用留数计算方法求无穷积分的一个注记

A Note About the Calculation of Residue for Infinite Integral

无穷积分存在与Cauchy主值存在是不等价的。无穷积分存在时,Cauchy主值存在且等于积分值,但Cauchy主值存在时,积分值不一定存在。在复变函数中,在无穷积分存在的前提下,运用留数定理求出无穷积分的Cauchy主值,从而得出积分值。如果不能确定无穷积分值存在,用留数定理求出无穷积分的办法是有不足之处的。文章导出了一个实用结论并给出实例。

The existence of infinite integral is not equivalent to the existence of Cauchy principle value.When the infinite integral exists,Cauchy principle value exists and is equal to the value of the integral.But when Cauchy principle value exists,we can not deduce infinite integral exists.In complex analysis,when infinite integral exists,we apply residue theorem to get Cauchy principle value as the value of infinite integral.But if we don't know whether infinite integral exists,the method above may not work.This paper gets a practical conclusion and gives an example.