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Title: Properties of the Complex Contour Integral
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Series: Complex Analysis
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YouTube-Title: Complex Analysis 19 | Properties of the Complex Contour Integral
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Original video for YT-Members (bright): Watch on YouTube
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Original video for YT-Members (dark): Watch on YouTube
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Forum: Ask a question in Mattermost
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Quiz: Test your knowledge
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Subtitle on GitHub: ca19_sub_eng.srt missing
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Download bright video: Link on Vimeo
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Download dark video: Link on Vimeo
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Timestamps (n/a)
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Subtitle in English (n/a)
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Quiz Content
Q1: Let $\gamma: [0,1] \rightarrow \mathbb{C}$ be piecewise continuously differentiable. What is not correct?
A1: $\gamma$ is continuous.
A2: There are $a_i \in [0,1]$ such that $\gamma|{[a_i,a{i+1}]}$ is continuous.
A3: There are $a_i \in [0,1]$ such that $\gamma|{[a_i,a{i+1}]}$ is differentiable.
A4: There are $a_i \in [0,1]$ such that $\gamma|{[a_i,a{i+1}]}$ is continuously differentiable.
A5: There are $a_i \in [0,1]$ such that $\gamma|{[a_i,a{i+1}]}$ is two-times differentiable.
Q2: Let $\gamma: [0,2 \pi] \rightarrow \mathbb{C}$ be given by $\gamma(t) = e^{it}$ and $f(z) = \frac{1}{z}$. What is $ \oint_\gamma f(z) , dz $?
A1: $0$
A2: $1$
A3: $2 \pi i$
A4: $i$
Q3: Let $\gamma: [a,b] \rightarrow \mathbb{C}$ be a piecewise continuously differentiable. What do we need to say that $\gamma$ is closed?
A1: $\gamma(a) = 0$
A2: $\gamma(a) = \gamma(b)$
A3: $\gamma(a) = \gamma(b) = 0$
A4: $\gamma(a) = 1$
Q4: Let $\gamma: [a,b] \rightarrow \mathbb{C}$ be a piecewise continuously differentiable and $\gamma^-$ the curve with reversed orientation. What is correct?
A1: $ \int_\gamma f(z) , dz = 0$
A2: $ \int_\gamma f(z) , dz = \int_{\gamma^-} f(z) , dz$
A3: $ \int_\gamma f(z) , dz = -\int_{\gamma^-} f(z) , dz$
A4: $ \int_\gamma f(z) , dz = 2 \int_{\gamma^-} f(z) , dz$
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Last update: 2024-10
