數(shù)學(xué)外文翻譯9頁(yè)

1、Power Series Expansion and Its ApplicationsIn the previous section, we discuss the convergence of power series, in its convergence region, the power series always converges to a function. For the simple power series, but also with itemized derivative, or quadrature methods, find this and function. T

2、his section will discuss another issue, for an arbitrary function, can be expanded in a power series, and launched into.Whether the power series as and function? The following discussion will address this issue.1 Maclaurin (Maclaurin) formulaPolynomial power series can be seen as an extension of rea

3、lity, so consider the function can expand into power series, you can from the function and polynomials start to solve this problem. To this end, to give here without proof the following formula.Taylor (Taylor) formula, if the function at in a neighborhood that until the derivative of order , then in

4、 the neighborhood of the following formula: (9-5-1)Among That for the Lagrangian remainder. That (9-5-1)-type formula for the Taylor.If so, get , (9-5-2)At this point, ().That (9-5-2) type formula for the Maclaurin.Formula shows that any function as long as until the derivative, can be equal to a po

5、lynomial and a remainder.We call the following power series (9-5-3)For the Maclaurin series.So, is it to for the Sum functions? If the order Maclaurin series (9-5-3) the first items and for, whichThen, the series (9-5-3) converges to the function the conditions.Noting Maclaurin formula (9-5-2) and t

6、he Maclaurin series (9-5-3) the relationship between the knownThus, whenThere,Vice versa. That if,Units must.This indicates that the Maclaurin series (9-5-3) to and function as the Maclaurin formula (9-5-2) of the remainder term (when).In this way, we get a function the power series expansion:. (9-5

7、-4)It is the function the power series expression, if, the function of the power series expansion is unique. In fact, assuming the function f(x) can be expressed as power series, (9-5-5)Well, according to the convergence of power series can be itemized within the nature of derivation, and then make

8、(power series apparently converges in the point), it is easy to get.Substituting them into (9-5-5) type, income and the Maclaurin expansion of (9-5-4) identical.In summary, if the function f(x) contains zero in a range of arbitrary order derivative, and in this range of Maclaurin formula in the rema

9、inder to zero as the limit (when n ,), then , the function f(x) can start forming as (9-5-4) type of power series.Power Series,Known as the Taylor series.Second, primary function of power series expansionMaclaurin formula using the function expanded in power series method, called the direct expansio

10、n method.Example 1 Test the functionexpanded in power series of .Solution because,Therefore,So we get the power series, (9-5-6)Obviously, (9-5-6)type convergence interval , As (9-5-6)whether type is Sum function, that is, whether it converges to , but also examine remainder . Because ()，且,Therefore,

11、Noting the value of any set ,is a fixed constant, while the series (9-5-6) is absolutely convergent, so the general when the item when , , so when n , there,From thisThis indicates that the series (9-5-6) does converge to, therefore ().Such use of Maclaurin formula are expanded in power series metho

12、d, although the procedure is clear, but operators are often too Cumbersome, so it is generally more convenient to use the following power series expansion method.Prior to this, we have been a function, and power series expansion, the use of these known expansion by power series of operations, we can

13、 achieve many functions of power series expansion. This demand function of power series expansion method is called indirect expansion.Example 2 Find the function,Department in the power series expansion.Solution because,And，（）Therefore, the power series can be itemized according to the rules of deri

14、vation can be,（）Third, the function power series expansion of the application exampleThe application of power series expansion is extensive, for example, can use it to set some numerical or other approximate calculation of integral value.Example 3 Using the expansion to estimatethe value of.Solution

15、 because Because of, (),So thereAvailable right end of the first n items of the series and as an approximation of . However, the convergence is very slow progression to get enough items to get more accurate estimates of value.此外文文獻(xiàn)選自于：Walter.Rudin.數(shù)學(xué)分析原理(英文版)M.北京：機(jī)械工業(yè)出版社.冪級(jí)數(shù)的展開及其應(yīng)用在上一節(jié)中，我們討論了冪級(jí)數(shù)的收斂性

16、，在其收斂域內(nèi)，冪級(jí)數(shù)總是收斂于一個(gè)和函數(shù)對(duì)于一些簡(jiǎn)單的冪級(jí)數(shù)，還可以借助逐項(xiàng)求導(dǎo)或求積分的方法，求出這個(gè)和函數(shù)本節(jié)將要討論另外一個(gè)問題，對(duì)于任意一個(gè)函數(shù)，能否將其展開成一個(gè)冪級(jí)數(shù)，以及展開成的冪級(jí)數(shù)是否以為和函數(shù)?下面的討論將解決這一問題一、 馬克勞林(Maclaurin)公式冪級(jí)數(shù)實(shí)際上可以視為多項(xiàng)式的延伸，因此在考慮函數(shù)能否展開成冪級(jí)數(shù)時(shí)，可以從函數(shù)與多項(xiàng)式的關(guān)系入手來(lái)解決這個(gè)問題為此，這里不加證明地給出如下的公式泰勒(Taylor)公式 如果函數(shù)在的某一鄰域內(nèi)，有直到階的導(dǎo)數(shù)，則在這個(gè)鄰域內(nèi)有如下公式：,(9-5-1)其中稱為拉格朗日型余項(xiàng)稱(9-5-1)式為泰勒公式如果令，就得到，

17、(9-5-2)此時(shí)，, ()稱(9-5-2)式為馬克勞林公式公式說明，任一函數(shù)只要有直到階導(dǎo)數(shù)，就可等于某個(gè)次多項(xiàng)式與一個(gè)余項(xiàng)的和我們稱下列冪級(jí)數(shù) (9-5-3)為馬克勞林級(jí)數(shù)那么，它是否以為和函數(shù)呢?若令馬克勞林級(jí)數(shù)(9-5-3)的前項(xiàng)和為，即，那么，級(jí)數(shù)(9-5-3)收斂于函數(shù)的條件為注意到馬克勞林公式(9-5-2)與馬克勞林級(jí)數(shù)(9-5-3)的關(guān)系，可知于是，當(dāng)時(shí)，有反之亦然即若則必有這表明，馬克勞林級(jí)數(shù)(9-5-3)以為和函數(shù)馬克勞林公式(9-5-2)中的余項(xiàng) (當(dāng)時(shí))這樣，我們就得到了函數(shù)的冪級(jí)數(shù)展開式：(9-5-4)它就是函數(shù)的冪級(jí)數(shù)表達(dá)式，也就是說，函數(shù)的冪級(jí)數(shù)展開式是唯一的事實(shí)

18、上，假設(shè)函數(shù)可以表示為冪級(jí)數(shù)， (9-5-5)那么，根據(jù)冪級(jí)數(shù)在收斂域內(nèi)可逐項(xiàng)求導(dǎo)的性質(zhì)，再令(冪級(jí)數(shù)顯然在點(diǎn)收斂)，就容易得到將它們代入(9-5-5)式，所得與的馬克勞林展開式(9-5-4)完全相同綜上所述，如果函數(shù)在包含零的某區(qū)間內(nèi)有任意階導(dǎo)數(shù)，且在此區(qū)間內(nèi)的馬克勞林公式中的余項(xiàng)以零為極限(當(dāng)時(shí))，那么，函數(shù)就可展開成形如(9-5-4)式的冪級(jí)數(shù)冪級(jí)數(shù),稱為泰勒級(jí)數(shù)二、 初等函數(shù)的冪級(jí)數(shù)展開式利用馬克勞林公式將函數(shù)展開成冪級(jí)數(shù)的方法，稱為直接展開法例1 試將函數(shù)展開成的冪級(jí)數(shù)解 因?yàn)? 所以，于是我們得到冪級(jí)數(shù)， (9-5-6)顯然，(9-5-6)式的收斂區(qū)間為，至于(9-5-6)式是否以為和函數(shù)，即它是否收斂于，還要考察余項(xiàng)因?yàn)?()，且，所以注意到對(duì)任一確定的值，是一個(gè)確定的常數(shù)，而級(jí)數(shù)(9-5-6)是絕對(duì)收斂的，因此其一般項(xiàng)當(dāng)時(shí)，所以當(dāng)時(shí)，有，由此可知這表明級(jí)數(shù)(9-5-6)確實(shí)收斂于，因此有 ()這種運(yùn)用馬克勞林公式將函數(shù)展開成冪級(jí)數(shù)的方法，雖然程序明確，但是運(yùn)算往往過于繁瑣，因此人們普遍采用下面的比較簡(jiǎn)便的冪級(jí)數(shù)展開法在此之前，我們已經(jīng)得到了函