Question

Cauchy principal in the definition of fractional laplacian

score 195 · Answer 1 · 2026-03-25 02:56:07

195

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Cauchy principal in the definition of fractional laplacian

Published on 25 Mar 2026 - 2:56

score 197 · Answer 2 · 2026-03-24 04:16:31

197

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Finding $\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}dx$

Published on 24 Mar 2026 - 4:16

score 101 · Answer 3 · 2026-03-25 10:43:46

Why does the Cauchy PV of $\mathsf E(1/X)$, $X\sim\mathcal N(\mu,\sigma^2)$ accurately reflect the sample mean when $|\sigma/\mu|$ is small?

score 41 · Answer 4 · 2026-03-25 14:01:16

41

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Why is this estimator with ill-defined moments useful? And why is the Cauchy PV of its expectation integral a reasonable measure of center?

Published on 25 Mar 2026 - 14:01

score 288 · Answer 5 · 2026-03-24 21:55:26

288

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Why do we use exponentials while integrating trigonometric functions in complex analysis

Published on 24 Mar 2026 - 21:55

score 71 · Answer 6 · 2026-03-24 18:11:43

71

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Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Published on 24 Mar 2026 - 18:11

score 176 · Answer 7 · 2026-03-24 16:57:30

176

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Сalculate the integral PV $\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}dx.$

Published on 24 Mar 2026 - 16:57

score 62 · Answer 8 · 2026-03-25 04:38:49

62

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Computing the asymptotics of a principal value integral

Published on 25 Mar 2026 - 4:38

score 63 · Answer 9 · 2026-03-24 23:11:38

63

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Meaning of the integral $I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}dy$

Published on 24 Mar 2026 - 23:11

score 31 · Answer 10 · 2026-03-24 22:33:21

31

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Legitimate or illegitimate manipulations within a principal value integral

Published on 24 Mar 2026 - 22:33

score 98 · Answer 11 · 2026-03-25 09:32:45

98

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Difficulty finding the principal value of $\int_{-\infty}^{\infty}\frac{\cos3x}{x-1}\,\mathrm{d}x$

Published on 25 Mar 2026 - 9:32

score 62 · Answer 12 · 2026-03-25 10:37:53

62

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Solve an integral equation in which there is Cauchy-type singularity

Published on 25 Mar 2026 - 10:37

score 203 · Answer 13 · 2026-03-24 20:29:30

203

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Keyhole contour for integral with pole on cut

Published on 24 Mar 2026 - 20:29

score 116 · Answer 14 · 2026-03-25 00:43:26

116

Views

An "asymmetric" Cauchy principal value distribution

Published on 25 Mar 2026 - 0:43

score 119 · Answer 15 · 2026-03-25 14:39:45

119

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Convergence of Hilbert transform of a converging sequence

Published on 25 Mar 2026 - 14:39

List Question

Cauchy principal in the definition of fractional laplacian

Finding $\text{PV}\int_0^\infty\frac{\sec\left(\pi B(xt-\lfloor xt+\frac12 \rfloor\right)-\sec\left(\pi B(x-\lfloor x+\frac12 \rfloor\right)}{x}dx$

Why does the Cauchy PV of $\mathsf E(1/X)$, $X\sim\mathcal N(\mu,\sigma^2)$ accurately reflect the sample mean when $|\sigma/\mu|$ is small?

Why is this estimator with ill-defined moments useful? And why is the Cauchy PV of its expectation integral a reasonable measure of center?

Why do we use exponentials while integrating trigonometric functions in complex analysis

Evaluate $PV \int_{-\infty}^{\infty} \frac{1-e^{iax}}{x^2}dx. a>0$

Сalculate the integral PV $\int_{0}^{\infty} \frac{dx}{x^\alpha(x-a)}dx.$

Computing the asymptotics of a principal value integral

Meaning of the integral $I(t)=\lim_{\epsilon\rightarrow 0^+} \int_0^t\frac{\phi(y)}{(\epsilon +iy)^n}dy$

Legitimate or illegitimate manipulations within a principal value integral

Difficulty finding the principal value of $\int_{-\infty}^{\infty}\frac{\cos3x}{x-1}\,\mathrm{d}x$

Solve an integral equation in which there is Cauchy-type singularity

Keyhole contour for integral with pole on cut

An "asymmetric" Cauchy principal value distribution

Convergence of Hilbert transform of a converging sequence

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