Question

$\xi(s)=\xi(1-s)$ functional equation from $\Gamma(s)\Gamma(s+1)\zeta(s)=\int_0^\infty t^{s-1}\sum_{\alpha \in \Bbb N} \Phi_{\alpha}(t)~ dt$

score 204 · Answer 1 · 2026-04-01 17:32:49

204

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$\xi(s)=\xi(1-s)$ functional equation from $\Gamma(s)\Gamma(s+1)\zeta(s)=\int_0^\infty t^{s-1}\sum_{\alpha \in \Bbb N} \Phi_{\alpha}(t)~ dt$

Published on 01 Apr 2026 - 17:32

score 169 · Answer 2 · 2023-04-14 12:09:13

169

Views

Simplifying a result obtained after integrating $\int_0^{\pi/2} \theta^2 \sqrt{\cot\left(\theta\right)} \ d\theta$ using complex methods

Published on 14 Apr 2023 - 12:09

score 175 · Answer 3 · 2026-02-24 05:20:00

175

Views

Convert $\frac1b\sum_{n=1}^\infty\frac{(b e^a)^n}{n!}B_{n-1}(an)$ to integral using $B_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x(e^t-1)}}{t^{n+1}}dt$

Published on 24 Feb 2026 - 5:20

score 17 · Answer 4 · 2026-04-01 16:20:09

17

Views

Evaluate $\int_{-\infty}^\infty \cos(\omega t)e^{-t^2} dt$

Published on 01 Apr 2026 - 16:20

score 51 · Answer 5 · 2026-04-07 19:44:38

51

Views

Choosing a path for reverse Cauchy integral formula

Published on 07 Apr 2026 - 19:44

score 90 · Answer 6 · 2026-04-01 16:22:07

90

Views

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Published on 01 Apr 2026 - 16:22

score 81 · Answer 7 · 2026-03-27 02:33:11

81

Views

Integrals involving Gamma Functions

Published on 27 Mar 2026 - 2:33

score 92 · Answer 8 · 2026-03-27 11:48:07

92

Views

Integrating $\frac {e^{-iz}}{z}$ over a semicircle counter-clockwise around $0$ of infinite radius $r$

Published on 27 Mar 2026 - 11:48

score 416 · Answer 9 · 2023-04-21 18:25:36

416

Views

Integral of $\int_0^1 \frac{dx}{(1-x^5)^{1/5}}$

Published on 21 Apr 2023 - 18:25

score 143 · Answer 10 · 2026-04-01 16:25:20

143

Views

How to choose the contour for real integrals of the form $\int_{-\infty}^{\infty} f(x) dx$?

Published on 01 Apr 2026 - 16:25

score 135 · Answer 11 · 2026-03-25 17:25:03

135

Views

Calculating an inverse Mellin transform

Published on 25 Mar 2026 - 17:25

score 70 · Answer 12 · 2026-04-01 21:37:21

70

Views

Justifying that semicircle contour integral goes to zero

Published on 01 Apr 2026 - 21:37

score 70 · Answer 13 · 2023-05-03 01:54:12

70

Views

Which types of real integrands accept contour integration?

Published on 03 May 2023 - 1:54

score 180 · Answer 14 · 2023-05-06 22:49:39

180

Views

Closed form of $\int_0^\infty \frac{x^n dx}{1+(x+a)^b}\,dx $ with $b \gg 1$

Published on 06 May 2023 - 22:49

score 120 · Answer 15 · 2026-04-01 18:10:05

120

Views

Evaluating $I(A, n) = \int_{0}^{2\pi} \dfrac{1}{A +\cos^n(\theta)} d\theta$

Published on 01 Apr 2026 - 18:10

List Question

$\xi(s)=\xi(1-s)$ functional equation from $\Gamma(s)\Gamma(s+1)\zeta(s)=\int_0^\infty t^{s-1}\sum_{\alpha \in \Bbb N} \Phi_{\alpha}(t)~ dt$

Simplifying a result obtained after integrating $\int_0^{\pi/2} \theta^2 \sqrt{\cot\left(\theta\right)} \ d\theta$ using complex methods

Convert $\frac1b\sum_{n=1}^\infty\frac{(b e^a)^n}{n!}B_{n-1}(an)$ to integral using $B_n(x)=\frac{n!}{2\pi i}\oint\frac{e^{x(e^t-1)}}{t^{n+1}}dt$

Evaluate $\int_{-\infty}^\infty \cos(\omega t)e^{-t^2} dt$

Choosing a path for reverse Cauchy integral formula

How to calculate the Laplace inversion of this function

Integrals involving Gamma Functions

Integrating $\frac {e^{-iz}}{z}$ over a semicircle counter-clockwise around $0$ of infinite radius $r$

Integral of $\int_0^1 \frac{dx}{(1-x^5)^{1/5}}$

How to choose the contour for real integrals of the form $\int_{-\infty}^{\infty} f(x) dx$?

Calculating an inverse Mellin transform

Justifying that semicircle contour integral goes to zero

Which types of real integrands accept contour integration?

Closed form of $\int_0^\infty \frac{x^n dx}{1+(x+a)^b}\,dx $ with $b \gg 1$

Evaluating $I(A, n) = \int_{0}^{2\pi} \dfrac{1}{A +\cos^n(\theta)} d\theta$

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