28.3 The Riemann Sphere

Definition 28.3.1 (Extended Complex Plane).label Let $\complex$ be the complex plane, then its one-point compactification $\complex_{\infty} = \complex \sqcup \bracs{\infty}$ is the extended complex plane.

Definition 28.3.2 (Holomorphic on $\complex_{\infty}$).label Let $E$ be a separated locally convex space and $f \in C(\complex_{\infty}; E)$, then $f$ is holomorphic at $\infty$ if $z \mapsto f(z^{-1})$ (under the identification that $1/0 = \infty$) is holomorphic at $0$.