设区域$D$上一条曲线$z=\gamma(t),a\leq t\leq b$,设起点$\gamma(a)=z_{0}$,现有一个定义在$D$上且在$z_{0}$处全纯且$f'(z_{0})\neq0$的函数$f(z)$,我们考虑曲线$\gamma$在他映射下的像$$w=\sigma(t)=f(\gamma(t)),a\leq t\leq b$$
那么$\sigma'(a)=f'(\gamma(a))\gamma'(a)$,因此$${\rm Arg}\sigma'(a)-{\rm Arg}\gamma'(a)={\rm Arg}f'(z_{0})$$
这说明经过$f(z)$的变换以后,曲线$w$在$w_{0}=f(z_{0})$处切线的倾斜角与$\gamma$在$z_{0}$处的切线的倾斜角之差为${\rm Arg}f'(z_{0})$.
因此如果有两条过$z_{0}$的曲线$z=\gamma_{1}(t),z=\gamma_{2}(t),a\leq t\leq b$且$$\gamma_{1}(a)=\gamma_{2}(a)=z_{0}$$
那么他们在$f(z)$作用后均过点$w_{0}=f(z_{0})$,且$${\rm Arg}\sigma_{1}'(a)-{\rm Arg}\gamma_{1}'(a)={\rm Arg}\sigma_{2}'(a)-{\rm Arg}\gamma_{2}'(a)={\rm Arg}f'(z_{0})$$
即有${\rm Arg}\sigma_{1}'(a)-{\rm Arg}\sigma_{2}'(a)={\rm Arg}\gamma_{1}'(a)-{\rm Arg}\gamma_{2}'(a)$,这说明在$f(z)$的作用下两条曲线$\gamma_{1},\gamma_{2}$在$z_{0}$处的夹角都等于他们的象集在$w_{0}$处的夹角,而且旋转方向不发生改变.
这就说明一个全纯函数在其导数不为零的点处是保角的!
而且根据$|\sigma'(a)|=|f'(z_{0})|\cdot|\gamma'(a)|$,该点的弧微分有个伸缩率$|f'(z_{0})|$.
