Definition:Laurent Series

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Not to be confused with Definition:Formal Laurent Series.

Definition

Let $f: \C \to \C$ be a complex function.

Let $z_0 \in U \subset \C$ such that $f$ is analytic in $U \setminus \left\{{z_0}\right\}$.

A Laurent series is a sum:

$\displaystyle \sum_{j \mathop = -\infty}^\infty a_j \left({z - z_0}\right)^j$

such that the sum converges to $f$ in $U \setminus \left\{{z_0}\right\}$.


Source of Name

This entry was named for Pierre Alphonse Laurent.


Historical Note

The Laurent series expansion of an analytic function, was established by Carl Friedrich Gauss in $1843$, but he never got round to publishing this work.

Karl Weierstrass independently discovered it during his work to rebuild the theory of complex analysis.