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瑕积分 - Wikipedia

瑕积分

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微积分学




瑕积分是被积函数带有瑕点的广义积分,参见如下定义。

目录

[编辑] 定义1

函数f(x)在(a,b]上连续,点a为f(x)的瑕点.取t>a,如果极限

\lim_{t \to a^+}\int_{t}^{b}f(x)dx

存在,则称此极限为函数f(x)在(a,b]上的反常积分

瑕积分仍然记作\int_{a}^{b}f(x)dx

[编辑] 定义2

设函数f(x)在[a,b)上连续,点b为f(x)的瑕点。取t>b,如果极限

\lim_{t \to b^-}\int_{a}^{t}f(x)dx

存在,则称此极限为函数f(x)在[a,b)上的反常积分。

[编辑] 定义3

设函数f(x)在[a,b]上除点c(a<c<b)外上连续,点c为f(x)的瑕点。如果两个瑕积分

\int_{a}^{c}f(x)dx\int_{c}^{b}f(x)dx

都收敛,则定义

\int_{a}^{b}f(x)=\int_{a}^{c}f(x)+\int_{c}^{b}f(x)

[编辑] 参见


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