• Title: Cauchy’s Integral Formula

  • Series: Complex Analysis

  • YouTube-Title: Complex Analysis 27 | Cauchy’s Integral Formula

  • Bright video: https://youtu.be/hll0DAilhoA

  • Dark video: https://youtu.be/54zwCAD5fds

  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ca27_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function where $\overline{B_r(z_0)} \subseteq D$. What is the correct Cauchy’s integral formula?

    A1: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f(\zeta) }{ \zeta - z } d\zeta $$ for all $z \in B_r(z_0)$.

    A2: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f(\zeta) }{ z - \zeta } d\zeta $$ for all $z \in B_r(z_0)$.

    A3: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f^\prime(\zeta) }{ \zeta - z } d\zeta $$ for all $z \in B_r(z_0)$.

    A4: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_r(z_0) } \frac{ f^\prime(\zeta) }{ \zeta - z } d \xi $$ for all $\zeta \in B_r(z_0)$.

    Q2: Let $f: \mathbb{C} \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = z^2$. What is correct by applying Cauchy’s integral formula?

    A1: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta^2 }{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.

    A2: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta}{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.

    A3: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ 1 }{ \zeta - z } d\zeta $$ for all $z \in B_1(0)$.

    A4: $$f(z) = \frac{1}{2 \pi i} \oint_{\partial B_1(0) } \frac{ \zeta }{ \zeta^2 - z } d\zeta $$ for all $z \in B_1(0)$.

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