Category:Laurent Series

This category contains results about Laurent Series.
Definitions specific to this category can be found in Definitions/Laurent Series.

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

Let $z_0 \in \C$ such that:

$f$ is analytic in $U := \set {z \in \C: r_1 \le \cmod {z - z_0} \le r_2}$

where $r_1, r_2 \in \overline \R$ are points in the extended real numbers.

A Laurent series is a summation:

$\forall z \in \C: r_1 < \cmod {z - z_0} < r_2: \map f z = \ds \sum_{n \mathop \in \Z} a_n \paren {z - z_0}^n$

where:

$a_n = \dfrac 1 {2 \pi i} \ds \int_C \map f z \paren {z - z_0}^{n + 1} \rd z$

$C$ is a circle with center $z_0$ and radius $r$ for $r_1 < r < r_2$

$\ds \int_C \map f z \paren {z - z_0}^{n + 1} \rd z$ is the contour integral over $C$.

such that the summation converges to $f$ in $U$.