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Title: Residue theorem
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 34 | Residue theorem
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Bright video: Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: ca34_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $f: \mathbb{C} \setminus {z_1, \ldots, z_k} \rightarrow \mathbb{C}$ be a holomorphic function and $\gamma$ a closed curve in $\mathbb{C} \setminus {z_1, \ldots, z_k}$. What is correct?
A1: $$ \oint_{\gamma} f(z) , dz = 2 \pi i \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$
A2: $$ \oint_{\gamma} f(z) , dz = \frac{1}{2 \pi i} \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$
A3: $$ 2 \pi i \oint_{\gamma} f(z) , dz = 2 \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$
A4: $$ \oint_{\gamma} f(z) , dz = 2 \pi i \sum_{j=1}^k \mathrm{Res}(f,z_j) $$
Q2: Let $f: \mathbb{C} \setminus {0 } \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$ and $\gamma$ be a closed curve in $\mathbb{C} \setminus {0 }$. What is $ \oint_{\gamma} f(z) , dz $?
A1: One needs more information.
A2: $ 1 $
A3: $ -2 $
A4: $ 0 $
A5: $ -1 $
A6: $ 2 \pi i$
Q3: Let $f: \mathbb{C} \setminus {0 } \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{2}{z} + \frac{e}{z^2}$ and $\gamma: [0,4 \pi]$ be given by $\gamma(t) = e^{i t}$. What is $ \oint_{\gamma} f(z) , dz $?
A1: $ 8 \pi i $
A2: $ -2 \pi i $
A3: $-1$
A4: $1$
A5: $0$
A6: $e$
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Last update: 2024-10
