• Title: Residue

  • Series: Complex Analysis

  • YouTube-Title: Complex Analysis 32 | Residue

  • Bright video: https://youtu.be/BuEoOYbiH04

  • Dark video: https://youtu.be/yLRx9F_vmLw

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  • Quiz: Test your knowledge

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  • Subtitle on GitHub: ca32_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function that has an isolated singularity at $z_0$. What is the correct definition of $\mathrm{Res}(f, z_0)$?

    A1: $ \frac{1}{2 \pi i} \oint_{\partial B_{\varepsilon}(z_0)} f(z) , dz $

    A2: $ \frac{1}{2 \pi} \oint_{\partial B_{\varepsilon}(z_0)} f(z) , dz $

    A3: $ 2 \pi i \oint_{\partial B_{\varepsilon}(z_0)} f(z) , dz $

    A4: $ \oint_{\partial B_{\varepsilon}(z_0)} f(z) , dz $

    Q2: Let $f: \mathbb{C}\setminus { z_0 } \rightarrow \mathbb{C}$ be given $f(z) = \frac{1}{z - z_0}$. What is the value of the residue $\mathrm{Res}(f, z_0)$?

    A1: $1$

    A2: $ \frac{1}{2 \pi} $

    A3: $ 2 \pi i $

    A4: $ 0 $

    Q3: Let $f: \mathbb{C}\setminus { 0, 1 } \rightarrow \mathbb{C}$ be given $f(z) = -\frac{1}{z(z-1)}$. What is the value of the residue $\mathrm{Res}(f, z_0)$?

    A1: $1$

    A2: $ \frac{1}{2 \pi} $

    A3: $ 2 \pi i $

    A4: $ 0 $

    A5: $ - 1 $

  • Last update: 2024-10

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