带有可能的瑕点的广义积分  

Improper Integral with Possible Flaws

作　　者：杨易霖 任咏红[1] YANG Yilin;REN Yonghong(School of Mathematics,Liaoning Normal University,Dalian 116029)

机构地区：[1]辽宁师范大学数学学院,辽宁大连116029

出　　处：《高等数学研究》2023年第6期61-65,70,共6页Studies in College Mathematics

基　　金：2022年度辽宁省普通高等教育本科教学改革研究一般项目(辽教通[2022]166号).

摘　　要：在数学分析的学习中,我们常常会遇到一类既不是Riemann积分又不是瑕积分的特殊积分,其被积函数在一闭区间上除去有限个点外有定义,但这些无定义的点又不是瑕点.该类积分通常被当作Riemann积分直接处理,一些学生会对此产生疑惑.本文将该类积分定义为带有可能的瑕点的广义积分,梳理了它与Riemann积分之间的联系,介绍了它的性质,并讨论了其在Fourier级数和含参量积分的函数极限问题中的应用.In the study of mathematical analysis,we often encounter a class of special integrals that are neither Riemann integrals nor defect integrals.The integrand has a definition on a closed interval except for a finite number of points,but these undefined points are not flaws.Such integrals are usually treated as Riemann integrals directly,and some students may have doubts about this.Naming this kind of integral by an improper integral with possible flaws,the relationship between it and Riemann integral is sorted out in this paper,its properties are introduced,and its applications in Fourier series and function limit problems with parametric integrals are discussed.

关 键 词：广义积分 RIEMANN积分 可能的瑕点 

分 类 号：O172.2[理学—数学]