• Title: Residue theorem

  • Series: Complex Analysis

  • YouTube-Title: Complex Analysis 34 | Residue theorem

  • Bright video: Watch on YouTube

  • Dark video: Watch on YouTube

  • Ad-free video: Watch Vimeo video

  • Forum: Ask a question in Mattermost

  • Quiz: Test your knowledge

  • PDF: Download PDF version of the bright video

  • Dark-PDF: Download PDF version of the dark video

  • Print-PDF: Download printable PDF version

  • Thumbnail (bright): Download PNG

  • Thumbnail (dark): Download PNG

  • Subtitle on GitHub: ca34_sub_eng.srt missing

  • Download bright video: Link on Vimeo

  • Download dark video: Link on Vimeo

  • Timestamps (n/a)
  • Subtitle in English (n/a)
  • Quiz Content

    Q1: Let $f: \mathbb{C} \setminus {z_1, \ldots, z_k} \rightarrow \mathbb{C}$ be a holomorphic function and $\gamma$ a closed curve in $\mathbb{C} \setminus {z_1, \ldots, z_k}$. What is correct?

    A1: $$ \oint_{\gamma} f(z) , dz = 2 \pi i \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$

    A2: $$ \oint_{\gamma} f(z) , dz = \frac{1}{2 \pi i} \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$

    A3: $$ 2 \pi i \oint_{\gamma} f(z) , dz = 2 \sum_{j=1}^k \mathrm{wind}(\gamma, z_j) \mathrm{Res}(f,z_j) $$

    A4: $$ \oint_{\gamma} f(z) , dz = 2 \pi i \sum_{j=1}^k \mathrm{Res}(f,z_j) $$

    Q2: Let $f: \mathbb{C} \setminus {0 } \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$ and $\gamma$ be a closed curve in $\mathbb{C} \setminus {0 }$. What is $ \oint_{\gamma} f(z) , dz $?

    A1: One needs more information.

    A2: $ 1 $

    A3: $ -2 $

    A4: $ 0 $

    A5: $ -1 $

    A6: $ 2 \pi i$

    Q3: Let $f: \mathbb{C} \setminus {0 } \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{2}{z} + \frac{e}{z^2}$ and $\gamma: [0,4 \pi]$ be given by $\gamma(t) = e^{i t}$. What is $ \oint_{\gamma} f(z) , dz $?

    A1: $ 8 \pi i $

    A2: $ -2 \pi i $

    A3: $-1$

    A4: $1$

    A5: $0$

    A6: $e$

  • Last update: 2024-10

  • Back to overview page


Do you search for another mathematical topic?