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 409 · Answer 1 · 2018-04-18 17:04:02

409

Views

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 18 Apr 2018 - 17:04

score 493 · Answer 2 · 2018-05-05 18:22:43

493

Views

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

Published on 05 May 2018 - 18:22

score 380 · Answer 3 · 2018-05-08 07:28:31

380

Views

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 08 May 2018 - 7:28

score 208 · Answer 4 · 2018-05-08 19:07:48

208

Views

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

Published on 08 May 2018 - 19:07

score 529 · Answer 5 · 2018-05-12 00:08:41

529

Views

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

Published on 12 May 2018 - 0:08

score 120 · Answer 6 · 2018-05-14 14:32:29

120

Views

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

Published on 14 May 2018 - 14:32

score 156 · Answer 7 · 2018-06-05 09:25:57

156

Views

Lebesgue dominated convergence with a principal value integral

Published on 05 Jun 2018 - 9:25

score 73 · Answer 8 · 2018-06-06 18:20:12

73

Views

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

Published on 06 Jun 2018 - 18:20

score 265 · Answer 9 · 2018-06-09 08:36:16

265

Views

Does Cauchy Principal Value always exist?

Published on 09 Jun 2018 - 8:36

score 353 · Answer 10 · 2018-07-07 15:03:26

353

Views

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

Published on 07 Jul 2018 - 15:03

score 351 · Answer 11 · 2018-07-09 21:20:34

351

Views

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

Published on 09 Jul 2018 - 21:20

score 1k · Answer 12 · 2018-07-26 08:46:23

1k

Views

Contour for Principal Value Integral

Published on 26 Jul 2018 - 8:46

score 164 · Answer 13 · 2018-08-14 22:00:22

164

Views

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

Published on 14 Aug 2018 - 22:00

score 121 · Answer 14 · 2018-08-19 06:25:17

121

Views

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

Published on 19 Aug 2018 - 6:25

score 241 · Answer 15 · 2018-09-24 23:12:07

241

Views

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

Published on 24 Sep 2018 - 23:12

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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