\section{The Riemann Sphere}
\label{section:riemann-sphere}


\begin{definition}[Extended Complex Plane]
\label{definition:extended-complex-plane}
    Let $\complex$ be the complex plane, then its one-point compactification $\complex_\infty = \complex \sqcup \bracs{\infty}$ is the \textbf{extended complex plane}. 
\end{definition}

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