利用留数计算实反常积分的反思  

Reflection on Calculation of Real Improper Integral by the Residue

作　　者：张来亮[1] 刘晓妍[1] 

机构地区：[1]山东科技大学公共课部,山东济南250031

出　　处：《河南教育学院学报（自然科学版）》2015年第1期51-52,共2页Journal of Henan Institute of Education(Natural Science Edition)

基　　金：山东科技大学教育教学研究"群星计划"重点资助项目(QX2013145)

摘　　要：在复变函数类教材中,均有在f(x)满足一定条件下,利用留数定理计算反常积分∫f(x)dx的内容,而常-∞+∞见的是f(x)取有理函数R(x)时的积分∫R(x)dx.有些教材不论R(x)在实轴上无奇点还是只有孤立简单极点都-∞认为收敛,混淆了两种不同的收敛概念;而不管R(x)在实轴上奇点的类型、不加以限制也得出相似的结论是不妥的.In the teaching materials of complex function class,the residue theorem is used to calculate improper integral ∫-∞^＋∞f（x） dx in certain satisfied conditions of f（x）,and f（x） is often taken to calculate rational function R（x）of integral ∫-∞^＋∞R（x） dx.Some materials conclude that R（x） is convergent regardless of no singularity or only isolated-∞simple pole convergence on the real axis,so two different concepts of convergence were confused.It is wrong to draw the similar conclusion regardless of the type of singularity of R（x） on the real axis without restrictions.

关 键 词：反常积分 收敛 P.V收敛 孤立奇点 留数 

分 类 号：O174[理学—数学]