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+---
+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