一道复数积分∫z³/(z²+1)dz
来源:学生作业帮助网 编辑:作业帮 时间:2024/06/30 01:07:06
![一道复数积分∫z³/(z²+1)dz](/uploads/image/z/14864102-62-2.jpg?t=%E4%B8%80%E9%81%93%E5%A4%8D%E6%95%B0%E7%A7%AF%E5%88%86%E2%88%ABz%26sup3%3B%EF%BC%8F%EF%BC%88z%26sup2%3B%2B1%29dz)
一道复数积分∫z³/(z²+1)dz
一道复数积分
∫z³/(z²+1)dz
一道复数积分∫z³/(z²+1)dz
见下图:
∫z³/(z²+1)dz=∫z²·z/(z²+1)dz=(1/2)∫z²/(z²+1)dz²=(1/2)∫(1-1/(z²+1)dz²=(1/2)[z²-∫1/(z²+1)dz²]=(1/2)z²-ln(z²+1)+C
∫z³/(z²+1)dz=∫[z(z²+1)-z]/(z²+1)dz
=∫zdz- ∫z/(z²+1)dz
=1/2z²-1/2∫1/(z²+1)dz²+1
=1/2z²-1/2ln(z²+1)+C
(*^__^*) 嘻嘻……
∫z³/(z²+1)dz=1/2∫z²/(z²+1)dz²=1/2∫(z²+1-1)/(z²+1)dz²=1/2∫[1-1/(z²+1)]dz²=1/2∫1dz²-1/2∫1/(z²+1)d(z²+1)=1/2z²-1/2ln(z²+1)