Solving Multiple Integrals Using Maple

The multiple integral problem is closely related to probability theory and quantum field theory. This paper uses the mathematical software Maple for the auxiliary tool to study three types of multiple integrals. We can obtain the infinite series forms of these three types of multiple integrals by using differentiation with respect to a parameter, differentiation term by term, and integration term by term. In addition, we provide some 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


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 Mozart, 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][2][3][4][5][6][7] can be adopted as references.
The multiple integral problem is closely related with probability theory and quantum field theory, and can refer to [8][9]. For this reason, the evaluation and numerical calculation of multiple integrals is important. In this study, we evaluate the following three types of n -tuple integrals We can obtain the infinite series forms of these three types of multiple integrals by using differentiation with respect to a parameter, differentiation term by term, and integration term by term ; these are the major results of this study (i.e., Theorems 1-3). In addition, we obtain three corollaries from the three theorems. For the study of related multiple integral problems can refer to [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. On the other hand, we propose some multiple integrals 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. The following is the flowchart of the research method used in this paper.

Main Results
Firstly, we introduce some notations and formulas used in this paper.

Notations
i. ,.., , where p is a positive integer, and 1 ! 0 = . Next, we introduce three important theorems used in this study.

Integration Term by Term ([27, p269]).
Suppose is a sequence of Lebesgue integrable functions defined on an inteval . If is convergent, then .
The following is the first major result in this study, we determine the infinite series form of the multiple integral (1). 2.6. Theorem 1 Suppose n is a positive integer, i m are non-negative integers, and Using differentiation with respect to a parameter and differentiation term by term, we differentiate i β by i m times on both sides of (6) for all Next, we determine the infinite series forms of the multiple integral (2).
Also, by differentiation with respect to a parameter and differentiation term by term, differentiating i β by i m times on both sides of (10) for all By differentiation with respect to a parameter and differentiation term by term, differentiating i β by i m times on both sides of (14) for all

Conclusions
As mentioned, evaluating the multiple integrals is important in probability theory and quantum field theory. In this study, we provide a new technique to solve three types of multiple integrals, and we hope this method can be applied in mathematical statistics or quantum physics. Simultaneously, the differentiation with respect to a parameter, the differentiation term by term and the integration term by term play significant roles in the theoretical inferences of this study. In fact, the application of the three theorems is extensive, and can be used to easily solve many difficult problems; we endeavor to conduct further studies on related applications.