Prove if an entire function f belongs to Aut(C), the set of all the Automorphisms of C if and only if there is a,b belongs to C where a is non zero with f(z)=az+b. I have shown that if f is a proper entire function then must be a polynomial. But it didn't help me at all. So what should I do to solve this problem.
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I would extend $f$ to $\hat{\mathbb{C}}=\mathbb{C}\cup{\infty}$, and show that $\infty$ is a simple pole of $f$. This immediately implies that $f$ is a linear polynomial. – Batominovski Nov 29 '17 at 18:08
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Equivalently, let $c= f^{-1}(0)$. Then $g(z) = \frac1{f(\frac{1}{z}-c)}$ is entire. – reuns Nov 29 '17 at 18:23