广义积分敛散性判别方法探讨

广义积分敛散性判别方法探讨
摘要

本文从分析被积函数本身所具的性质出发，借助传统的广义积分敛散性判别方法，发现1系列更简捷适用的'新判别方法：单变量函数广义积分方面，通过考察被积函数，结合正项级数与正函数情形下无穷区间上广义积分的内在联系，给出了几个与正项级数敛散性判别法相类似的广义积分敛散性判别方法；多变量函数广义积分方面，着重讨论了广义2重积分和广义3重积分，结合被积函数的特点，运用比较判别法和柯西判别法，本文给出了判别广义2重积分收敛的1种新方法．将单变量函数广义积分和广义2重积分中常用的比较判别法和柯西判别法推广到了广义3重积分，利用2次型理论解决了1类广义3重积分的判敛和计算问题．
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