Classify all invertible meromorphic functions

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I am studying complex analysis. And I see a topic, namely meremorphic functions

But I cannot find any enough information about this function. So I have insufficient knowledge about this topic.

And I have found a question like following below with its answer.

Please can someone explain the answer of this question in more detail explanatorily? I Will be happy. Thank you so much for helping:)


Question: classify all invertible meromorphic functions from $\Bbb C\cup \{\infty\}$ to $\Bbb C \cup \{\infty\}$


The question with its answer:

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