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author | Justin Gassner <justin.gassner@mailbox.org> | 2023-09-12 07:36:33 +0200 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-01-13 20:41:27 +0100 |
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tree | dc42d2ae9b4a8e7ee467f59e25c9e122e63f2e04 /pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md | |
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diff --git a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md new file mode 100644 index 0000000..b49cdf4 --- /dev/null +++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md @@ -0,0 +1,60 @@ +--- +title: The Calculus of Residues +parent: One Complex Variable +grand_parent: Complex Analysis +nav_order: 4 +# cspell:words +#published: false +--- + +# {{ page.title }} + +{: .definition-title } +> Definition (Residue) +> +> TODO + +Calculation of Residues + +If $f$ has a simple pole at $c$, then +$\Res(f,c) = \lim_{z \to c} (z-c) f(z)$. + +If $f$ has a pole of order $k$ at $c$, then + +$$ +\Res(f,c) = \frac{1}{(k-1)!} g^{(k-1)}(c), \quad \text{where} \ g(z) = (z-c)^k f(z). +$$ + +If $g$ and $h$ are holomorphic near $c$ and $h$ has a simple zero at $c$, +then $f = g/h$ has a simple pole at $c$ and + +$$ +\Res(f,c) = \frac{g(c)}{h'(c)} +$$ + + + + +{: .theorem-title } +> Residue Theorem (Basic Version) +> {: #residue-theorem-basic-version } +> +> Let $f$ be a function meromorphic in an open subset $G \subset \CC$. +> Let $\gamma$ be a contour in $G$ such that +> the interior of $\gamma$ is contained in $G$ +> and contains finitely many poles $c_1, \ldots, c_n$ of $f$. +> Then +> +> +> $$ +> \int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k) +> $$ +> {: .katex-display .mb-0 } + +{% proof %} +{% endproof %} + +TODO +- argument principle +- Rouché's theorem +- winding number |