泰勒公式的几种证明及应用外文翻译原文.docx

1、泰勒公式的几种证明及应用外文翻译原文 Several proof and application of Taylor formulaAbstract:Taylor formula is an important formula in higher mathematics, itplays a very important role in theoretical and methodological.In theelementary function, polynomial is the most simple function, because theoperations of polynom

2、ial function only add, subtract, multiply threeoperations.Obviously,the study of functional state and function value ofapproximate calculation ofgreat significance,If wecan replacepolynomialfunctionapproximationforrational fractionfunction,especially theunreasonable functionand elementary transcende

3、ntalfunction to the thought of a approximation,while error can meet therequirements.The introduction :As everybody knows, polynomial function is one ofthe functions whose structure are very simple and convenient to makecalculation,it is an important content to use polynomial approximationfunction of

4、 approximate calculation and theoretical analysis .As we knowit is very convenient to usepolynomial approximationf xf ?x0xx0 replacing the function value f x andxx0 itsnearby, but its accuracy often can not meet our actual demand, it request-s us to find another polynomial ,the famous mathematician

5、Taylor putsforward the famous theorem,it is a good way to solve the polynomialapproximation by replacing some of the more complex functions such acomplicated problem with Taylor formulaWith more than Palin type of Taylor formulaTheorem 1:If the function f in the pointxountil N derivative,there aref

6、?x f ?x0 fx0?x x0 f?x0?x? x0?2f ?x0?x x0?noxx0?nn2!n!Prove:Assuming-thatTnf ?x0 fx0?x x0 f?x0?x? x0?2f?x0?x? x0?n?n?12!n!Rnf ?x?Tnx?Qnxx x0?nRn ?x 0As long as lim,By the relation( 1) we can getQ ?x?x?x0nRn ?x0 Rn?x0? Rn?nx00,Qn ?x0 Qn?x0 Qn?n?1x00,Qn?n x? n!, if there are f ?nx0,So in a neighborhood

7、 of the point0xo ,there are N-1 derivative function ,Allowed after usingL Hospitalsrule N-1 times,gettinglim Rn ?x lim Rn?x lim Rn?n?1xn?1x?x?xQ0nQ ?x?nx?x0 Qn ?x?x?x0f ?n?1x f ?n?1 x? fnx0?x x0?n?n?1?2?x? x0 lim00x?xf ?n?1x f ?n?1x0f? 1 limn!,?nx0 0xx0x?x0f ?x f ?x0 fx0?x x0 f?x0?x? x0?2f ?x0?x x0?

8、no?x x0?nn?2!n!Power Series Expansion and Its Applications.In the previous section, we discuss the convergence of power series, inits convergence region, the power series always converges to a function.of the simple power series with itemized derivative, or quadraturemethods finding this function. T

9、his section will discuss another issue, foran arbitrary function f x , if can be expanded in a power series, andlaunched into.Whether the power series f x as and function? Thefollowing discussion will address this issue.McLaughlin formulaPolynomial power series can be seen as an extension of reality

10、, soconsidering the function fx whether can expand into power series, youcanstarttosolvethisproblemfromthe functionfxandpolynomialsTo this end, giving the following formula here withoutproof.Taylor Taylor formula, ifthe function f xatxx0in aneighborhood thatuntil thederivative oforder n+1, thenin th

11、eneighborhood of the following formula:f xf x0 f ?x0xx0 fx0 xx0?f x0 xx0rnxnn!n22!9-5-1Amongrnx fn?1? xx0n?1.That rnx for the Lagrangiann?1!remainder.We can call that formula for Taylor Taylor formula.Ifx00,gettingf xf 0? x x2?xnrnx9-5-2rn?1x f n?1? xn?1 f n?1?x xn?1,0 ?1 .We call that typen?1!n?1!f

12、ormula9-5-2 for the McLaughlin formulaThe application of Taylor formulaTaylor formula of bond pricing research object mainly focused oncredit bonds, or from the Angle of the pricing issue, many scholarsanalyzed the interactive relations between all kinds of risks and itsinfluence on the bond pricing

13、 through the financial system from theAngle of theory andpractice,theyissue bonds pricing.should beaccording to the issuance of bonds factors such as the ease with which torelease quantity, At present, the domestic and foreign experts and scholarsof Taylor formula research in finance, research the A

14、ngle of each are notidenticalPetra Gerlach - Kristen 2004 to the introduction of thelong-term bond yields a forward-looking Taylor rule, pointing long-termbond rates reflect -ing the future inflation expectations, the formulation ofshort-term interest rate policy plays an important role; Mr Xie thin

15、kTaylor rule can provide a reference for Chinas monetary policy measure,measures the tightness of monetary policy; Jiang Lu think to some extent,the Taylor rule for specific movements in interbank lending rates in Chinaalso has a certain guiding role, can be used to guide the central bankmonetary po

16、licy; Xia suggest to give up the money supply target becausethe based currency i cant not be controlled, the instability of moneymultiplier, money velocity drops, resulted in the money supply and price,job growth and other macro target correlation is not high, so , mainly onthe rate of inflationMr.

17、Cao and ZuoYang Model framework,SVECexamined the peoples bank of China during the use of various monetarypolicy tools cointegration relation between multiple policyobjectives.between January 1996 and February 1996,Refers to thecentral bank monetary policy rules for monetary policy decisions andopera

18、tion guiding principle, the Taylor rule is a widely used and effectiverules.In March 1995, the goal of monetary policy is to keep the currencystable and thereby promotes economic growth promulgated by the Chinapeoples bank specified in article 3,the ultimate goal of monetary policyin our country is

19、clear from the law of keeping the currency stabletheintermediate target of monetary policy and operational goals changesfrom lending to the monetary base to money supplied.The scholar thinks,on our country money supply, Compared with the intermediary target ofmonetary policy implementation situation

20、 , the price stability the ultimateobjectiveis notsignificant Therefore, ifChinesemoney supplyintermediary goals are still useful, and whether there is a better policytools becoming a focus of the current domestic scholars.On the otherhand, with the increasingly market-oriented economy in our countr

21、y, themonetary policy in macroeconomic regulation is becoming more andmore significant role. But how can monetary policy adjust the economy,is discretion or policy rules and adopt what kind of policy rules, domesticscholars also debate for a long time.from Taylor rule, through empiricaltest of China

22、s monetary policy is to answer these questions, we use thequarterly data from 2000 to 2011 to examine the Taylor rule the originalinterest rate smoothing, Taylor rule and forward-looking Taylor rule.Model test consequence shows that the Taylor rule of the original type ofgoodness-of-fit is low, not

23、a good description of our country interest ratetrajectory; After joining interest rate smoothing factors, results in bettergoodness-of-fit and its value is very big, 0.8826, so the value of interestrate rules can carry on the good fitting to the monetary policy in ourcountry, and smooth of coefficie

24、nt of 0.7313 and its t statistic is verysignificant, indicating that at present in our countrys interest rates havetendency rate smoothing, and this kind of behavior is trying to be thepeoples bank of China, to keep interest rates steady is also one of Chinasmonetary policy objectives.Taylor formula

25、 expansion application in physics: harmonic vibration ofthe corresponding potential energywith x 2 forms, and canbecalculated precisely in mathIn physics first of all, paying attention tobalance state can be considered to be fixedin order to deal with thegeneral situation,. In order to achieve the e

26、ffect of move, we always adda perturbation to equilibrium, the object vibrationIn this case, thepotential field tends to be complex, so it is very difficult to solve thevibration of the concrete form. But if we use Taylor formula to deal withthe problem ,it becomes much simpler. The small vibration

27、theory tells usthat in theoretical mechanics, nearthe equilibrium to make Taylorexpansion potential energy for x form of power series, zero advisable to 0,anitem dueto theequilibrium statecorrespondingto theimum/minimum also is 0, starting from the second item is not zero. Ifthe accurate to the seco

28、ndary approximation, the potential energy in theform of the completely same with simple harmonic motion, so it is easyto solve. This processing method has a broad application in quantummechanics, solid physicsDealing with this kind problem in addition tothe Taylor series,and he usesoften Fourier ser

29、iesand Legendrepolynomial in the process.Withtherapiddevelopmentofcomputerandcommunicationtechnology , the use of computers for approximate calculation in manyfields such as science and engineering technology, has become one of theindispensableimportantsegmentsinthescientificresearchandengineering d

30、esign , that is to say, the approximate calculation is a veryimportant scientific research methodTaylor formula is a polynomialfitting, and polynomial is a simple function, it is very easy and convenientfor us to study it, especially for computer programmingTaylor theorem created the theory of finit

31、e difference, making anysingle variable function spread power series, at the same time alsomaking Taylor became the founder of the finite difference theory. In thebook Taylor discussed theapplications of calculus for aseries ofproblems, the result of the string lateral vibration is particularly important,he deduced basic frequency formula by solving equations , pioneeredthe study of string townthe advent of Taylor formula making manydifficult problems which cannot be solved before getting hope and manyof them turning into reality