• Title: Holomorphic Functions are C-infinity Functions

  • Series: Complex Analysis

  • YouTube-Title: Complex Analysis 28 | Holomorphic Functions are C-infinity Functions

  • Bright video: https://youtu.be/X_g5d5_e7iQ

  • Dark video: https://youtu.be/G1MhLleHz28

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  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ca28_sub_eng.srt missing

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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: $$ f^{\prime \prime}(z) = \frac{2!}{ 4 \pi i} \oint_{\partial B_1(0) } \frac{ f(\zeta) }{ (\zeta - z)^{3} } , d \zeta $$

  • Last update: 2024-10

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