17 lines
672 B
TeX
17 lines
672 B
TeX
\section{The Riemann Sphere}
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\label{section:riemann-sphere}
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\begin{definition}[Extended Complex Plane]
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\label{definition:extended-complex-plane}
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Let $\complex$ be the complex plane, then its one-point compactification $\complex_\infty = \complex \sqcup \bracs{\infty}$ is the \textbf{extended complex plane}.
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\end{definition}
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\begin{definition}[Holomorphic on $\complex_\infty$]
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\label{definition:holomorphic-on-sphere}
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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 $0$.
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\end{definition}
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