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DiscussiononthenatureofLegendrepolynomial

Abstract

LegendrepolynomialsarederivedbysolvingLegendre'sequations.Legendreequationisakindofdifferentialequationthatisoftenencounteredinphysicsandothertechnicalfields.Asearlyas1785,Legendrestudiedtheattractionbetweenspheresandthemotionofplanets.HeintroducedLegendre'sequationandobtaineditssolutionbymeansofseriessolution,whichwascalledLegendrepolynomial.Inthisproject,IwillexploreafewnicepropertiesofLegendrepolynomials,whicharethesimplestclassicalpolynomials.

Introduction

Legendrepolynomialsplayanimportantroleinpracticalmathematicalcalculation.Wecanuseittoprovemanyotherlawsandconclusionsmoreeasily.Andintheprocessofproving,wecanfindmoreperfectembodimentofmathematicalbeautyinmatrixtheory.Therefore,ourpaperonLegendrepolynomialsisveryimportantforustounderstandit.

MainResults

PART(a):Proofofthefollowingtheorem.

Theorem1 IfisasequenceofLegendrepolynomials,then

(i)formabasisfor.

(ii),i.e.,isorthogonaltoeverypolynomialofdegreelessthan.

Proof

(i)

SinceisasequenceofLegendrepolynomials,arelinearlyindependentlywitheachother.isapolynomialspaceofdimensionn,andtherearenlinearlyindependentpolynomials.Suchthateachpolynomialincanberepresentedby.

Henceformabasisfor.

(ii)

Letbeapolynomialin,sinceformabasisfor.Suchthatcanbewrittenintheformwhichis.Becauseisorthogonalto,isorthogonaltoeverypolynomialofdegreelessthan.

PART(b):Constructionfromdefinition.

Startingfromthepolynomial,useDefinition1toconstructthefirstfourLegendrepolynomials.

Weknowthatisasequenceoforthogonalpolynomials,sothatwhenever.Itcanbeobtainedfromtheabovethatforeachandforeach.

Wecanconstructanequationsetasfollows.

Suppose,and

Solvetheaboveequationset,.

Thensuppose,theequationsetis

Solvetheaboveequationset,.

Similartotheworkingabove,wecanobtainthat.

Hence,thefirstfourLegendrepolynomialswereconstructed.

PART(c):Constructionfromrecursionrelation.

Legendrepolynomialscanbegeneratedbythefollowingthreerecursiverelationships,

whereisdefinedtobezero.Checkthatthefirstfourpolynomialsdefinedbyabovearethepolynomialsinpartb.

Since,wecanobtain.Solvetheequation,then.

Thencalculatewithand

Solvetheequation,then.

Similartotheworkingabove,wecanobtainthat.

Hence,thefirstfourpolynomialsdefinedbyabovearethepolynomialsinpartb.

PART(d):Constructionfromthegeneratingfunction.

Let

ThefunctiondefinedaboveiscalledthegeneratingfunctionforLegendrepolynomials.Legendrepolynomialsarethecoefficientsintheexpansionofthisfunctioninpowersof.Expandasthepowerseriesinpowersof,andshowthatthefirstfourcoefficientsaregivenbypartb.

Fromthequestion,wecanobtainthat

ThecoefficientsofformLegendrepolynomials.Derivatewithonbothsidesoftheequation,weobtainthat

Organizetheaboveequation,weobtainthat

Comparativecoefficientoftheequation,weobtainthefollowingrecursiveformula

Itissamewiththerecursiveformulaweobtaininpartc.TakeTaylorexpansionof,weobtainthatand.

Withtherecursiveformulaand,weobtain,.

PART(e):Constructionfromcertaindifferentialequations.

ThegeneralformulaforLegendrepolynomialscanbewrittenas

Checkthatthefirstfourpolynomialsdefinedbyabovearethepolynomialinpartb.Verifythatthepolynomialgivenbyabovesatisfiesthefollowingdifferentialequationforeach:

or,equivalently,

whichariseswhenseparatingthevariablesinLaplace’sequationinsphericalcoordinates.

Withthefirstequation,weobtainthat

Hence,thefirstfourpolynomialsdefinedbyabovearethepolynomialinpartb.

Toverifythatthepolynomialgivenbyabovesatisfiesthefollowingdifferentialequationforeach.

PART(f):IntroductionofoneapplicationofLegendrePolynomials

WecanuseLegendrepolynomialstosolvethefixedsolutionofLaplaceEquation.

ThroughthestudyofLegendreequation,wegetthesolutionofLegendrepolynomialpowerseries.ItstudiessomenaturesofLegendrePolynominals.MakinguseofthenatureofLegendrePolynominals，itisveryeasytosolvethefixedsolutionofLaplaceEquation.ThismethodisobviouslybetterthanusinggreenfunctiontofindthefixedsolutionofLaplaceequation.

Laplace'sequation,alsoknownastheharmonicequationandthepotentialequation,isapartialdifferentialequation,soitisnamedafterLaplace,aFrenchmathematicianwhofirstproposedit.

ThroughthestudyofLegendrepolynomials,wefindmanypropertiesofLegendrepolynomials.InsolvingLaplace'sequation,thefollowingpropertiesareused.

Firstofall,property1:Legendrepolynomialshaveuniformexpressions.

Pnx=12nn!dn{(x2-1)n}dxn

Property2:Pnxisanevenfunctionwhenniseven;Whennisodd,Pnxistheoddfunction.

Property3:therecurrenceformulaforLegendrepolynomialsis

P'n+1x-xP'nx=(n+1)PnxxP'nx-P'n+1x=nPnx

Property4:Legendrepolynomialsareorthogonalontheinterval[-1,1].

Property5:thesquarerootof-11P2nxdxiscalledthemagnitudeoftheLegendrepolynomial.And

-11P2nxdx=22n+1

Byflexiblyapplyingtheabovefivecharacteristics,wecaneasilysolvethedefinitesolutionofLaplaceequation.

ConclusionandAckno