Meromorphic functions on the Riemann sphere

A map $f:\widehat{\mathbb{C}}\to \widehat{\mathbb{C}}$ is meromorphic on the Riemann sphere when it is not identically $\mathrm{\infty }$, is holomorphic wherever it takes finite values, and has only pole-type singularities where it takes the value $\mathrm{\infty }$, all in the standard charts of The standard holomorphic charts on the Riemann sphere, with holomorphy and poles at infinity.

Equivalently:

• at each finite point $a\in \mathbb{C}$, either $f(a)\in \mathbb{C}$ and $f$ is holomorphic near $a$ in the ordinary sense, or $f(a)=\mathrm{\infty }$ and $a$ is a pole of the scalar-valued function on a punctured neighbourhood of $a$ in the sense of Meromorphic functions on a plane domain;
• at $\mathrm{\infty }$, when $f(\mathrm{\infty })\in \mathbb{C}$, the source-chart expression is holomorphic at $0$ and takes the value $f(\mathrm{\infty })$ there; when $f(\mathrm{\infty })=\mathrm{\infty }$, the target infinity-chart expression, equal to $1\mathrm{/}f(1\mathrm{/}w)$ for $w\mathrm{\ne }0$ and defined to be $0$ at $w=0$, is holomorphic at $0$.

Thus a meromorphic function on $\widehat{\mathbb{C}}$ is exactly a sphere-valued map, not identically $\mathrm{\infty }$, whose local chart expressions are ordinary meromorphic functions.