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Title: Cauchy Principal Value
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Series: Real Analysis
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Chapter: Riemann Integral
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YouTube-Title: Real Analysis 64 | Cauchy Principal Value
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Bright video: Watch on YouTube
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Dark video: 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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Exercise Download PDF sheets
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Subtitle on GitHub: ra64_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 $f: [0,2]\setminus{1} \rightarrow [0,1]$ be an unbounded function. What is the correct definition of the improper Riemann integral $$ \int_0^2 f(x), dx$$
A1: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 1 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 1 + \varepsilon}^{2} f(x), dx $$
A2: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 1 + \varepsilon}^{2} f(x), dx $$
A3: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{ 2 + \varepsilon}^{2} f(x), dx $$
A4: $$ \lim_{\varepsilon \searrow 0 } \int_0^{ 2 - \varepsilon} f(x), dx + \lim_{\varepsilon \searrow 0 } \int_{\varepsilon}^{2} f(x), dx $$
Q2: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x}$. Does the improper Riemann integral exist?
A1: Yes!
A2: No!
Q3: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x}$. Does the Cauchy principal value exist?
A1: Yes!
A2: No!
Q4: $f: [-1,1] \setminus {0} \rightarrow [0,1]$ be given by $ f(x) = \frac{1}{x^3}$. What is the Cauchy principal value $$\mathrm{p.v.} \int_{-1}^1 \frac{1}{x^3} , dx $$ of this function?
A1: $0$
A2: $1$
A3: $\frac{1}{2}$
A4: $\frac{1}{3}$
A5: $3$
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Last update: 2025-01
