反常二重积分收敛性的判定

On Criteria for Convergence of Improper Double Integral

华东师范大学数学系编《数学分析(下册)》教材在第21.8节介绍了反常二重积分收敛的定义、判定定理,作者发现教材中对本节内容的处理不够清晰,特别是没有给出定理21.19关于反常二重积分收敛等价于绝对收敛的直观解释.本文优化了该节的内容,理顺了反常二重积分收敛的判定方法,证明了无界区域上的二重积分转化为累次积分的定理,构造例子说明了反常一重积分收敛与反常二重积分收敛的本质区别.通过分析例子表明,在本文框架下判定反常二重积分收敛性及计算积分值是非常有效的.

The definition and criteria for convergence of improper double integral were introduced in Section 21.8 of ＂Mathematical Analysis＂, written by Department of Mathematics, East China Normal University. We found that contents in this section are unclear; especially the conclusion of Theorem 21.19 is not intuitively explained. In this paper, we optimize contents of this section, rationalize the criteria for convergence of improper double integral, obtain a theorem to transform improper double integral on unbounded domain into an iterated integral, give the intuitive explanation of Theorem 21. 19 and calculate some examples by using the framework in this paper. By analyzing the examples, our framework is very effective to judge the convergence and calculate the value of improper double integral.