What is the definition for holomorphic functions on the Riemann sphere?

Question

I'm trying to study complex analysis in general setting, but i have troubles with defining things in this general setting.

I have skimmed definitions in Ahlfors, Conway and wikipedia, but these all merely give definitions in $\mathbb{C}$.

What does it exactly mean by "a function $f$ is holomorphic on an open subset $U$ of $\overline{\mathbb{C}}$"?

What does it mean by "Poles at infinity"? (Wikipedia describes this very shortly and it's stated there that the definition of "holomorphic on an open subset of $\overline{\mathbb{C}}$" is required, but the definition is not given.)

I'm curious why all texts i saw introduce definitions in $\mathbb{C}$ even though there are useful theorems in the general setting. (e.g. Great piccard)

Answer 1

Short version: If you add $\infty$ to the complex plane and endow it with the 'correct' topology, you get (topologically) a sphere. You can get a complex structure along the following lines: the sphere can be covered by two sets (each one the sphere without one point, think of two antipodal points for the two sets) which you can identify with $\mathbb{C}$ and the "image" of $\mathbb{C}$ under the map $z\mapsto 1/z$.

A function $f$ on the sphere is then holomorphic if it is in each of these open sets. If you have a function on $\mathbb{C}$ and want to check whether it extends to the sphere check the behaviour of $f(1/z)$ near the origin.

(Think of the sphere as the union of the sphere minus the northpole and the sphere minus the South pole. Identify the first set with $\mathbb{C}$, the second with $(\mathbb{C} - \{0\})\cup\{\infty\}$ and identify both sets on $\mathbb{C} - \{0\}$ using $z\mapsto 1/z$)

This is only an outline of the idea, making this rigorous takes some work.

Answer 2

The Riemann sphere $\overline{\mathbb C}$ is the topological space $\mathbb C \cup \{\infty\}$ (the one point compactification of $\mathbb C$. The group of $2\times 2$ invertible matrices over $\mathbb C$ act as continuous permutations of this set via the usual linear fractional transformations: $$z \mapsto (a z + b)/(c z + d) .$$
(Think this through; you have to work out how to apply some elementary algebra to the symbol $\infty$, but this is always possible in an unambiguous manner. As one example, note that $z \mapsto 1/z$ will interchange $0$ and $\infty$.)

We are going to use this structure to make $\overline{\mathbb C}$ into a Riemann surface.

What does this mean? A Riemann surface is a topological space on which (a) for any open set $U$, we know what it means for a function $f:U \to \mathbb C$ to be holomorphic; (b) each point has an open neighborhood which is holomorphically isomorphic to an open disk in $\mathbb C$. (We know what this means because of (a).)

How do we do it?

Well, if an open set $U$ of $\overline{\mathbb C}$ is actually contained in $\mathbb C$, we know what it means for a function on $U$ to be holomorphic.

Now, suppose given an open set $U$ contained instead in $\overline{\mathbb C} \setminus \{0\}$. Then if we apply $1/z$ to $U$, we obtain an open subset of $\mathbb C$; and we will define a function $f$ on $U$ to be holomorphic precisely if its composite with $1/z$ is holomorphic in the usual sense (which now makes sense, because the domain of $f(1/z)$ is an open subset of $\mathbb C$).

Suppose that $U$ is contained in the overlap $\mathbb C \cap (\overline{\mathbb C} \setminus \{0\}) = \mathbb C \setminus \{0\}$; then we have two competing notions of holomorphic. If $f$ is a function on $U$, then thinking of $U$ as an open subset of $\mathbb C$, we should say $f$ is holomorphic iff $f$ is holomorphic in the usual sense. But thinking of $U$ as a subset of $\overline{\mathbb C} \setminus \{0\}$, we should say $f$ is holomorphic if $f(1/z)$ is holomorphic in the usual sense.

But $z \mapsto 1/z$ is a holomorphic automorphism of $\mathbb C\setminus \{0\}$, and so $f(z)$ is holomorphic in the usual sense iff $f(1/z)$ is! Thus the two competing notions coincide, and we have an unambiguous definition of holomorphic.

Now if $U$ is any open subset of $\overline{\mathbb C}$ and $f: U \mapsto C$, we can define $f$ to be holomorphic if the restriction of $f$ to each of $U\cap \mathbb C$ and $U \cap (\overline{\mathbb C} \setminus \{0\})$ are holomorphic.

So now we have achieved point (a). I leave it to you to verify (b).

As a good exercise, check that (with these definition) the linear fractional transformations are holomorphic automorphisms of $\overline{\mathbb C}$. (I.e. if $F$ is a linear fractional transformation, and $f: U \to \mathbb C$ is a function on some open subset $U$ of $\overline{\mathbb C}$, then $f$ is holomorphic as a function on $U$ iff $f\circ F$ is holomorphic as a function on $F^{-1}(U)$.)

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The details of this will be in any book on Riemann surfaces. Note that more basic complex analysis books likely won't cover it in detail; they may instead just have a quasi-formal discussion about using $f(1/z)$ near $0$ as a tool for investigating $f$ near $\infty$.

The general idea of covering a topological space by charts, and looking at the transition maps between overlapping charts, comes up in the theory of Riemann surfaces (and more generally of complex manifolds), as well as in the theory of topological and smooth manifolds. Most people first learn it in the context of topological and smooth manifolds, so you may want to look at an introductory discussion in that context, just because there are probably more such discussions available in that context, and so there is a better chance that you might find one that suits you.