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A Study on the Differential Problem of Trigonometric Functions with Maple

Chii-Huei Yu

Department of Management and Information, Nan Jeon University of Science and Technology, Tainan City, Taiwan

Abstract: This article uses the mathematical software Maple for the auxiliary tool to study the differential problem of two types of trigonometric functions. We can obtain the Fourier series expansions of any order derivatives of these two types of functions by using binomial theorem and differentiation term by term theorem, and hence greatly reduce the difficulty of calculating their higher order derivative values. On the other hand, we propose two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple.

Keywords: derivatives, trigonometric functions, Fourier series expansions, binomial theorem, differentiation term by term theorem, Maple

Received December 08, 2013; Revised December 28, 2013; Accepted January 15, 2014

Citation: Chii-Huei Yu, A Study on the Differential Problem of Trigonometric Functions with Maple. American Journal of Systems and Software, 2 (1). 9-13. January 2014. doi:10.12691/ajss-2-1-2

1. Introduction

As information technology advances, whether computers can become comparable with human brains to perform abstract tasks, such as abstract art similar to the paintings of Picasso and musical compositions similar to those of Beethoven, is a natural question. Currently, this appears unattainable. In addition, whether computers can solve abstract and difficult mathematical problems and develop abstract mathematical theories such as those of mathematicians also appears unfeasible. Nevertheless, in seeking for alternatives, we can study what assistance mathematical software can provide. This study introduces how to conduct mathematical research using the mathematical software Maple. The main reasons of using Maple in this study are its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. By employing the powerful computing capabilities of Maple, difficult problems can be easily solved. Even when Maple cannot determine the solution, problem-solving hints can be identified and inferred from the approximate values calculated and solutions to similar problems, as determined by Maple. For this reason, Maple can provide insights into scientific research. Inquiring through an online support system provided by Maple or browsing the Maple website (www.maplesoft.com) can facilitate further understanding of Maple and might provide unexpected insights. For the instructions and operations of Maple, [1-7] can be adopted as references.

In calculus and engineering mathematics curricula, evaluating the -th order derivative value of a function at , in general, needs to go through two procedures: firstly determining the -th order derivative of , and then taking into . These two procedures will make us be faced with increasingly complex calculations when calculating higher order derivative values of this function (i.e. is large), and hence to obtain the answers by manual calculations is not easy. In this paper, we mainly study the differential problem of the following two types of trigonometric functions

[picture] | (1)

[picture] | (2)

where are real numbers. We can obtain the Fourier series expansions of any order derivatives of these two types of trigonometric functions by using binomial theorem and differentiation term by term theorem; these are the major results of this study (i.e., Theorems 1 and 2), and hence greatly reduce the difficulty of calculating their higher order derivative values. For the study of related differential problems can refer to [8-24]. In addition, we provide two examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these solutions by using Maple. This type of research method not only allows the discovery of calculation errors, but also helps modify the original directions of thinking from manual and Maple calculations. Therefore, Maple provides insights and guidance regarding problem-solving methods.

2. Main Results

Firstly, we introduce a notation and two formulas used in this study.

2.1. Notation

Suppose are non-negative integers, . Define and .

2.2. Euler’s Formula

, where , is any real number.

2.3. Taylor Series Expansion of Exponential Function

, where is any real number.

The followings are two important theorems used in this paper, we can refer to [25,26] respectively.

2.4. Binomial Theorem

Suppose are complex numbers, and is any non-negative integer. Then

2.5. Differentiation Term by Term Theorem

For all non-negative integers , if the functions satisfy the following three conditions： (i) there exists a point such that is convergent, (ii) all functions are differentiable on open interval , (iii) is uniformly convergent on . Then is uniformly convergent and differentiable on . Moreover, its derivative .

Before deriving the first major result of this study, we need a lemma.

2.6. Lemma 1

Suppose y are real numbers, and is any non-negative integer. Then the trigonometric function

[picture] | (3)

for all .

Proof

Next, we determine the Fourier series expansions of any order derivatives of the trigonometric function (1).

2.7. Theorem 1

Suppose are real numbers, and is any positive integer. Let the domain of the trigonometric function

be . Then the -th order derivative of ,

[picture] | (4)

for all .

Proof Because

[picture] | (5)

By differentiation term by term theorem, differentiating -times with respect to on both sides of (5), we obtain

for all .

To prove the second major result of this study, we also need a lemma.

2.8. Lemma 2

Let the assumptions be the same as Lemma 1, then

[picture] | (6)

for all .

Proof

Finally, we find the Fourier series expansions of any order derivatives of the trigonometric function (2).

2.9. Theorem 2

If the assumptions are the same as Theorem 1. Let the domain of the trigonometric function

be . Then the -th order derivative of ,

[picture] | (7)

for all .

Proof Because

[picture] | (8)

Also, using differentiation term by term theorem, differentiating -times with respect to on both sides of (8), we have

for all .

3. Examples

In the following, for the differential problem of the two types of trigonometric functions in this study, we provide two examples and use Theorems 1, 2 to determine the Fourier series expansions of any order derivatives of these functions and evaluate some of their higher order derivative values. On the other hand, we employ Maple to calculate the approximations of these higher order derivative values and their solutions for verifying our answers.

3.1. Example 1

Suppose the domain of the trigonometric function

[picture] | (9)

is . By Theorem 1, we obtain any -th order derivative of ,

[picture] | (10)

for all .

Therefore, the 13 -th order derivative value of at ,

[picture] | (11)

Next, we use Maple to verify the correctness of (11).

>f:=x->exp(3*cos(2*x-Pi/4));

>evalf((D@@13)(f)(5*Pi/4),22);

>evalf(-2^13*sum(3^k/(2^k*k!)*sum(k!/(m!*(k-m)!)*(k-2*m)^13*sin((k-2*m)*Pi/4),m=0..k),k=0..infinity),22);

The above answer obtained by Maple appears I (), it is because Maple calculates by using special functions built in. The imaginary part is very small, so can be ignored.

3.2. Remark 1

In Example 1, the first two derivatives of are

[picture] | (12)

[picture] | (13)

If we continue to differentiate , we will be faced with a complex calculation. Thus, Theorem 1 gives us a convenient method to evaluate the derivatives of , and hence reduces the difficulty of calculating the higher order derivative values of .

3.3. Example 2

If the domain of the trigonometric function

[picture] | (14)

is . Using Theorem 2, we can evaluate any -th order derivative of ,

[picture] | (15)

for all .

Thus, the 10 -th order derivative value of at ,

[picture] | (16)

>g:=x->exp(5*sin(7*x+4*Pi/3));

>evalf((D@@10)(g)(-Pi/6),28);

>evalf(-7^10*sum(5^k/(2^k*k!)*sum(k!/(m!*(k-m)!)*(k-2*m)^10*cos((k-2*m)*Pi/3),m=0..k),k=0..infinity),28);

The imaginary part of the above answer obtained by Maple is very small, so can be ignored.

3.4. Remark 2

In Example 2, the first two derivatives of are

[picture] | (17)

[picture] | (18)

Also, if we continue to differentiate , the calculation will be complicated. Therefore, Theorem 2 provide a good method to find any derivatives of , and reduces the difficulty of calculating its higher order derivative values.

4. Conclusion

In this study, we provide a new technique to evaluate any order derivatives of some trigonometric functions. We hope this technique can be applied to solve another differential problems. On the other hand, the binomial theorem and the differentiation term by term theorem play significant roles in the theoretical inferences of this study. In fact, the applications of these two theorems are extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications. In addition, Maple also plays a vital assistive role in problem-solving. In the future, we will extend the research topic to other calculus and engineering mathematics problems and solve these problems by using Maple. These results will be used as teaching materials for Maple on education and research to enhance the connotations of calculus and engineering mathematics.

References

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