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fix(meromorphic): clarify Laurent is a requirement
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Closes #324.
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vEnhance committed Nov 16, 2023
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11 changes: 4 additions & 7 deletions tex/complex-ana/meromorphic.tex
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Expand Up @@ -66,23 +66,20 @@ \section{Meromorphic functions}

\begin{definition}
Let $f \colon U \to \CC$ as usual.
A \vocab{meromorphic} function is a
function which is holomorphic on $U$
A \vocab{meromorphic} function is a function which is holomorphic on $U$
except at an isolated set $S$ of points
(meaning it is holomorphic as a function $U \setminus S \to \CC$).
For each $p \in S$, called a \vocab{pole} of $f$,
the function $f$ must admit a \vocab{Laurent series},
meaning that
the function $f$ is further required to admit a \vocab{Laurent series}, meaning that
\[
f(z) =
\frac{c_{-m}}{(z-p)^m}
+ \frac{c_{-m+1}}{(z-p)^{m-1}}
+ \dots
+ \frac{c_{-1}}{z-p} + c_0 + c_1 (z-p) + \dots
\]
for all $z$ in some open neighborhood of $p$,
other than $z = p$.
Here $m$ is a positive integer (and $c_{-m} \neq 0$).
for all $z$ in some open neighborhood of $p$, other than $z = p$.
Here $m$ is a positive integer, and $c_{-m} \neq 0$.
\end{definition}
Note that the trailing end \emph{must} terminate.
By ``isolated set'', I mean that we can draw
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