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Gameline Complex Analysis, P. 265 #8 is like this,

Show that every conformal self-map of the complex plane $ \mathbb C$ is linear.

Hint: The isolated singularity of $f(z)$ must be the simple pole.

First of all, how do I argue that the singularity is not essential or removable? Second of all, how do I argue it is a pole and is simple?

I can see there is a singularity at $ \infty$ because function is not really defined there. Some hints please!

Addendum Now I can see that the singularity can not be removable because of the Liouville's Theorem. If the singularity at $\infty$ were removable then that would make function bounded and hence analyticity in the entire complex plane tells function is constant but which is impossible because function is bijective.

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    The key point is that $f(z)$ is bijective, so if it had a pole of order $2$ or an essential singularity at infinity can you see how injectivity would fail?2012-12-27
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    Not really, would you please elaborate?2012-12-27
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    If it has a pole of order 2, then the the principle part of the Laurent series has two negative terms, does that imply that $z_{0}$ maps to $\infty $ twice or what? Confused!!2012-12-27
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    The following are links to related questions: [one](http://math.stackexchange.com/questions/29758/entire-1-1-function) [two](http://math.stackexchange.com/questions/39479/entire-bijection-of-mathbbc-with-2-fixed-points?lq=1) [three](http://math.stackexchange.com/questions/53303/why-cant-an-analytic-function-be-injective-in-a-neighborhood-of-an-essential-si?lq=1) [four](http://math.stackexchange.com/questions/168629/relation-between-linearity-and-injectivity-of-an-entire-function?lq=1) [five](http://math.stackexchange.com/questions/163780/transcendental-entire-function-aut-mathbbc?lq=1)2012-12-27
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    Thanks Jonah Meyer. That really helps.2012-12-27

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