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The Scientific World Journal
Volume 2013 (2013), Article ID 536280, 7 pages
http://dx.doi.org/10.1155/2013/536280
Research Article

Taylor’s Expansion for Composite Functions

1Nhatrang Educational College, 01 Nguyen Chanh Street, Nhatrang City, Vietnam
2Department of Mathematics, University of Architecture of HoChiMinh City, 196 Pasteur Street, District 3, HoChiMinh City, Vietnam

Received 4 August 2013; Accepted 28 August 2013

Academic Editors: A. Agouzal and J.-S. Chen

Copyright © 2013 Le Thi Phuong Ngoc and Nguyen Anh Triet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We build a Taylor’s expansion for composite functions. Some applications are introduced, where the proposed technique allows the authors to obtain an asymptotic expansion of high order in many small parameters of solutions.

1. Introduction

Let and , .

Let two functions and, which depend on and , be defined as follows: where where notation stands for the norm in .

The function has the form of a composite function as follows: where where notation stands for the scalar product in .

Under the above assumptions, in order to obtain an asymptotic expansion of high order in many small parameters of solutions in recent papers [14], we need to solve the following problem.

Problem 1. Establish the functions (independent of ) such that for being small enough, it means that where the constant is independent from .
By Taylor’s expansion for , it implies that

However, for which is given in (3), it is very difficult to calculate , and so, the Problem 1 cannot proceed. Whichmethod can be used for solving the Problem 1? To answer, let us note that the function has the form of a composite function;this viewsuggests thatwe need to construct the Taylor-Maclaurin expansion of the composite function. So, first in Section 2 we solve the following problem.

Problem 2. Let be an open subset of and . Letand.Seek the representation formula for , such that for being small enough, where are calculated from the values of the given functions and of their derivatives at a suitable point.

Next in Section 3, as an application of the method used, we study the Problem 1. This technique is also a great help for the authors to obtain an asymptotic expansion as they want in recent papers [14].

2. Solving the Problem 2

We use the following notations. For a multi-index and , we put

The following lemma is useful to solve the Problems 1 and 2.

Lemma 3. Let and . Then where the coefficients depending on are defined by the recurrent formulas

The proof of Lemma 3 can be found in [2].

Now, using Taylor’s expansion of the function around the point , we obtain that

Similarly, we use Maclaurin’s expansion of as follows:

Substituting (13) into (12), we get where

Applying Lemma 3, with , it implies that

Hence,

Consequently,

Clearly, the Problem 2 is solved with

3. Solving the Problem 1

As an application of the method used in Section 2 for , and , for eachfixed , , for eachfixed , for eachfixed , we now investigate the Problem 1.

For eachfixed , using Taylor’s expansion of the function around the point up to order we obtain that where is small enough, .

Next, the precise structure of the representation formulas for will be needed below to continue.

For each fixed we have the following.

(i) The representation formula for .

We rewrite as follows in which

It is similar to ; we write where

For eachfixed we have the following.

(ii) The representation formula for .

Write

Similarly, in which where

Then(22), (24), (26), (28), and (30) imply that

We also need the following lemma.

Lemma 4. For all , then where

Proof of Lemma 4. We have
Note that , so
Combining (35) and (36) yields where
Lemma 4 is proved.

Substituting (37) into (20), we obtain where

Since , hence . Then from (39) and (40)1, we get for is small enough; consequently the Problem 1 is solved.

Remark 5. (i) This result improves the one in [1], for .
(ii) In case of , where , this result is also obtained in [3].

Acknowledgments

The authors wish to express their sincere thanks to the referees for their valuable comments and important remarks. This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under Grant no. B2013-18-05.

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