# 留数定理

## 定理

${\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {I} (\gamma ,a_{k})\operatorname {Res} (f,a_{k}).}$

${\displaystyle \oint _{\gamma }f(z)\,dz=2\pi i\sum _{k=1}^{n}\operatorname {Res} (f,a_{k}).}$

## 例子

${\displaystyle \int _{-\infty }^{\infty }{e^{itx} \over x^{2}+1}\,dx}$

${\displaystyle \int _{C}{f(z)}\,dz=\int _{C}{e^{itz} \over z^{2}+1}\,dz.}$

 ${\displaystyle {\frac {e^{itz}}{z^{2}+1}}\,\!}$ ${\displaystyle {}={\frac {e^{itz}}{2i}}\left({\frac {1}{z-i}}-{\frac {1}{z+i}}\right)\,\!}$ ${\displaystyle {}={\frac {e^{itz}}{2i}}{\frac {1}{z-i}}-{\frac {e^{itz}}{2i(z+i)}},\,\!}$

f(z)在z = i留数是：

${\displaystyle \operatorname {Res} _{z=i}f(z)={e^{-t} \over 2i}.}$

${\displaystyle \int _{C}f(z)\,dz=2\pi i\cdot \operatorname {Res} _{z=i}f(z)=2\pi i{e^{-t} \over 2i}=\pi e^{-t}.}$

${\displaystyle \int _{\mbox{straight}}+\int _{\mbox{arc}}=\pi e^{-t}\,}$

${\displaystyle \int _{-a}^{a}=\pi e^{-t}-\int _{\mbox{arc}}.}$

${\displaystyle \int _{\mbox{arc}}{e^{itz} \over z^{2}+1}\,dz\leq \int _{\mbox{arc}}\left|{e^{itz} \over z^{2}+1}\right|\,|dz|=\int _{\mbox{arc}}{|e^{itz}| \over |z^{2}+1|}\,|dz|=\int _{\mbox{arc}}{1 \over |z^{2}+1|}\,|dz|\leq \int _{\mbox{arc}}{1 \over a^{2}-1}\,|dz|={\frac {\pi a}{a^{2}-1}}\rightarrow 0\ {\mbox{as}}\ a\rightarrow \infty .}$

${\displaystyle \int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{-t}.}$

${\displaystyle \int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{t},}$

${\displaystyle \int _{-\infty }^{\infty }{e^{itz} \over z^{2}+1}\,dz=\pi e^{-\left|t\right|}.}$

（如果t = 0，这个积分就可以很快用初等方法算出来，它的值为π。）

## 参考文献

1. ^ 史济怀; 刘太顺. 复变函数. 合肥: 中国科学技术大学出版社. 1998/12/1. ISBN 9787312009990.