设区域$D$上一条曲线$z=gamma(t),aleq tleq b$,设起点$gamma(a)=z_{0}$,现有一个定义在$D$上且在$z_{0}$处全纯且$f'(z_{0}) eq0$的函数$f(z)$,我们考虑曲线$gamma$在他映射下的像$$w=sigma(t)=f(gamma(t)),aleq tleq b$$
那么$sigma'(a)=f'(gamma(a))gamma'(a)$,因此$${ m Arg}sigma'(a)-{ m Arg}gamma'(a)={ m Arg}f'(z_{0})$$
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${ m Arg}f'(z_{0})$.
因此如果有两条过$z_{0}$的曲线$z=gamma_{1}(t),z=gamma_{2}(t),aleq tleq b$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$${ m Arg}sigma_{1}'(a)-{ m Arg}gamma_{1}'(a)={ m Arg}sigma_{2}'(a)-{ m Arg}gamma_{2}'(a)={ m Arg}f'(z_{0})$$
即有${ m Arg}sigma_{1}'(a)-{ m Arg}sigma_{2}'(a)={ m Arg}gamma_{1}'(a)-{ m Arg}gamma_{2}'(a)$,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$|f'(z_{0})|$.