-
Title: Cauchy’s Theorem (general version)
-
Series: Complex Analysis
-
YouTube-Title: Complex Analysis 25 | Cauchy’s Theorem (general version)
-
Bright video: https://youtu.be/4kstnqz9OFs
-
Dark video: https://youtu.be/MKXWguG6z68
-
Ad-free video: Watch Vimeo video
-
Original video for YT-Members (bright): https://youtu.be/f3fNmR995ik
-
Original video for YT-Members (dark): https://youtu.be/iXOFqNsETzI
-
Forum: Ask a question in Mattermost
-
Quiz: Test your knowledge
-
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: ca25_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 $D = \mathbb{C} \setminus { z \in \mathbb{R} \mid z \leq 0 }$ and $\gamma$ be a closed curve in $D$. What is the winding number $ \mathrm{wind}(\gamma, 0) $?
A1: $ 0 $
A2: $ 1$
A3: $ -1$
A4: $ 2$
A5: One needs more information!
Q2: Let $\gamma$ be a closed curve. What is the definition of the exterior $\mathrm{Ext}(\gamma)$?
A1: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) = 0 }$
A2: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) \neq 0 }$
A3: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) = 1 }$
A4: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, 0) = z }$
A5: $ \mathrm{Ext}(\gamma) = { z \in \mathbb{C} \setminus \mathrm{Ran}(\gamma) \mid \mathrm{wind}(\gamma, z) \geq 0 }$
Q3: Let $f : D \rightarrow \mathbb{C}$ be holomorphic and $\gamma: [a,b] \rightarrow D$ be a closed curve. Which claim is in general not correct?
A1: $ \oint_{\gamma} f(z) , dz = 0 $, no matter which open domain one chooses for $D$.
A2: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a star domain.
A3: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a convex set.
A4: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a slitted plane.
A5: $ \oint_{\gamma} f(z) , dz = 0 $ if $D$ is a disc.
-
Last update: 2024-10
