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I have some trouble finding the definition of meromorphic function on Riemann sphere. On the complex plane, it is just the ratio of two entire functions. So do we define the meromorphic function on the Riemann sphere with the same spirit, namely something like a ratio of two entire function on Riemann sphere? Then how do we define a entire function on Riemann sphere?

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You define meromorphic functions on $\mathbb C$ as functions whose singularities are at worst poles. Then you prove that they are quotients of entire functions.

On the Riemann sphere $\hat{\mathbb C}$, entire functions are just constant, so this approach won't work. One way to define meromorphic functions on $\hat{\mathbb C}$ (or any Riemann surface) as holomorphic maps to $\hat{\mathbb C}$ (excluding the constant function $\infty$).

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  • $\begingroup$ Is it equivalent to that defining it as quotient of functions that are either constant or holomorphic function on the complex plane with pole at infinity? $\endgroup$
    – jack
    Nov 14 '15 at 20:04
  • $\begingroup$ I'm not sure what you mean, but saying "pole at infinity" just means mapping $\infty$ to $\infty$ so you basically get the holomorphic functions you had before. Think of $\frac{1}{z}$, for an example. $\endgroup$ Nov 14 '15 at 20:09
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    $\begingroup$ Since I see that we have the theorem saying meromorphic function on the Riemann sphere are rational functions. And we have that holomorphic functions in complex plane with pole at infinity are polynomial. So I think that defining the meromorphic function as the quotient of such functions would make more sense. $\endgroup$
    – jack
    Nov 14 '15 at 20:12
  • $\begingroup$ Ok. Then yes, quotients of entire functions on $\mathbb C$ with a pole at infinity is ok, but these are not entire functions on $\hat{\mathbb C}$, so you just need to be careful with terminology. $\endgroup$ Nov 14 '15 at 20:18

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