泰勒公式的几种证明及应用外文翻译原文.docx

泰勒公式的几种证明及应用外文翻译原文.docx

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泰勒公式的几种证明及应用外文翻译原文

SeveralproofandapplicationofTaylorformula

Abstract:

Taylorformulaisanimportantformulainhighermathematics,it

playsaveryimportantroleintheoreticalandmethodological.Inthe

elementaryfunction,polynomialisthemostsimplefunction,becausethe

operationsofpolynomialfunctiononlyadd,subtract,multiplythree

operations.Obviously,thestudyoffunctionalstateandfunctionvalueof

approximatecalculationofgreatsignificance,Ifwecanreplace

polynomialfunctionapproximationforrationalfractionfunction,

especiallytheunreasonablefunctionandelementarytranscendental

functiontothethoughtofa"approximation",whileerrorcanmeetthe

requirements.

Theintroduction:

Aseverybodyknows,polynomialfunctionisoneof

thefunctionswhosestructureareverysimpleandconvenienttomake

calculation,itisanimportantcontenttousepolynomialapproximation

functionofapproximatecalculationandtheoreticalanalysis.Asweknow

itisveryconvenienttouse

polynomialapproximation

fxf?

x0xx0replacingthefunctionvaluefxandxx0its

nearby,butitsaccuracyoftencannotmeetouractualdemand,itrequest

-sustofindanotherpolynomial,thefamousmathematicianTaylorputs

forwardthefamoustheorem,itisagoodwaytosolvethepolynomial

approximationbyreplacingsomeofthemorecomplexfunctionssucha

complicatedproblemwithTaylorformula

WithmorethanPalintypeofTaylorformulaTheorem1:

IfthefunctionfinthepointxountilNderivative,thereare

f?

xf?

x0fx0?

?

xx0f?

x0?

?

x?

x0?

2f?

x0?

?

xx0?

noxx0?

?

nn2!

n!

Prove:

Assuming-that

Tnf?

x0fx0?

?

xx0f?

x0?

?

x?

x0?

2f?

x0?

?

x?

x0?

n

?

n?

1

2!

n!

Rnf?

x?

?

Tnx?

Qnxxx0?

n

Rn?

x0

Aslongaslim

Bytherelation

（1）wecanget

Q?

x?

x?

x

0

n

Rn?

x0Rn?

?

x0?

Rn?

nx00

Qn?

x0Qn?

?

x0Qn?

n?

1x00,

Qn?

nx?

n!

iftherearef?

nx0,Soinaneighborhoodofthepoint

0

xo,thereareN-1derivativefunction,AllowedafterusingLHospitals

ruleN-1times,getting

limRn?

xlimRn?

?

xlimRn?

n?

1xn?

1x?

x?

x

Q

0

n

Q?

x?

nx?

x0Qn?

x?

x?

x

0

f?

n?

1

xf?

n?

1x?

fnx0?

?

xx0?

n?

n?

1?

?

2?

x?

x0lim

0

0

x?

xf?

n?

1

xf?

n?

1x0f?

1lim

n!

?

nx00

xx0

x?

x0

f?

xf?

x0fx0?

?

xx0f?

x0?

?

x?

x0?

2f?

x0?

?

xx0?

no?

?

xx0?

nn?

2!

n!

PowerSeriesExpansionandItsApplications.

Intheprevioussection,wediscusstheconvergenceofpowerseries,in

itsconvergenceregion,thepowerseriesalwaysconvergestoafunction.

ofthesimplepowerserieswithitemizedderivative,orquadrature

methodsfindingthisfunction.Thissectionwilldiscussanotherissue,for

anarbitraryfunctionfx,ifcanbeexpandedinapowerseries,and

launchedinto.Whetherthepowerseriesfxasandfunction?

The

followingdiscussionwilladdressthisissue.

McLaughlinformula

Polynomialpowerseriescanbeseenasanextensionofreality,so

consideringthefunctionfxwhethercanexpandintopowerseries,you

canstarttosolvethisproblemfromthefunctionfxand

polynomialsTothisend,givingthefollowingformulaherewithout

proof.

TaylorTaylorformula,ifthefunctionfx

at

xx0

ina

neighborhoodthatuntilthederivativeofordern+1,theninthe

neighborhoodofthefollowingformula:

fxfx0f?

x0xx0fx0xx0

?

…fx0xx0rnx

n

n!

n

2

2!

9-5-1

Among

rnxf

n?

1

?

xx0n?

1.ThatrnxfortheLagrangian

n?

1!

remainder.WecancallthatformulaforTaylorTaylorformula.

Ifx00,getting

fxf0?

xx

2

?

…x

nrnx

9-5-2

rn?

1xfn?

1?

xn?

1fn?

1?

xxn?

1

0?

1.Wecallthattype

n?

1!

n?

1!

formula9-5-2fortheMcLaughlinformula

TheapplicationofTaylorformula

Taylorformulaofbondpricingresearchobjectmainlyfocusedon

creditbonds,orfromtheAngleofthepricingissue,manyscholars

analyzedtheinteractiverelationsbetweenallkindsofrisksandits

influenceonthebondpricingthroughthefinancialsystemfromthe

Angleoftheoryandpractice,they

issuebondspricing.shouldbe

accordingtotheissuanceofbondsfactorssuchastheeasewithwhichto

releasequantity,Atpresent,thedomesticandforeignexpertsandscholars

ofTaylorformularesearchinfinance,researchtheAngleofeacharenot

identicalPetraGerlach-Kristen2004totheintroductionofthe

long-termbondyieldsaforward-lookingTaylorrule,pointinglong-term

bondratesreflect-ingthefutureinflationexpectations,theformulationof

short-terminterestratepolicyplaysanimportantrole;MrXiethink

TaylorrulecanprovideareferenceforChina'smonetarypolicymeasure,

measuresthetightnessofmonetarypolicy;JiangLuthinktosomeextent,

theTaylorruleforspecificmovementsininterbanklendingratesinChina

alsohasacertainguidingrole,canbeusedtoguidethecentralbank

monetarypolicy;Xiasuggesttogiveupthemoneysupplytargetbecause

thebasedcurrencyican'tnotbecontrolled,theinstabilityofmoney

multiplier,moneyvelocitydrops,resultedinthemoneysupplyandprice,

jobgrowthandothermacrotargetcorrelationisnothigh,so,mainlyon

therateofinflationMr.CaoandZuoYangModelframework,SVEC

examinedthepeople'sbankofChinaduringtheuseofvariousmonetary

policytoolscointegrationrelationbetweenmultiplepolicy

objectives.betweenJanuary1996andFebruary1996,

Referstothe

centralbankmonetarypolicyrulesformonetarypolicydecisionsand

operationguidingprinciple,theTaylorruleisawidelyusedandeffective

rules.

InMarch1995,thegoalofmonetarypolicyistokeepthecurrency

stableandtherebypromoteseconomicgrowthpromulgatedbytheChina

people'sbank"specifiedinarticle3,"theultimategoalofmonetarypolicy

inourcountryisclearfromthelawofkeepingthecurrencystablethe

intermediatetargetofmonetarypolicyandoperationalgoalschanges

fromlendingtothemonetarybasetomoneysupplied.Thescholarthinks,

onourcountrymoneysupply,Comparedwiththeintermediarytargetof

monetarypolicyimplementationsituation,thepricestabilitytheultimate

objectiveisnotsignificantTherefore,ifChinesemoneysupply

intermediarygoalsarestilluseful,andwhetherthereisabetterpolicy

toolsbecomingafocusofthecurrentdomesticscholars.Ontheother

hand,withtheincreasinglymarket-orientedeconomyinourcountry,the

monetarypolicyinmacroeconomicregulationisbecomingmoreand

moresignificantrole.Buthowcanmonetarypolicyadjusttheeconomy,

isdiscretionorpolicyrulesandadoptwhatkindofpolicyrules,domestic

scholarsalsodebateforalongtime.fromTaylorrule,throughempirical

testofChina'smonetarypolicyistoanswerthesequestions,weusethe

quarterlydatafrom2000to2011toexaminetheTaylorruletheoriginal

interestratesmoothing,Taylorruleandforward-lookingTaylorrule.

ModeltestconsequenceshowsthattheTaylorruleoftheoriginaltypeof

goodness-of-fitislow,notagooddescriptionofourcountryinterestrate

trajectory;Afterjoininginterestratesmoothingfactors,resultsinbetter

goodness-of-fitanditsvalueisverybig,0.8826,sothevalueofinterest

raterulescancarryonthegoodfittingtothemonetarypolicyinour

country,andsmoothofcoefficientof0.7313anditststatisticisvery

significant,indicatingthatatpresentinourcountry'sinterestrateshave

tendencyratesmoothing,andthiskindofbehavioristryingtobethe

people'sbankofChina,tokeepinterestratessteadyisalsooneofChina's

monetarypolicyobjectives.

Taylorformulaexpansionapplicationinphysics:

harmonicvibrationof

thecorrespondingpotentialenergywithx^2forms,andcanbe

calculatedpreciselyinmathInphysicsfirstofall,payingattentionto

balancestatecanbeconsideredtobe"fixed"inordertodealwiththe

generalsituation,.Inordertoachievetheeffectof"move",wealwaysadd

aperturbationtoequilibrium,theobjectvibrationInthiscase,the

potentialfieldtendstobecomplex,soitisverydifficulttosolvethe

vibrationoftheconcreteform.ButifweuseTaylorformulatodealwith

theproblem,itbecomesmuchsimpler.Thesmallvibrationtheorytellsus

thatintheoreticalmechanics,neartheequilibriumtomakeTaylor

expansionpotentialenergyforxformofpowerseries,zeroadvisableto0,

anitemduetotheequilibriumstatecorrespondingtothe

imum/minimumalsois0,startingfromtheseconditemisnotzero.If

theaccuratetothesecondaryapproximation,thepotentialenergyinthe

formofthecompletelysamewithsimpleharmonicmotion,soitiseasy

tosolve.Thisprocessingmethodhasabroadapplicationinquantum

mechanics,solidphysicsDealingwiththiskindprobleminadditionto

theTaylorseries,andheusesoftenFourierseriesandLegendre

polynomialintheprocess.

Withtherapiddevelopmentofcomputerandcommunication

technology,theuseofcomputersforapproximatecalculationinmany

fieldssuchasscienceandengineeringtechnology,hasbecomeoneofthe

indispensableimportantsegmentsinthescientificresearchand

engineeringdesign,thatistosay,theapproximatecalculationisavery

importantscientificresearchmethodTaylorformulaisapolynomial

fitting,andpolynomialisasimplefunction,itisveryeasyandconvenient

forustostudyit,especiallyforcomputerprogrammingTaylortheoremcreatedthetheoryoffinitedifference,makingany

singlevariablefunctionspreadpowerseries,atthesametimealso

makingTaylorbecamethefounderofthefinitedifferencetheory.Inthe

bookTaylordiscussedtheapplicationsofcalculusforaseriesof

problems,theresultofthestringlateralvibrationisparticularlyimportant,

hededucedbasicfrequencyformulabysolvingequations,pioneered

thestudyofstringtowntheadventofTaylorformulamakingmany

difficultproblemswhichcannotbesolvedbeforegettinghopeandmany

ofthemturningintoreality