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TheLaplace’sandSteepestDescentsMethods wRe(w)Im(w)Re(f(w))0 Steepest descent pathSteepest ascent path Figure1:Typicalplotoftherealpartof)overasimplesaddlepointCrossingthispointwecan“ndasteepestascentpath(dashedlines)andasteepestdescentpath.Intheremainingofthepaper,wewillcallasymptoticpointstoboth,thecriticalpointsof)andtheendpointsof(ifagivenpointisboth,acriticalpointandanendpoint,wecallitcriticalpoint).Ifthefunctionhasnotone,butseveralasymptoticpoints:,thenthedominantcontributionto)comesfromthemostrelevantasymptoticpoint.Thisisde“nedastheasymptoticpointlocatedon(oronifthedeformationisnecessary)atwhich))ismaximal.Thentheasymptoticexpansion)maybederivedapplyingtheLaplacesmethod(when(whena,b)isreal)ortheSteepestDescentsmethod(when)iscomplex)atthatrelevantasymptoticpoint.Itmayhappensthatthisrelevantasymptoticpointisnotunique.Inthiscase,wemustapplyLaplacesmethodorSteepestDescentsmethodatalloftherelevantasymptoticpoints.Itisimpossibletoresumeinfewwordstheimportancethatthesemethodshavehadtoderiveasymptoticapproximationsofintegralsingeneral,andofspecialfunctionsinparticular.Inthisway,itisverydiculttogiveanexhaustivelistofthethousandsofpublicationsdevotedtoconstruct,generalizeorapplytospeci“cexamplesthesemethods.Considerthatmostoftheknownasymptoticexpansionsofspecialfunctionsderivedfromintegralshavebeenobtainedbyusingoneofthesemethods.Themaintechnicalcomplicationintheapplicationofthesemethodsliesintheidenti“cationofthesteepestdescentspaths(intheSteepestDescentsmethod)andinthechangeofvariable(inbothmethods).Anothersourceofcomplicationistheexistenceofmorethanonerelevantasymptoticpoint.Recently,someauthorshavesuggestedbymeansofspeci“cexamplesthatthesemethodsmaybesimpli“ed.Theideaisjusttoexpand)andperhapsalsopartofin(1)attherelevantasymptoticpoint(s)of)without TheLaplace’sandSteepestDescentsMethodsexceptat.Iffa,b)isrealandtimesdierentiableon[a,b]andanda,b],thenmaybeacriticalpointofmultiplicityeven:oranendpointif0if0ifthiscase=1).If)iscomplex(analyticbetweenand),thenasaddlepointofofmultiplicityiii)Thefunction)hasaTayorexpansionat )when)iscontinuouson)whenv)Theintegral(2)andtheintegralsvergeabsolutelyanduniformlywithrespectto.Thesameholdsfortheintegralsifthedeformationisrequired.)isrealandsmoothenoughon,and,Laplacesmethodsaysthatwemustperformin(2)thechangeofvariablede“nedbyandthenapproximate))at=0).If)and)areanalyticbetween,theSteepestDescentsmethodsaysthatwemustdeformthecontourthesteepestdescentpath(whichrunsthrow)andthenapplyLaplacesmethodtotheresultingintegralover.Inanycase,bothmethodstellusthatthemaincontributionoftheintegrandin(2)tothevalueoftheintegralcomesfromtheneighbourhoodofthepoint.Weusethisideahereinamoresimpleway:weapproximate)atbutwithoutanychangeofvariableneitheradeformationofthecontour.Wejustintroducetheexpansion(3)in(2)andinterchangesumandintegral: TheLaplace’sandSteepestDescentsMethods.Then,weneedtoapproximate)atbothofthemsimultaneouslyandwiththesameaccuracy.ThismeansthatweneedthesimultaneousTaylorexpansionof)at.Adetailedexpositionofthismaybefoundin[6].Weneedtoreplaceconditioniii)andiv)insection2by:iii)Thefunction)hasatwo-pointTaylorexpansionat,..., and,for k!(nŠk)!(Š1)n+1ng(nŠk)(w21)knŠk)(w1) Theremaindertermveri“es)when)when)iscontinuouson)when)when1andIntroducingtheexpansion(9)in(2)andinterchangingsumandintegralweobtainobtainakk(z)+bkk(z)]+dw,wdwdw. TheLaplace’sandSteepestDescentsMethodsThemaincontributionoftheintegrandin(2)tothevalueoftheintegralcomesfromtheneighbourhoodofalloftherelevantasymptoticpoints.Then,weneedtoapproximate)atallofthemsimultaneously.Butasitwillbeclearbelow,thedegreeofaccuracyintheapproximationof)atmustdependonthemultiplicityofaszeroof).Thismeansthatweneedthemulti-pointTaylorexpansionof)atthepointsrepeatedrepeated.Adetailedexpositionofthismaybefoundin[7].Then,conditionsiii)andiv)insection2mustbereplacedby:iiiŽ)Thefunction)hasamulti-pointTaylorexpansionat)arepolynomialsofdegree)when.Coecientsmaybecalculatedtakingintoaccountthatthepolynomialintherighthandsideof(19)istheinterpolationpolynomialof)atwithdata,...,N.Orelse[7]: k,j,lk,j,l (+l)!d+l dw+l g(w) Ns=1,sj(wŠws) w=wj+Nr=1,rj1 (Š1)!dŠ1 dw+l g(w) )iscontinuouson,...,NIntroducingtheexpansion(19)in(2)andinterchangingsumandintegralweobtain TheLaplace’sandSteepestDescentsMethodswherethecontourencirclestheorigininthecounterclockwisedirectionandcontainsnopolesof2).Theshift12inthevariableisintroducedinordertohavere”ectionsymmetryinthepolynomials.Wewanttoapproximate)forlargeandwrite(28)intheform 2=n! dw,n)=1)=(sinh(2).Theuniquesaddlepointof)isandhasmultiplicity1.Applyingthemethoddescribedinsection2weobtaintheasymptoticexpansion(4)-(6)of2)forlarge=0).Forsimplicityweonlywriteexplicitlythe“rsttwotermsoftheexpansion: 2=nnznŠ1 2sinh(11+4 2coth1 2zcoth1 2z1 8nz2+O1 ThisisanasymptoticexpansionforlargeforthosevaluesofforwhichtheSteepestDescentsmethodcanbeappliedtotheintegral(29).Thesteepestascentanddescentpathsof)are(see“g.2(a)):Arg((steepestascentpath)+Arg( sin(+Arg(Arg(Arg((steepestdescentpath)Therealpartof)in(29)isnegativeoverastraightpathjoiningtheendsofthesteepestdescentpathatthein“nity(dashedlinein“g.2(a)).Therefore,thecontourmaybedeformedintothesteepestdescentpathwheneverthepoles)arenotinside.Thishappensfor5.2JacobipolynomialsConsiderthefollowingintegralrepresentationoftheJacobipolynomialsfor 2Š1)n (1+ (1+ wherethefunctionx,w)isde“nedbyx,wlog(1+)+log(1 TheLaplace’sandSteepestDescentsMethods Re(w)Im(w)C z1 2 i Im(w) Re(w)Steepest ascent pathSteepest descent path1-x i1-x-1-x 21-x i2-C (a)(b)Figure2:(a)Steepestdescentpathofthefunctionforpositive.Thecondition2guaranteesthatthepolesofthefunctionsinh(2)lieoutsidethesteepestdescentpathandthecontourmaybedeformedinto.(b)Steepestdescentandascentpathsofthefunction(32).Thebranchpointsofthefunction)=(1(1+lieoutsidethesteepestdescentpathandthecontourmaybedeformedinto.6UniformityaspectsofthemethodIfthefunction)intheintegrand(2)containsarealparameter,saythesaddlepointsofw,a)maydependon.Then,itmayhappenthatwhencrossesacriticalvalue,anasymptoticpointof)changesitscharacterfromnon-relevanttorelevantorvice-versa,orthattworelevantasymptoticpointscoalesce,.Thatis,theasymptoticbehaviouroftheintegralmaychangedrasticallyatcertainvaluesof.Itiswellknownthat,inthiscase,thestandardLaplacesorSteepestDescentsmethodsmustbemodi“edtogetasymptoticexpansionsof(2)uniformlyvalidforvaryingaryingchap.7].Thatmodi“cationsdependonthekindofcoalescenceofasymptoticpoints.Wearespeakingaboutthewellknownuniformasymptoticmethods.Forexample,iftwosaddlepointscoalesce,then(2)maybeapproximatedbyAiryfunctions;ifasaddlepointcoalescewithandendpoint,(2)mayapproximatedbyparaboliccylinderfunctions,Themethoddescribedinsections2-4doesnotneedanymodi“cationbe-causeitisalreadyauniformmethod.Considerthefollowingtwopossibilities(notmutuallyexcluding)forthecriticalpointsofw,a)whencrossesacertaincriticalvaluea)Twoormorerelevantasymptoticpoints(criticalorendpoints)coalesce.Inthiscasethemethoddescribedinsection4isuniform.Thisisbecausetheexpansionof)issimultaneousatallofthoseasymptoticpoints TheLaplace’sandSteepestDescentsMethodsCoecients2,areregularwhen1)if0orif=0and[6].Forexample, =1,=1, =1, =1.Thefunctions),)and)in(23)are(1+ (1+ xb+1 (b xŠ(b,(2)0(x0eŠxt(1+t)bt2dt=ex xb+1 (b x2Š2(b +(Therefore,forlarge,b,c,x xb+1(bc+a(0)1 (b x(bŠ1+a(0)2 (b x2(bŠ2(b x(b+O1 uniformlyforfora0,a1]with2if=0.7ConcludingremarksTheasymptoticexpansionoftheintegral(2)forlargeisdeterminedbythenumber,multiplicityandnatureoftherelevantasymptoticpointsoftheexponent).Then,onlythevalueof)aroundthosepointsisofimportanceandasimultaneousapproximationof)atthosepointsisinorder.Butthisapproximationmusthaveanappropriatecontactorderateachofthosepointsaccordingtotheirmultiplicity.Thisisachievedbythemulti-pointTaylorexpansionof)atthosepointscountingmultiplicitiesultiplicitiesThemethoddescribedinthispaperonlyinvokestheclassicalLaplacesorSteepestDescentsmethodstoshowtheasymptoticbehaviourofthetermsoftheexpansionandoftheremainder.Theasymptoticexpansionisderived TheLaplace’sandSteepestDescentsMethodsInthestandardLaplacesorSaddlepointmethods,theasymptoticexpan-sionoftheintegralhasaformsimilarto(4).Thesequence)inthosemethodsisusuallyverysimple:itisproportionalto.Butthecoecientsareusuallyverycomplicated:theyaretheTaylorcoecientsofacom-plicatedfunctionobtainedafterachangeofvariable(includingajacobian).Ontheotherhand,thesequence)inthismethodismorecomplicated(formula(5))butthecoecientsaresimpler:theyaretheTaylorcoe-cientsofthefunction)at(formula(3)).Wecansaythat,withrespecttothestandardmethods,inthismethodthereisamovementofdif-“cultyfromthecoecientstothesequence).Thecomplexityofthesequence)willdependofcourseonthepreciseexampleanalyzed(inalltheexamplesconsidered[3],[4],[5]andtheexample6.1thesequenceturnsouttobeverysimple).Iftheintegrationpathinthede“nitionof)in(2)containrelevantandnon-relevantasymptoticpointsanduniformityisnotaproblem,wedonotneedtoconsideralloftheasymptoticpointssimultaneouslyinthemethod.Wejustneedtodividetheintegrationpathinseveralpiecescontaininglessasymptoticpoints.Ofcourse,thedominantasymptoticbehaviourcomesfromthepieceofcontainingthedominantasymptoticpoints.Thecontributionoftheremainingpieceswillbeexponentiallysubdominant.Ithasbeenpointedoutinsection6thatthismethodisintrinsicallyuni-form.ClassicaluniformmethodsbasedonstandardLaplacesorSteepestDescentsmethodsarestrongmodi“cationsofthecorrespondingnon-uniformmethods.Moreover,thereareseveralkindofmodi“cations(andhenceofuniformmethods)dependingonthekindofcoalescenceofasymptoticpoints(saddlepointnearanendpoint,twocoalescingsaddlepoints,).ThisissobecauseeveryuniformversionofthestandardLaplacesorSteepestDescentsmethodsrequiresanspeci“cchangeofvariable.Butinthemethoddescribedabovewedonotperformanychangeofvariable,justexpand)attheasymptoticpoints,whatevertheirdependenceoftheparameterdescribedinsection6is.Then,themethodisintrinsicallyuniform,andmoreover,itisuniqueindependentlyofthekindofcoalescenceofasymptoticpoints.Anothersourceofasymptoticpoints(relevantforanasymptoticanalysis)arethe(possible)singularitiesof).Webelievethatinthiscasewejustmayintroducethisnewkindofasymptoticpointsinthelistofasymptoticpointsandapplytheaboveideas.Wethinkthatinthiscasethemulti-pointTaylorexpansionof)mustbereplacedbyamulti-pointLaurentorTaylor-Laurentexpansion[7].Thisissubjectoffurtherinvestigation.