Does entire function extend to the holomorphic function of Riemann sphere? 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 $\bar{f}|_\mathbb{C}=f$?
 A: Proposition. Let $f:\mathbb{C}\to\mathbb{C}$ be an entire function. The following are equivalent:


*

*$f$ extends continuously (and hence holomorphically) to a function $f:S^2\to S^2$;

*$f$ is either constant or $z\mapsto f(1/z)$ has a pole at $0$;

*$f$ is either constant or a proper map.

*$f$ is a polynomial;


Proof. Let us assume that $f$ is non-constant (otherwise all statements are true); then by Liouville's Theorem, $f$ is unbounded.
The equivalence of the first three statements is easy. Indeed, as $f$ is unbounded, $f$ extending continuously to $\infty$ is equivalent to $\lim_{z\to\infty} f(z)=\infty$, which in turn is equivalent to $f$ being a proper map. 
If $f$ is a non-constant polynomial, then clearly again $\lim_{z\to\infty} f(z)=\infty$, so 4. implies 1. To prove that 1. implies 4., assume 1. and observe that the set of zeros of $f$ is a discrete and closed subset of $S^2$, so this set is finite. Let $g$ be a polynomial with the same zeros (and same multiplicities) as $f$ and consider the function $h:=f/g$. It is not difficult to show that the function $h$ is constant, and hence $f$ is a polynomial.
(Alternatively, if you know about Laurent series, just observe the relationship between the Laurent series of $f(1/z)$ around zero and the Taylor series that represents $f$ in the complex plane. So the singularity of the former function is non-essential if and only if the Taylor series is finite; i.e., if and only if $f$ is a polynomial.) $\blacksquare$  
A: No, the function $\exp$, for example has an essential singularity at $\infty$ and has no holomorphic extension. The same is true for the simple function $f(z) = z$ which has a pole of first order at $\infty$ and so on. Basically, the behavior of $f(z)$ at $\infty$ is the same than the behavior of $f(1/z)$ at zero.
