Question

The identity $\int_0^\infty x^\alpha/(\exp(x)-1)dx=\zeta(\alpha+1)\Gamma(\alpha+1)$

score 357 · Answer 1 · 2014-02-13 18:34:30

357

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The identity $\int_0^\infty x^\alpha/(\exp(x)-1)dx=\zeta(\alpha+1)\Gamma(\alpha+1)$

Published on 13 Feb 2014 - 18:34

score 48 · Answer 2 · 2014-02-14 05:25:49

48

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$_2F_1\left(\frac34,\frac54,2,x^2\right)$ in terms of elliptic integrals $E$ and $K$

Published on 14 Feb 2014 - 5:25

score 1k · Answer 3 · 2026-03-28 01:48:43

1k

Views

Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?

Published on 28 Mar 2026 - 1:48

score 61 · Answer 4 · 2014-02-14 21:22:14

61

Views

Does $\int^1_0\frac{(1-s)^\alpha}{s^\beta} \operatorname d \!s$ converge?

Published on 14 Feb 2014 - 21:22

score 324 · Answer 5 · 2014-02-17 21:18:45

324

Views

Integral with the incomplete upper gamma function

Published on 17 Feb 2014 - 21:18

score 1k · Answer 6 · 2026-04-11 06:13:03

1k

Views

Prove this integral, the Dirichlet's formula

Published on 11 Apr 2026 - 6:13

score 714 · Answer 7 · 2014-02-18 12:49:18

714

Views

taylor series expansion for a rational function

Published on 18 Feb 2014 - 12:49

score 161 · Answer 8 · 2014-02-18 19:03:14

161

Views

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Published on 18 Feb 2014 - 19:03

score 223 · Answer 9 · 2026-04-11 02:38:56

223

Views

Equality of a function and Taylor Series

Published on 11 Apr 2026 - 2:38

score 64 · Answer 10 · 2014-02-19 19:02:30

64

Views

Do Bessel functions have any "nice" input/output pairs?

Published on 19 Feb 2014 - 19:02

score 245 · Answer 11 · 2026-04-08 05:52:41

245

Views

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

Published on 08 Apr 2026 - 5:52

score 75 · Answer 12 · 2014-02-20 12:48:45

75

Views

ODE second order and Bessel function

Published on 20 Feb 2014 - 12:48

score 296 · Answer 13 · 2026-04-12 09:52:28

296

Views

a limit of the digamma function

Published on 12 Apr 2026 - 9:52

score 55 · Answer 14 · 2026-04-13 01:03:09

55

Views

How to estimate this special integral?

Published on 13 Apr 2026 - 1:03

score 117 · Answer 15 · 2026-04-10 15:29:55

117

Views

Solving equation with LambertW function?

Published on 10 Apr 2026 - 15:29

List Question

The identity $\int_0^\infty x^\alpha/(\exp(x)-1)dx=\zeta(\alpha+1)\Gamma(\alpha+1)$

$_2F_1\left(\frac34,\frac54,2,x^2\right)$ in terms of elliptic integrals $E$ and $K$

Proving $\text{Li}_3\left(-\frac{1}{3}\right)-2 \text{Li}_3\left(\frac{1}{3}\right)= -\frac{\log^33}{6}+\frac{\pi^2}{6}\log 3-\frac{13\zeta(3)}{6}$?

Does $\int^1_0\frac{(1-s)^\alpha}{s^\beta} \operatorname d \!s$ converge?

Integral with the incomplete upper gamma function

Prove this integral, the Dirichlet's formula

taylor series expansion for a rational function

Structure of the functional space $\int_ {- \infty} ^ \infty f (x) dx = 1 $

Equality of a function and Taylor Series

Do Bessel functions have any "nice" input/output pairs?

How to find a closed form for the derivatives of $F(x)=\frac1x\int_0^x\frac{1-\cos t}{t^2}\,dt,$ $F(0)=\frac12$?

ODE second order and Bessel function

a limit of the digamma function

How to estimate this special integral?

Solving equation with LambertW function?

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