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I want to use the theorem any holomorphic function on Rieman sphere is a rational function, and hence a polynomial. So, I want to show that $f$ is holomorphic on Rieman sphere. So first thing I am trying to do is show that $\infty$ is a removable singularity, since if it is removeable $f$ can extend to a holomorphic function on Rieman sphere then I can apply the theorem I mentioned and I am done. But I am having trouble showing that.

Any hint and nudge toward right direction would be great!

Thanks!

Remu X
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    Get rid of the zeros (which are only finite in number) and apply Liouville to $\frac 1 f$. [The question is a duplicate]. – Kavi Rama Murthy Dec 09 '22 at 05:20
  • @geetha290krm thatnks for the comment, when you say the set of zeros is finite, is it because of the following: Suppose $E$ is the set of zeros of $f$ in $\mathbb{C}$ and assume to contrary that $E$ is not finite, then there exists accumulation point $x$ of sequence in $E$ and hence $x\in \mathbb{C}$. Since $\mathbb{C}$ is open and connected we must have $f$ is identically zero, which contradicts $f$ goes to infinity.? – Remu X Dec 09 '22 at 05:26
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    Yes, that is correct. – Kavi Rama Murthy Dec 09 '22 at 05:28
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    Check these: https://math.stackexchange.com/q/1079463/42969, https://math.stackexchange.com/q/966392/42969, https://math.stackexchange.com/q/2924936/42969, https://math.stackexchange.com/q/3772707/42969. – Martin R Dec 09 '22 at 05:37

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