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Let $f\colon \mathbb{C}\to \mathbb{C}$ be a nonconstant entire function.

Is it true that there exists $\bar{f}\colon S^2\to S^2$ with $\bar{f}|_\mathbb{C}=f$?

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    possible duplicate of [Nomenclature in complex analysis](http://math.stackexchange.com/questions/64266/nomenclature-in-complex-analysis)2011-11-15
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    Isn't it true that $f\colon \mathbb{C}\to \mathbb{C}$ defined by $z\mapsto e^z$ can be extended to the holomorphic function of $S^2$?2011-11-15
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    No, since $\exp{1/z}$ has an essential singularity at $0$, there is no way to extend this function to a continuous function in $0$, let alone a holomorphic one (the image of every punctured neighborhood of $0$ is dense in $\mathbb{C}$ by Casorati-Weierstrass).2011-11-15
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    I got ya. Thank you man.2011-11-15
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    What do you mean by "there exists $\bar f:\ S^2\to S^2\ $"? You could define $\bar f(\infty):=0$.2011-11-15
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    Any holomorphic function on the Riemann sphere is going to be constant, as a result of the MMP and the sphere being compact.2011-12-27

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