Using Maple to Study Multiple Improper Integrals

Chii-Huei Yu

10 IJRIT International Journal of Research in Information Technology, Volume 1, Issue 8, August, 2013, Pg. 10-14

International Journal of Research in Information Technology (IJRIT)

www.ijrit.com

ISSN 2001-5569

Using Maple to Study Multiple Improper Integrals

1

Chii-Huei Yu

1Assistant Professor, Department of Management and Information, Nan Jeon University of Science and Technology,

Tainan City, Taiwan

1[email protected]

Abstract

Multiple improper integral problems are closely related with probability theory and quantum field theory. Therefore, the evaluation and numerical calculation of multiple improper integrals is an important issue. This paper takes the mathematical software Maple as the auxiliary tool to study some type of multiple improper integrals. We can obtain the closed form of this type of multiple improper integrals by using differentiation with respect to a parameter and Leibniz differential rule. 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. 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.

Keywords: multiple improper integrals, closed form, differentiation with respect to a parameter, Leibniz differential rule, Maple.

1. Introduction

The computer algebra system (CAS) has been widely employed in mathematical and scientific studies. The rapid computations and the visually appealing graphical interface of the program render creative research possible.

Maple possesses significance among mathematical calculation systems and can be considered a leading tool in the CAS field. The superiority of Maple lies in its simple instructions and ease of use, which enable beginners to learn the operating techniques in a short period. In addition, through the numerical and symbolic computations performed by Maple, the logic of thinking can be converted into a series of instructions. The computation results of Maple can be used to modify previous thinking directions, thereby forming direct and constructive feedback that can aid in improving understanding of problems and cultivating research interests. 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, we can refer to [1-7].

In this paper, we mainly study the multiple improper integral problem. This problem is closely related with probability theory and quantum field theory, and can refer to [8-9]. Therefore, the evaluation and numerical

Chii-Huei Yu

11 calculation of multiple improper integrals are important issues. In this paper, we determine the following type of multiple improper integrals

∫

₁^∞⋅ ⋅⋅

∫

1^∞[ln(x1+x2+⋅ ⋅⋅+x_k)]ⁿ(x1+x2+⋅ ⋅⋅+x_k)^rdx1dx2⋅ ⋅⋅dx_k (1) where r is a real number, n, are positive integers, and k r<−k. We can obtain the closed form of this type of multiple improper integrals by using differentiation with respect to a parameter and Leibniz differential rule; this is the major result of this study (i.e., Theorem A). As for the related study of multiple improper integrals can refer to [10-15]. 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. 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 definition used in this paper.

2.1 Definition.

Suppose n is a positive integer, a₁,a₂,andβ_i are real numbers for all i=1,...,n, and let a₁<a₂. Assume the multivariable function g(x₁,x₂,⋅ ⋅⋅,x_n,a) is defined on [β₁,∞)×[β₂,∞)×⋅ ⋅⋅×[β_n,∞)×[a₁,a₂]. If, for any

>0

ε , there exists a real number R≥β_i(i=1,...,n) such that

∫

^∞⋅ ⋅⋅

∫

^∞ ⋅ ⋅⋅ ⋅ ⋅⋅ <ε

tn t g x xn a dx dxn

1 ( 1, , , ) 1 , for all ]

, [a₁ a₂

a∈ , whenever t_i ≥R for all i=1,...,n. Then we say that

∫

^{∞ ⋅ ⋅⋅}

∫

^{∞ ⋅ ⋅⋅ ⋅ ⋅⋅}

n g x xn a dx dxn

β β₁ ( ¹, , , ) ¹ is uniformly convergent on [a₁,a₂]. Next, we introduce two important theorems used in this study.

2.2 Differentiation with respect to a parameter.([16])

Suppose I = [β₁,∞)×[β₂,∞)×⋅ ⋅⋅×[β_n,∞)×[a₁,a₂], and the multivariable functiong(x₁,x₂,⋅ ⋅⋅,x_n,a) is defined on I. Assume g(x₁,x₂,⋅ ⋅⋅,x_n,a)and its partial derivative (x₁,x₂, ,x ,a)

a g

⋅⋅ n

∂ ⋅

∂ are continuous functions

on I. If

∫

_β^∞_n^{⋅ ⋅⋅}

∫

_β^∞ g x ^{⋅ ⋅⋅} xn a dx₁^{⋅ ⋅⋅}dxn

1 ( 1, , , ) are convergent for all a∈[a₁,a₂]^and

∫

^{∞ ⋅ ⋅⋅}

∫

^∞ _∂^{∂ ⋅ ⋅⋅ ⋅ ⋅⋅}

n x xn a dx dxn

a g

β β 1

1 ( 1, , , ) is uniformly convergent on [a₁,a₂]. Then

= ) (a

G

∫

^{∞ ⋅ ⋅⋅}

∫

^{∞ ⋅ ⋅⋅ ⋅ ⋅⋅}

n g x xn adx dxn

β β 1

1 ( 1, , , ) is differentiable on the open interval (a₁,a₂), and its derivative

= ) (a daG

d

∫

^{∞ ⋅ ⋅⋅}

∫

^∞ _∂^{∂ ⋅ ⋅⋅ ⋅ ⋅⋅}

n x xn a dx dxn

a g

β β 1

1 ( 1, , , ) for all a∈(a₁,a₂)^. 2.3 Leibniz differential rule. ([17])

Let n be a positive integer, and f(x),g(x)are functions such that their m -th order derivatives f^(m)(x),g^(m)(x) exist for all m=1,...,n. Then the n -th order derivative of the product function f(x)g(x),

) ( )

(fg ⁽ⁿ⁾ x ^{( )}( ) ^{( )}( )

0

x g x m f

n _{n m m}

n m

−

=

∑

_







= 

Chii-Huei Yu

12 , where

)!

(

!

! m n m

n m

n

= −







 .

The following is the main result in this study, we obtain the closed form of the type of multiple improper integrals (1).

2.4 Theorem A. Suppose r is a real number, n, are positive integers, and k r<−k. Then the multiple improper integral

k r

k n

k x x x dx dx dx

x x

x + +⋅ ⋅⋅+ + +⋅ ⋅⋅+ ⋅ ⋅⋅

⋅⋅

∫

₁^∞⋅

∫

1^∞[ln( 1 2 )] ( 1 2 ) 1 2













+ +

⋅⋅

⋅ + +

+ +

−

 ⋅







⋅ 

−

= _{+ + +}

=

−

+

∑

¹ ₁ ² _{1 1}

0 (ln ) ( 1) ! ( 1) ( 2) ( )

) 1

( ^m _{m m} ^k_m

n

m

m n r

k k

k r

A r

A r

m A m k

k n (2)

where A₁,A₂,⋅ ⋅⋅,A_k are constants which satisfy

k r

A r

A r

A k r r

r

k

+ +

⋅⋅

⋅ + + + + + =

⋅⋅

⋅ +

+1)( 2) ( ) 1 2

(

1 _{1 2}

.

2.4.1 Proof. Because

r k

k dx dx dx x

x

x + +⋅ ⋅⋅+ ⋅ ⋅⋅

⋅⋅

∫

₁^∞⋅

∫

1^∞( 1 2 ) 1 2

k x

x r

k dx dx

x x

r x  ⋅ ⋅⋅









 + +⋅ ⋅⋅+

⋅⋅ +

⋅

=

∫

^∞

∫

^{∞ =∞}

=

+ 2

1 1

1

1 1 1 2

1 )

1( 1

k r

k dx dx

x

r ⋅ ⋅ ⋅⋅ +x +⋅ ⋅⋅+ ⋅ ⋅⋅

+

= −

∫

1^∞

∫

1^{∞ +} 2

1

2 )

1 1 (

1

k x

x

k r dx dx

x r x

r  ⋅ ⋅⋅



 



 + +⋅ ⋅⋅+

⋅⋅ +

⋅ + ⋅

= −

∫

1^∞

∫

1^{∞ + =}₌^∞ 3

2 2 1

2 ) 2

1 2( 1 1

1

k r

k dx dx

x r x

r ⋅ ⋅ ⋅⋅ + +⋅ ⋅⋅+ ⋅ ⋅⋅

+ +

= −

∫

1^∞

∫

1^{∞ +} 3

2 3

2

) 2

) ( 2 )(

1 (

) 1 (

k k r k k

dx x

k k r r

r_{+ +}⁻ _⋅⁻_{⋅⋅ + −} ^⋅

∫

^{∞ − + + −}

= ¹ ₁ ( 1 ) ¹

) 1 (

) 2 )(

1 (

) 1 (











 − +

⋅ +

− +

⋅⋅

⋅ + +

= − ^=∞

=

− + xk

xk k r k k

x k k

r k

r r

r ₁

1

) 1 1 (

) 1 (

) 2 )(

1 (

) 1

(

) ( ) 2 )(

1 (

) 1 (

k r r

r

k^{r k}

k

+

⋅⋅

⋅ + +

⋅

= − ⁺



 





+ +

⋅⋅

⋅ + + + +

⋅

−

= ⁺

k r

A r

A r

k^{r k} A ^k

k

2 ) 1

1

( ^{1 2} (3)

Chii-Huei Yu

13 Using differentiation with respect to a parameter and Leibniz differential rule, differentiating n -times with respect to r on both sides of (3), we obtain

k r

k n

k x x x dx dx dx

x x

x + +⋅ ⋅⋅+ + +⋅ ⋅⋅+ ⋅ ⋅⋅

⋅⋅

∫

₁^∞⋅

∫

1^∞[ln( 1 2 )] ( 1 2 ) 1 2 ) ( 2

) 1 (

0 ( ) 1 2

) 1 (

m m k

n n r

m k k

k r

A r

A r

k A m

k n 



 





+ +

⋅⋅

⋅ + + + +







⋅ 

−

= ⁻

∑

=













+ +

⋅⋅

⋅ + +

+ +

−

 ⋅







⋅ 

−

= _{+ + +}

=

−

+

∑

¹ ₁ ² _{1 1}

0 (ln ) ( 1) ! ( 1) ( 2) ( )

) 1

( ^m _{m m} ^k_m

n

m

m n r

k k

k r

A r

A r

m A m k

k n ■

3. Examples

In the following, aimed at the type of multiple improper integrals in this study, we provide two examples and use Theorem A to determine their closed forms. On the other hand, we employ Maple to calculate the approximations of these multiple improper integrals and their closed forms for verifying our answers.

3.1 Example 1. By Theorem A, we obtain the following double improper integral

∫ ∫

₁^{∞ ∞ + + −}5 1 2

2 1 1

2 4

1 )] ( )

[ln(x x x x dx dx













− −

− −

 ⋅







=  _{+ +}

=

−

−

∑

⁴ _{1 1}

0

4 3

) 3 (

1 )

4 (

! 1 ) 1 ( ) 2 4 (ln

2 m m

m m

m m

m



 



 + + + +

= 10368

2 781 864ln ) 175 2 144(ln ) 37 2 36(ln ) 7 2 12(ln

1 8

1 _{4 3 2}

(4)

Next, we use Maple to verify the correctness of (4).

>evalf(Doubleint((ln(x1+x2))^4*(x1+x2)^(-5),x1=1..infinity,x2=1..infinity),14);

0.052895388838728

>evalf(1/8*(1/12*(ln(2))^4+7/36*(ln(2))^3+37/144*(ln(2))^2+175/864*ln(2)+781/10368),14);

0.052895388838729

3.2 Example 2. Using Theorem A, we can determine the following triple improper integral

∫ ∫ ∫

₁^{∞ ∞ ∞ + + + + −}6 1 2 3

3 2 1 1 1

3 3 2

1 )] ( )

[ln(x x x x x x dx dx dx













+ −

− −

− −

 ⋅







⋅ 

−

= _{+ + +}

=

−

−

∑

³ _{1 1 1}

0

3 6

3 3

) 3 (

2 / 1 )

4 (

1 )

5 (

2 /

! 1 ) 1 ( ) 3 3 (ln 3

) 1

( ^m _{m m m}

m

m m

m

Chii-Huei Yu

14





 



 + + +

= 2160000

39743 3

36000 ln ) 1489 3 1200 (ln ) 47 3 60(ln

1 27

1 _{3 2}

(5)

We also use Maple to verify the correctness of (5).

>evalf(Tripleint((ln(x1+x2+x3))^3*(x1+x2+x3)^(-6),x1=1..infinity,x2=1..infinity,x3=1..infinity),14);

0.0049337411954653

>evalf(1/27*(1/60*(ln(3))^3+47/1200*(ln(3))^2+1489/36000*ln(3)+39743/2160000),14);

0.0049337411954653

4. Conclusions

As mentioned, the differentiation with respect to a parameter and the Leibniz differential rule 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. On the other hand, 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. References

[1] M. L. Abell and J. P. Braselton, Maple by Example, 3rd ed., New York: Elsevier Academic Press, 2005.

[2] D. Richards, Advanced Mathematical Methods with Maple, New York: Cambridge University Press, 2002.

[3] C. T. J. Dodson and E. A. Gonzalez, Experiments in Mathematics Using Maple, New York: Springer-Verlag, 1995.

[4] R. J. Stroeker and J. F. Kaashoek, Discovering Mathematics with Maple : An Interactive Exploration for Mathematicians, Engineers and Econometricians, Basel: Birkhauser Verlag, 1999.

[5] F. Garvan, The Maple Book, London: Chapman & Hall/CRC, 2001.

[6] J. S. Robertson, Engineering Mathematics with Maple, New York: McGraw-Hill, 1996.

[7] C. Tocci and S. G. Adams, Applied Maple for Engineers and Scientists, Boston: Artech House, 1996.

[8] F. Streit, “On multiple integral geometric integrals and their applications to probability theory,” Canadian Journal of Mathematics, vol. 22, pp. 151-163, 1970.

[9] L. H. Ryder, Quantum Field Theory, 2nd ed., New York: Cambridge University Press, 1996.

[10] C. -H. Yu, “Application of Maple on multiple improper integral problems, ” Proceedings of 2012 Optoelectronics and Communication Engineering Workshop, National Kaohsiung University of Applied Sciences, Taiwan, pp. 275-280, October 2012.

[11] C. -H. Yu,“Application of Maple: taking the double improper integrals as examples, ” Proceedings of 2013 Information Education and Technology Application Seminar, Overseas Chinese University, Taiwan, pp. 1-5, March 2013.

[12]C. -H. Yu, “ Evaluation of two types of multiple improper integrals, ” Proceedings of 2012 Changhua, Yunlin and Chiayi Colleges Union Symposium, Da-Yeh University, Taiwan, M-7, December 2012.

[13] C. -H. Yu, “Using Maple to study the multiple improper integral problem,” Proceedings of IIE Asian Conference 2013, National Taiwan University of Science and Technology, Taiwan, vol. 1, pp. 625-632, July 2013.

[14] C. -H. Yu, “ Application of Maple on evaluating multiple improper integrals,” Proceedings of 6th IEEE/International Conference on Advanced Infocomm Technology, National United University, Taiwan, no.

00282, July 2013.

[15]C. -H. Yu,“A study on the multiple improper integral problems,” Journal of Hsin Sheng, in press.

[16] S. Lang, Undergraduate Analysis, New York: Springer-Verlag, p289, 1983.

[17]T. M. Apostol, Mathematical Analysis, 2nd ed., Boston: Addison-Wesley Publishing Co., Inc., p121, 1975.