math121a-f23:september_18_monday
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math121a-f23:september_18_monday [2023/09/18 18:23] pzhou |
math121a-f23:september_18_monday [2026/02/21 14:41] (current) |
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| ==== Residue Theorem ==== | ==== Residue Theorem ==== | ||
| - | **Thm:** let $f$ has a pole (of some order) at $z_0$, and let $C$ be a small circle of radius $\epsilon$ around $z_0$ positively oriented (i.e in clockwise direction) so that $f$ is holomorphic on $C$, and $C$ encloses only one pole. Then $$\oint_{C} f(z) dz = (2\pi i) Res_{z_0} f. $$ | + | **Thm:** let $f$ has a pole (of some order) at $z_0$, and let $C$ be a small circle of radius $\epsilon$ around $z_0$ positively oriented (i.e in counter-clockwise direction) so that $f$ is holomorphic on $C$, and $C$ encloses only one pole. Then $$\oint_{C} f(z) dz = (2\pi i) Res_{z_0} f. $$ |
| Proof sketch: write $f$ as a Laurent expansion, then integrate term by term. | Proof sketch: write $f$ as a Laurent expansion, then integrate term by term. | ||
math121a-f23/september_18_monday.1695061429.txt.gz · Last modified: 2026/02/21 14:44 (external edit)
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