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Projectone
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
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