广义积分敛散性判别方法探讨

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翻新时间：2023-03-31

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广义积分敛散性判别方法探讨

摘要

关键词：广义积分；收敛；发散

The Research of the Criterion of Convergence and Divergence of Generalized Integral

ABSTRACT

By means of traditional criterions of generalized integral’s convergence and divergence, this paper, from the analysis of integrated function’s nature, discovers a series of new criterions, which are briefer and more suitable: on the aspect of generalized integral for functions of a single variable, through the investigation of integrated function, together with the inner relationship between positive series and generalized integral in infinite interval under the condition of positive function, it gives several criterions of generalized integral’s convergence and divergence, which are similar to positive series’ criterions of convergence and divergence. On the aspect of generalized integral for functions of several variables, it emphasizes on generalized integral for functions of two variables and generalized integral for functions of three variables. By using the comparison test and the Cauchy test, along with the characters of integrated function, this paper gives a new way to determine the convergence of generalized integral for functions of two variables as well as a new way to determine its divergence. The comparison test and the Cauchy test which are often used in generalized integral for functions of a single variable and that of two variables are generalized to generalized integral for functions of three variables, and it uses the quadratic form theory to determine and calculate generalized integral for functions of three variables.

Key words: generalized integral; convergence; divergence