李乌维尔定理

设 $f$ 为整函数且在全复平面有界：$|f(z)|\le M$。

由柯西积分公式可得柯西估计：对任意 $R>0$ 与任意 $z_0$，

$$|f'(z_0)|\le \frac{M}{R}.$$

令 $R\to\infty$ 得 $f'(z_0)=0$。由于 $z_0$ 任意，故 $f'\equiv 0$，因此 $f$ 为常数。

这就是李乌维尔定理。

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英文解析

Let $f$ be an entire function that is bounded on the entire complex plane: $|f(z)|\le M$.

From Cauchy's integral formula, we obtain the Cauchy estimate: for any $R>0$ and any $z_0$,

$$|f'(z_0)|\le \frac{M}{R}.$$

Letting $R\to\infty$ yields $f'(z_0)=0$. Since $z_0$is arbitrary, it follows that $f'\equiv 0$; therefore, $f$is constant.

This is Liouville's Theorem.