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Title: Antiderivatives
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 20 | Antiderivatives
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Bright video: https://youtu.be/00ab7OTsl6s
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Dark video: https://youtu.be/-oOhmEUXNDg
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Ad-free video: Watch Vimeo video
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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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Print-PDF: Download printable PDF version
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Thumbnail (bright): Download PNG
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Thumbnail (dark): Download PNG
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Subtitle on GitHub: ca20_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 $f: \mathbb{C} \rightarrow \mathbb{C}$ given by $f(z) = z$. Which of the following functions is not an antiderivative of $f$?
A1: $F(z) = z^2$
A2: $F(z) = \frac{1}{2} z^2$
A3: $F(z) = \frac{1}{2} z^2 + 2$
A4: $F(z) = \frac{1}{2} z^2 - 1$
Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ and $g: \mathbb{C} \rightarrow \mathbb{C}$ two holomorphic functions. Which statement is not correct?
A1: $ \frac{d}{dz}(f \circ g)(z) = f( g(z) ) g^\prime(z) $
A2: $ \frac{d}{dz}(f \circ g)(z) = f^\prime( g(z) ) g^\prime(z) $
A3: $ \frac{d}{dz} f(g(z)) = f^\prime( g(z) ) g^\prime(z) $
A4: $ \frac{d}{dz} f(g(z)) =\frac{df}{dz} ( g(z) ) \frac{dg}{dz}(z) $
Q3: Let $f: U \rightarrow \mathbb{C}$ be a holomorphic function and $\gamma$ a closed curve in $U$. Which statement is correct?
A1: $$ \oint_{\gamma} f(z) , dz = 0 $$
A2: $$ \oint_{\gamma} f(z) , dz \neq 0 $$
A3: One needs more information!
Q4: Let $f: \mathbb{C} \setminus { 0 } \rightarrow \mathbb{C}$ be the holomorphic function given by $f(z) = \frac{1}{z^4}$. Which statement is correct for a closed curve $\gamma$?
A1: $$ \oint_{\gamma} f(z) , dz = 0 $$
A2: $$ \oint_{\gamma} f(z) , dz \neq 0 $$
A3: One needs more information!
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Last update: 2024-10
