# 刘维尔定理 (复分析)

## 简介

${\displaystyle \lim _{z\to z_{0}}{\frac {f(z)-f(z_{0})}{z-z_{0}}}}$

${\displaystyle z\in \mathbb {C} ,\;\;|f(z)|\leqslant M,}$

## 证明

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k}}$

${\displaystyle a_{k}={\frac {f^{(k)}}{k!}}={1 \over 2\pi i}\oint _{C_{r}}{f(\zeta ) \over \zeta ^{k+1}}\,d\zeta }$

${\displaystyle |a_{k}|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {|f(\zeta )|}{|\zeta ^{k+1}|}}|\,d\zeta |\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M}{r^{k+1}}}|\,d\zeta |\leq {\frac {M}{r^{k}}},}$

${\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k}=a_{0}}$

## 应用与推论

### 整函数的大小关系

${\displaystyle |f|\leqslant |g|}$说明，函数${\displaystyle h}$的模长总小于等于1。另一方面，由于${\displaystyle |f|\leqslant |g|}$，所以${\displaystyle {\frac {f(z)}{g(z)}}}$奇点都是可去奇点，可以依照上面的方式拓延为整函数${\displaystyle h}$。所以${\displaystyle h}$作为一个有界的整函数，根据刘维尔定理，必然是常数函数。这说明${\displaystyle f}$${\displaystyle g}$成比例关系。

### 次线性整函数

${\displaystyle \forall z\in \mathbb {C} ,\;\;|f(z)|\leqslant M|z|.}$

${\displaystyle |f'(z)|={\frac {1}{2\pi }}\left|\oint _{C_{r}}{\frac {f(\zeta )}{(\zeta -z)^{2}}}\mathrm {d} \zeta \right|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {\left|f(\zeta )\right|}{\left|(\zeta -z)^{2}\right|}}\left|\mathrm {d} \zeta \right|\leq {\frac {1}{2\pi }}\oint _{C_{r}}{\frac {M\left|\zeta \right|}{\left|(\zeta -z)^{2}\right|}}\left|\mathrm {d} \zeta \right|={\frac {MI_{r}^{z}}{2\pi }}}$

${\displaystyle I_{r}^{z}=\oint _{C_{r}}{\frac {\left|\zeta \right|}{\left|(\zeta -z)^{2}\right|}}\left|\mathrm {d} \zeta \right|=\int _{0}^{2\pi }{\frac {\left|z+r\cdot e^{it}\right|}{r}}\mathrm {d} t}$

${\displaystyle r=|z|}$，则${\displaystyle \left|z+r\cdot e^{it}\right|\leqslant |z|+r=2r.}$ 所以${\displaystyle I_{r}^{z}\leqslant \int _{0}^{2\pi }2\mathrm {d} t=4\pi }$，因此

${\displaystyle |f'(z)|\leqslant 2M.}$

## 参考文献

• Knopp, K. Theory of Functions Parts I and II, Two Volumes Bound as One, Part II. New York: Dover, p. 74, 1996.
• Krantz, S. G. "Entire Functions and Liouville's Theorem." §3.1.3 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 31-32, 1999.