Question

PV of $\int_{-\infty}^\infty \frac{x^n~ e^{-ax^2} dx}{(x-x_1)(x-x_2)},~ (n=0,1,2,3,...)$

score 410 · Answer 1 · 2026-03-25 06:13:12

410

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PV of $\int_{-\infty}^\infty \frac{x^n~ e^{-ax^2} dx}{(x-x_1)(x-x_2)},~ (n=0,1,2,3,...)$

Published on 25 Mar 2026 - 6:13

score 494 · Answer 2 · 2026-03-25 06:12:08

494

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Using Contour Integration to Prove $\lim_{\epsilon\to 0^{+}}\int_{-1}^{1}x^{\epsilon-1}\,\mathrm{d}x=-\pi\mathrm{i}$

Published on 25 Mar 2026 - 6:12

score 381 · Answer 3 · 2026-03-25 06:13:09

381

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How to show that $\mathscr{P} \int_{-\infty}^{\infty} \frac{d\omega}{\sqrt{\omega^2}} e^{ - i \omega t }$ is equal to $- \log(t^2)$

Published on 25 Mar 2026 - 6:13

score 209 · Answer 4 · 2026-03-25 06:10:53

209

Views

Problem with Hilbert Transform of $\operatorname{sgn}(t)$

Published on 25 Mar 2026 - 6:10

score 530 · Answer 5 · 2026-03-25 06:12:39

530

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Find Cauchy principal values of $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$

Published on 25 Mar 2026 - 6:12

score 121 · Answer 6 · 2026-03-25 06:11:57

121

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Showing that $\int_0^{\infty} \frac{\cos(x)-e^{-x}}{x} dx = 0 $

Published on 25 Mar 2026 - 6:11

score 157 · Answer 7 · 2026-03-25 06:10:38

157

Views

Lebesgue dominated convergence with a principal value integral

Published on 25 Mar 2026 - 6:10

score 74 · Answer 8 · 2026-03-25 06:13:10

74

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Conditional convergence of $\Bbb E[X^{-1}]$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ as $\operatorname{pr}(X>0)\to 1$

Published on 25 Mar 2026 - 6:13

score 266 · Answer 9 · 2026-03-25 06:09:08

266

Views

Does Cauchy Principal Value always exist?

Published on 25 Mar 2026 - 6:09

score 354 · Answer 10 · 2026-03-25 06:12:01

354

Views

Solution for the Laplace Transform $\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\mu}\right\}(\beta)$

Published on 25 Mar 2026 - 6:12

score 352 · Answer 11 · 2026-03-25 06:13:12

352

Views

Convergence of Parte Finie of $x \mapsto \frac{1}{x^2}$

Published on 25 Mar 2026 - 6:13

score 1k · Answer 12 · 2026-03-27 13:44:08

1k

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Contour for Principal Value Integral

Published on 27 Mar 2026 - 13:44

score 165 · Answer 13 · 2026-03-25 07:48:01

165

Views

Evaluation of $\int_{0}^{\infty}\frac{\sin(bx)}{x^2}\mathrm{d}x$

Published on 25 Mar 2026 - 7:48

score 122 · Answer 14 · 2026-03-25 06:12:09

122

Views

Principal value of $\int^\infty_{-\infty}\frac1{x^2}dx$: counter-intuitive?

Published on 25 Mar 2026 - 6:12

score 242 · Answer 15 · 2026-03-25 06:13:02

242

Views

How to calculate the particle derivative of the principal value of this integral?

Published on 25 Mar 2026 - 6:13

List Question

PV of $\int_{-\infty}^\infty \frac{x^n~ e^{-ax^2} dx}{(x-x_1)(x-x_2)},~ (n=0,1,2,3,...)$

Using Contour Integration to Prove $\lim_{\epsilon\to 0^{+}}\int_{-1}^{1}x^{\epsilon-1}\,\mathrm{d}x=-\pi\mathrm{i}$

How to show that $\mathscr{P} \int_{-\infty}^{\infty} \frac{d\omega}{\sqrt{\omega^2}} e^{ - i \omega t }$ is equal to $- \log(t^2)$

Problem with Hilbert Transform of $\operatorname{sgn}(t)$

Find Cauchy principal values of $\int_{-\infty}^{\infty}\frac{x}{(x^2+4)(x^2-2x+5)}\,dx$

Showing that $\int_0^{\infty} \frac{\cos(x)-e^{-x}}{x} dx = 0 $

Lebesgue dominated convergence with a principal value integral

Conditional convergence of $\Bbb E[X^{-1}]$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ as $\operatorname{pr}(X>0)\to 1$

Does Cauchy Principal Value always exist?

Solution for the Laplace Transform $\mathscr{L}\left\{\frac{t^{\alpha-1}}{t-\mu}\right\}(\beta)$

Convergence of Parte Finie of $x \mapsto \frac{1}{x^2}$

Contour for Principal Value Integral

Evaluation of $\int_{0}^{\infty}\frac{\sin(bx)}{x^2}\mathrm{d}x$

Principal value of $\int^\infty_{-\infty}\frac1{x^2}dx$: counter-intuitive?

How to calculate the particle derivative of the principal value of this integral?

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