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'Laurent series'
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| Title of object: |
Laurent series |
| Canonical Name: |
LaurentSeries |
| Type: |
Definition |
| Created on: |
2001-12-28 06:26:00-05 |
| Modified on: |
2001-12-28 06:28:19-05 |
Revision comment (for changes between this and next version):
| Changes for correction #961 ('uniqueness'). |
Preamble:
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{graphicx}
\usepackage{xypic}
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Content:
A {\em Laurent series} centered about $a$ is a series of the form
\sum_{k=-\infty}^\infty c_k (z-a)^k
where $c_k, a, z \in \mathbb{C}$.
One can prove that the above series converges everywhere inside the set
D := \{z \in \mathbb{C} \mid R_1 < |z-a| < R_2 \}
where
R_1 := \limsup_{k \rightarrow\infty} |c_{-k}|^{1/k}
R_2 := 1/\left(\limsup_{k \rightarrow\infty} |c_{k}|^{1/k}\right).
(This set may be empty)
Every Laurent series has an associated function, given by
f(z) := \sum_{k=-\infty}^\infty c_k (z-a)^k,
whose domain is the set of points in $\mathbb{C}$ on which the series converges. This function is analytic inside the annulus $D$, and conversely, every analytic function on an annulus is equal to some Laurent series. |
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