• Title: Residue for Poles

  • Series: Complex Analysis

  • YouTube-Title: Complex Analysis 33 | Residue for Poles

  • Bright video: https://youtu.be/1kJJwFSgUng

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

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  • Subtitle on GitHub: ca33_sub_eng.srt missing

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  • Quiz Content

    Q1: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$. Which claim is correct?

    A1: $ f(z) = z^{-2} (1+ z) $

    A2: $ f(z) = z^{-1} (1+ z) $

    A3: $ f(z) = z^{-2} (1+ z^2) $

    A4: $ f(z) = z^{2} (1+ z^{-1}) $

    Q2: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by $f(z) = \frac{1}{z} + \frac{1}{z^2}$. What is the order of the pole $z_0 = 0$?

    A1: $ 2 $

    A2: $ 1 $

    A3: $ -2 $

    A4: $ 0 $

    A5: $ -1 $

    A6: One needs more information.

    Q3: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function with a pole of order $1$ at $z_0$. What is the correct formula to calculate the residue?

    A1: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} (z-z_0) f(z) $$

    A2: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) $$

    A3: $$ \mathrm{Res}(f, z_0) = \frac{1}{2} \lim_{z \rightarrow z_0} (z-z_0) f(z) $$

    A4: $$ \mathrm{Res}(f, z_0) = \frac{1}{3!} \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) $$

    Q4: Let $f: D \rightarrow \mathbb{C}$ be a holomorphic function given by two holomorphic functions in the form $f(z) = \frac{h(z)}{g(z)}$ with a pole of oder 1 at $z_0$. This means that $g$ has a zero of order 1 at $z_0$. Which statement is correct?

    A1: $$ \mathrm{Res}(f, z_0) = \lim_{z \rightarrow z_0} (z-z_0) \frac{h(z)}{g(z) - g(z_0)} = \frac{h(z_0)}{g^\prime(z_0)}$$

    A2: $$ \mathrm{Res}(f, z_0) = 1$$

    A3: $$ \mathrm{Res}(f, z_0) = \frac{h(z_0)}{g(z_0)} $$

    A4: $$ \mathrm{Res}(f, z_0) = \frac{1}{3!} \lim_{z \rightarrow z_0} \frac{d}{dz} (z-z_0) f(z) = g^\prime(z_0) $$

  • Last update: 2024-10

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