Category:Definitions/Laurent Series
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This category contains definitions related to Laurent Series.
Related results can be found in Category: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 $C$.
Pages in category "Definitions/Laurent Series"
The following 7 pages are in this category, out of 7 total.