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Question

Let $f$ be holomorphic in a domain $D\subset \Bbb{C}$. Then $f$ has a zero of order $m$ in $z_0\in D \iff \frac{1}{f}\in H({D \setminus f^{-1}(0)}) \text{ has a pole of order $m$ in } z_0$.

My attempt: I have proved the "$\implies$" direction.

For the other implication, we suppose that $$\min\left\{v\in \Bbb{N} : \frac{(z-z_0)^v}{f}\text{ is bounded near }z_0\right\}=m$$

We need to find a $g\in H(D)$ with $g(z_0)\neq 0$ such that $f = (z-z_0)^m g$.

I haven't been able to do this. Please tell me what I could do.

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  • $\begingroup$ Hmm, for me this is the definition of a pole, what's yours? $\endgroup$ – GPerez Jun 4 '15 at 13:19
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If $\frac{1}{f}$ has a pole of order $m$ at $z_0$, there exists a holomorphic function $g$ in a neighborhood $U$ of $z_0$ such that $\frac{1}{f(z)} = \frac{g(z)}{(z - z_0)^m}$ with $g(z_0) \neq 0$. Since $g$ is continuous and $g(z_0)\neq 0$, $g(z)\neq 0$ in a neighborhood $V$ of $z_0$. Now $U\cap V$ is a neighborhood of $z_0$ such that $f(z) = (z - z_0)^m\cdot \frac{1}{g(z)}$ for all $z\in U\cap V$, and $\frac{1}{g}$ is holomorphic on $U\cap V$ since $g$ is holomorphic and zero-free in $U\cap V$. Furthermore, $\frac{1}{g(z_0)}\neq 0$. Hence, $f$ has a zero of order $m$ at $z_0$.

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  • $\begingroup$ In the definition of "zero of order m", it is required that $\frac{1}{g}$ is holomorphic on $D$, isn't it? We only have that it's holomorphic on $U\cap V$ $\endgroup$ – iwriteonbananas Jun 5 '15 at 7:26
  • $\begingroup$ @iwriteonbananas no, it's required that $\frac{1}{g}$ is holomorphic in a neighborhood of $z_0$. $\endgroup$ – kobe Jun 5 '15 at 7:29
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Hint: If $\frac{1}{f}$ has a pole $z_0$ of order $m$ then there exists a function $\varphi (z)$ analytic and nonzero at $z_0$ such that $$\frac{1}{f(z)} = \frac{\varphi (z)}{(z- z_0)^m}$$ then $$f(z) = (z-z_0)^m \frac{1}{\varphi(z)}$$

take $g (z)= \frac{1}{\varphi (z)} $.

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  • $\begingroup$ ok, but why is $g$ holomorphic in $D$? And why is $g(z_0) \neq 0$ ? $\endgroup$ – iwriteonbananas Jun 5 '15 at 7:20
  • $\begingroup$ $g (z_0) \neq 0$ because $\varphi (z_0) \neq 0$. It is holomorphic in $D$ because $\varphi (z)$ is analytic (differentiate termwise). $\endgroup$ – Aaron Maroja Jun 5 '15 at 12:43

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