# 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.
• Ahlfors, Lars, Complex Analysis, McGraw Hill, 1979, ISBN 0-07-085008-9
• Mitronivić, Dragoslav; Kečkić, Jovan, The Cauchy method of residues: Theory and applications, D. Reidel Publishing Company, 1984, ISBN 90-277-1623-4
• Lindelöf, Ernst, Le calcul des résidus et ses applications à la théorie des fonctions, Editions Jacques Gabay, 1905 (1989), ISBN 2-87647-060-8