvaluesoftheparameterwehavetheexpansion (b)1F1(a;b;x)xa(a) (1+ ,wherethecoecientsaredenedin(1.3);compare(1.1). R.B.ParisTable2:Valuesof)and)in(3.1)and(2.10)usinganoptimaltruncationofthealgebraicexpansionandindex=5fordierentvaluesofandparameters 50( 75( F(x)E(x) F(x)E(x) 5 3)+8 4)+5 5)+6 6)+3 7)+4 8)+2 9)+3 10)+1 11)+2 13)+9 50( 75( F(x)E(x) F(x)E(x) 5 1.240840(3)1.247194(3) 3)+3 4.340627(6)4.341510(6) 5)+1 1.977241(8)1.977300(8) 7)+1 1.006538(10)1.006546(10) 10)+7 5.450153(13)5.450167(13) 12)+4 WemakethreeremarksonTheorems1and2.First,theleadingbehaviouroftheexponentiallysmallexpansionontheStokeslineisO()fornotequaltohalf-integervalues,butisO( )when 2,±3 Secondly,theso-calledStokesmultiplier(givenbythequantityincurlybracesin(2.11)and(2.13))isequaltocostoleadingorder.Andthirdly,when,weseefrom(2.13)thattheexponentialseriescanalsobewrittenintheform )isdenedin(1.2).Whenisnon-integer,however,thereisanadditionalcontributionfromtheseriesinvolvingthecoecientsTheexpansionfora,b,xe)followsfrom(2.2)and(2.6),withchosentobetheoptimaltruncationindexaccordingto(2.3),asa,b,xe(1+ ieia )(1+ 2M1j=0()jAjxji 21j=0()jBjxj+O(xM)(3.2)