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Let $z_0\in\mathbb{C}$ and $R>0$. Suppose $f(z):D_r(z_0) − \{z_0\}$ is holomorphic and that there exists $C>0$ and some $\epsilon> 0$ such that $|f(z)| \leq C|z − z_0| ^{−1+\epsilon}$ for all $z$ near $z_0$. Show that the singularity of $f$ at $z_0$ is removable. That is, there exists a unique holomorphic function $F:D_r(z_0)\to\mathbb{C}$ such that $F(z)=f(z)$ for $z\in D_r(z_0) − \{z_0\}$.

I am not seeking for solutions. I want to understand the existence of the unique holomorphic function. Can someone help me out?

thanks in advance.

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  • $\begingroup$ Hint: Try to think in terms of the Laurent series expression of $f$ around $z_0$ then $\endgroup$ Jun 8 '20 at 15:27
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HINT:

THEOREM 1.2[Conway p.103] If $f$ has an isolated singularity at the point $z = a$, then $a$ is a removable singularity if and only if $\lim_{z \to a}(z-a)f(z) = 0$

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