Definition:Laurent Series

Not to be confused with Definition:Formal Laurent Series.

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Definition

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

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

A Laurent series is a summation:

$\ds \sum_{j \mathop = -\infty}^\infty a_j \paren {z - z_0}^j$

such that the summation converges to $f$ in $U \setminus \set {z_0}$.

Also known as

A Laurent series is also commonly known as a Laurent expansion.

Also see

• Results about Laurent series can be found here.

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.