解析函数的辐角相等

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enhanced phase portraits

In fact, analytic functions are (almost) uniquely determined by their (pure) phase portraits, but this is not so for general functions. For example, the functions $f$ (analytic) and $g$ (not analytic) defined by \begin{eqnarray}\label{example} f(z)=\frac{z-1}{z^2+z+1}, \qquad g(z)=(z-1)\left(\overline{z}^2+\overline{z}+1\right) \end{eqnarray} have the same phase (except at their zeros and poles) though they are completely different.

$f,g$为解析函数,如何证明$\arg f=\arg g$推出$f=g$呢