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author | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
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committer | Justin Gassner <justin.gassner@mailbox.org> | 2024-02-14 07:24:38 +0100 |
commit | 28407333ffceca9b99fae721c30e8ae146a863da (patch) | |
tree | 67fa2b79d5c48b50d4e394858af79c88c1447e51 /pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md | |
parent | 777f9d3fd8caf56e6bc6999a4b05379307d0733f (diff) | |
download | site-28407333ffceca9b99fae721c30e8ae146a863da.tar.zst |
Update
Diffstat (limited to 'pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md')
-rw-r--r-- | pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md | 36 |
1 files changed, 13 insertions, 23 deletions
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 index b49cdf4..a2fa53d 100644 --- a/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md +++ b/pages/complex-analysis/one-complex-variable/the-calculus-of-residues.md @@ -3,16 +3,13 @@ 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 +{% definition Residue %} +TODO +{% enddefinition %} Calculation of Residues @@ -32,24 +29,17 @@ $$ \Res(f,c) = \frac{g(c)}{h'(c)} $$ +{% theorem * 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 - - -{: .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 } +$$ +\int_{\gamma} f(z) \, dz = 2 \pi i \sum_{k=1}^n \Res(f,c_k) +$$ +{% endtheorem %} {% proof %} {% endproof %} |