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Title: Cauchy’s Theorem (general version)
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 25 | Cauchy’s Theorem (general version)
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Bright video: Watch on YouTube
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Dark video: Watch on YouTube
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): 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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Dark-PDF: Download PDF version of the dark video
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ca25_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 a closed curve in $D$. What is the winding number $ \mathrm{wind}(\gamma, 0) $?
A1: $ 0 $
A2: $ 1$
A3: $ -1$
A4: $ 2$
A5: One needs more information!
Q2: Let $\gamma$ be a closed curve. What is the definition of the exterior $\mathrm{Ext}(\gamma)$?
A1: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) = 0 }$
A2: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) \neq 0 }$
A3: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) = 1 }$
A4: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, 0) = z }$
A5: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) \geq 0 }$
Q3: Let $f : D \rightarrow \mathbb{C}$ be holomorphic and $\gamma: [a,b] \rightarrow D$ be a closed curve. Which claim is in general not correct?
A1: $ \oint_{\gamma} f(z) , dz = 0 $, no matter which open domain one chooses for $D$.
A2: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a star domain.
A3: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a convex set.
A4: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a slitted plane.
A5: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a disc.
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Last update: 2024-10
