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Title: Holomorphic Functions are C-infinity Functions
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 28 | Holomorphic Functions are C-infinity Functions
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Bright video: https://youtu.be/X_g5d5_e7iQ
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Dark video: https://youtu.be/G1MhLleHz28
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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: ca28_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 $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function. Does the second-order derivative $f^{\prime \prime}$ exist at each point?
A1: Yes, always.
A2: No, never.
A3: One need more information.
Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a complex-differentiable function. Does the second-order derivative $f^{\prime \prime}$ exist at each point?
A1: Yes, always.
A2: No, never.
A3: One need more information.
Q3: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a complex-differentiable function. Which formula for $f^{\prime \prime}$ is correct?
A1: $$ f^{\prime \prime}(z) = \frac{1}{ \pi i} \oint_{\partial B_1(0) } \frac{ f(\zeta) }{ (\zeta - z)^{3} } , d \zeta $$
A2: $$ f^{\prime \prime}(z) = \frac{2!}{ \pi i} \oint_{\partial B_1(0) } \frac{ f(\zeta) }{ (\zeta - z)^{2} } , d \zeta $$
A3: $$ f^{\prime \prime}(z) = \frac{2!}{ 4 \pi i} \oint_{\partial B_1(0) } \frac{ f(\zeta) }{ (\zeta - z)^{2} } , d \zeta $$
A4: $$ f^{\prime \prime}(z) = \frac{2!}{ 4 \pi i} \oint_{\partial B_1(0) } \frac{ f(\zeta) }{ (\zeta - z)^{3} } , d \zeta $$
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Last update: 2024-10
