The Application of Parseval’s Theorem to Integral Problems

Chii-Huei Yu

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The Application of Parseval’s Theorem to Integral Problems

Chii-Huei Yu

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

Abstract

This paper uses the mathematical software Maple as an auxiliary tool to study six types of definite integrals. We can obtain the infinite series forms of these definite integrals by using Parseval’s theorem. On the other hand, we provide some examples to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying the answers by using Maple.

Cite this article:

  • Yu, Chii-Huei. "The Application of Parseval’s Theorem to Integral Problems." Applied Mathematics and Physics 2.1 (2014): 4-9.
  • Yu, C. (2014). The Application of Parseval’s Theorem to Integral Problems. Applied Mathematics and Physics, 2(1), 4-9.
  • Yu, Chii-Huei. "The Application of Parseval’s Theorem to Integral Problems." Applied Mathematics and Physics 2, no. 1 (2014): 4-9.

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1. Introduction

As the 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 use the mathematical software Maple to conduct mathematical research. The main reasons of using Maple in this paper are its simple instructions and ease of use, which enable the 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 approximations calculated and solutions to similar problems, as determined by Maple. For this reason, Maple can provide insights into scientific research.

In calculus and engineering mathematics courses, we learnt many methods to solve the integral problems including the change of variables method, the integration by parts method, the partial fractions method, the trigonometric substitution method, and so on. In this study, we evaluate the following six types of definite integrals which are not easy to obtain their answers using the methods mentioned above.

(1)
(2)
(3)
(4)
(5)
(6)

Where is a real number. We can obtain the infinite series forms of these definite integrals by using Parseval’s theorem; these are the major results of this paper (i.e., Theorems 1 and 2). The study of related integral problems can refer to [1-15][1]. Additionally, we propose some definite integrals to do calculation practically. The research methods adopted in this study involved finding solutions through manual calculations and verifying these answers 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. For this reason, Maple provides insights and guidance regarding problem-solving methods.

2. Main Results

Firstly, we introduce a notation, a definition and some formulas used in this article.

2.1. Notation

Let z = a + ib be a complex number, where , a, b are real numbers. We denote a the real part of z by Re(z), and b the imaginary part of z by Im(z).

2.2. Definition

Suppose f (x) is a continuous function defined on [0, 2π], the Fourier series expansion of f (x) is , where , , for all positive integers k.

2.3. Formulas
2.3.1. Euler’s Formula

, where x is any real number.


2.3.2. DeMoivre’s Formula

, where n is any integer, and x is any real number.


2.3.3. ([16])

, where α, β are real numbers.


2.3.4. ([16])

, where α, β are real numbers.


2.3.5. Taylor Series Expansion of Hyperbolic Secant Function ([17])

, where z is a complex number, , En are the Euler numbers for all non-negative integers n.


2.3.6. Taylor Series Expansion of Hyperbolic Cosecant Function ([17])

, where z is a complex number, , and Bn are the Bernoulli numbers for all positive integers n.

Next, we introduce an important theorem used in this study.

2.4. Parseval’s Theorem ([18])

Suppose f (x) is a continuous function defined on [0, 2π], f (0) = f (2π). If the Fourier series expansion of f (x) is , then .

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

2.5. Lemma 1

Suppose α, β are real numbers with . Then

(7)
(8)

2.5.1. Proof

(By Formula 2.3.4.)

And

In the following, we determine the infinite series forms of the definite integrals (1), (2) and (3).

2.6. Theorem 1

If r is a real number with . Then the definite integrals

(9)
(10)
(11)

2.6.1. Proof Because

(By (7))

(By Formula 2.3.5.)

(By DeMoivre’s formula)

(By Euler’s formula)

(12)

Thus, using Parseval’s theorem, we obtain

Similarly, because

(13)

By Parseval’s theorem, we have

On the other hand, from the summation of (9) and (10) and using (8),

Before deriving the second major result of this paper, we also need a lemma.

2.7. Lemma 2

Suppose α, β are real numbers with . Then

(14)
(15)

2.7.1. Proof

(By Formula 2.3.3.)

And

Finally, we find the infinite series forms of the definite integrals (4), (5) and (6).

2.8. Theorem 2

If r is a real number with . Then the definite integrals

(16)
(17)
(18)

2.8.1. Proof Because

(By (14))

(By Formula 2.3.6.)

(19)

Using Parseval’s theorem, we have

Similarly,

(By (14))

(By Formula 2.3.6.)

(20)

Therefore, using Parseval’s theorem, we obtain

In addition, from the summation of (16) and (17) and by (15), we have

3. Examples

In the following, for the six types of definite integrals in this study, we provide some definite integrals and employ Theorems 1 and 2 to determine their infinite series forms. In addition, we use Maple to calculate the approximations of these definite integrals and their solutions for verifying our answers.

3.1. Example 1

Taking into (9), we obtain the definite integral

(21)

Next, we use Maple to verify the correctness of (21).

>evalf(int((cosh(1/2*cos(x))*cos(1/2*sin(x)))^2/((sinh(1/2*cos(x)))^2+(cos(1/2*sin(x)))^2)^2,x=0..2*Pi),18);

>evalf(2*Pi+Pi*sum((euler(2*n))^2/((2*n)!)^2*(1/2)^(4*n),n=1..infinity),18);

Similarly, if in (10), the definite integral

(22)

>evalf(int((sinh(1/sqrt(5)*cos(x))*sin(1/sqrt(5)*sin(x)))^2/((sinh(1/sqrt(5)*cos(x)))^2+(cos(1/sqrt(5)*sin(x)))^2)^2,x=0..2*Pi),18);

>evalf(Pi*sum((euler(2*n))^2/((2*n)!)^2*(1/sqrt(5))^(4*n),n=1..infinity),18);

Finally, taking into (11), we have

(23)

>evalf(int(1/((sinh(4/5*cos(x)))^2+(cos(4/5*sin(x)))^2),x=0..2*Pi),18);

>evalf(2*Pi+2*Pi*sum((euler(2*n))^2/((2*n)!)^2*(4/5)^(4*n),n=1..infinity),18);

3.2. Example 2

If in (16), then

(24)

>evalf(int((sinh(2*cos(x))*cos(2*sin(x)))^2/((sinh(2*cos(x)))^2+(sin(2*sin(x)))^2)^2,x=0..2*Pi),18);

>evalf(Pi*(1/36+4*sum((1-2^(2*n-1))^2*(bernoulli(2*n))^2/((2*n)!)^2*2^(4*n-2),n=2..infinity)),18);

In addition, if in (17), then

(25)

>evalf(int((cosh(sqrt(7)*cos(x))*sin(sqrt(7)*sin(x)))^2/((sinh(sqrt(7)*cos(x)))^2+(sin(sqrt(7)*sin(x)))^2)^2,x=0..2*Pi),18);

>evalf(Pi*(1183/1764+4*sum((1-2^(2*n-1))^2*(bernoulli

(2*n))^2/((2*n)!)^2*(sqrt(7))^(4*n-2),n=2..infinity)),18);

On the other hand, if in (18), we have

(26)

>evalf(int(1/((sinh(9/5*cos(x)))^2+(sin(9/5*sin(x)))^2),x=0..2*Pi),18);

>evalf(Pi*(3229/4050+8*sum((1-2^(2*n-1))^2*(bernoulli

(2*n))^2/((2*n)!)^2*(9/5)^(4*n-2),n=2..infinity)),18);

4. Conclusion

In this article, we provide a new technique to determine some definite integrals. We hope this technique can be applied to solve another definite integral problems. On the other hand, the Parseval’s theorem plays a significant role in the theoretical inferences of this study. In fact, the applications of this theorem 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.

5. Appendix

In the following, we introduce some Maple's commands.

5.1. evalf( ); calculating the approximation.

Example. Evaluating the approximation of . > evalf(sqrt(2));

5.2. sum( ); Determining the summation.

Example. Evaluating the infinite series . > sum(1/n^2,n=1..infinity);

5.3. int( ); Determining the integral.

Example. Finding the integral . > int(cos(x),x);

5.4. bernoulli(n); Evaluating the n-th Bernoulli number.

Example. Determining the 12-th Bernoulli number. > bernoulli(12);

5.5. euler(n); Calculating the n-th Euler number.

Example. Evaluating the 8-th Euler number. > euler(8);

5.6. laplace(f(t),t,s); Evaluating the Laplace transform of .

Example. Finding the Laplace transform of . > laplace(cos(a*t),t,s);

5.7. invlaplace(F(s),s,t); Evaluating the inverse Laplace transform of .

Example. Determining the inverse Laplace transform of . > invlaplace(a/(s^2-a^2),s,t);

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