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Title: Keyhole contour
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 26 | Keyhole contour
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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: ca26_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 $D = \mathbb{C} \setminus { z \in \mathbb{R} \mid z \leq 0 }$ and $\gamma$ be given by a keyhole contour in $D$. Is Cauchy’s integral theorem applicable?
A1: Yes, we have $ \oint_{\gamma} f(z) , dz = 0 $.
A2: No, we have $ \oint_{\gamma} f(z) , dz \neq 0 $.
A3: One needs more information!
Q2: Let $g: B_2(0) \setminus {0 } \rightarrow \mathbb{C}$ be holomorphic. For every $\varepsilon < 2$, define the curve $\gamma_{\varepsilon}: [0,2\pi] \rightarrow \mathbb{C}$ given by $\gamma_{\varepsilon}(t) = \varepsilon \exp(i t)$. Which claim is, in general, correct?
A1: $$ \oint_{\gamma_1} f(z) , dz = \oint_{\gamma_{\varepsilon}} f(z) , dz $$ for all $0 < \varepsilon < 2$.
A2: $$ \oint_{\gamma_1} f(z) , dz \neq \oint_{\gamma_{\varepsilon}} f(z) , dz $$ for all $0 < \varepsilon < 2$.
A3: $$ \oint_{\gamma_{\varepsilon}} f(z) , dz = 0 $$ for all $0 < \varepsilon < 2$.
A4: $$ \oint_{\gamma_{\varepsilon}} f(z) , dz = 2 i \pi $$ for all $0 < \varepsilon < 2$.
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Last update: 2024-10
