Meromorphic function as a holomorphic map to the Riemann sphere

A meromorphic function on a Riemann surface is equivalently a holomorphic map to the Riemann sphere, with every pole mapped to infinity. Composition with a rational self-map of the sphere is again such a holomorphic map.

A nonconstant meromorphic function on a compact Riemann surface with exactly one simple pole gives a degree-one holomorphic map to the sphere. A degree-one map is a biholomorphism. The surface therefore has genus zero and the function is a global rational coordinate.