Question

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

score 1.7k · Answer 1 · 2014-07-30 21:58:04

1.7k

Views

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

Published on 30 Jul 2014 - 21:58

score 215 · Answer 2 · 2014-08-05 19:32:24

215

Views

Hypergeometric function ratios: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

Published on 05 Aug 2014 - 19:32

score 185 · Answer 3 · 2026-03-26 03:10:46

185

Views

Continued Fraction Expansion

Published on 26 Mar 2026 - 3:10

score 420 · Answer 4 · 2026-03-26 02:56:19

420

Views

How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$

Published on 26 Mar 2026 - 2:56

score 300 · Answer 5 · 2014-08-18 07:49:25

300

Views

Integral's Closed-form expression in terms of hypergeometric function

Published on 18 Aug 2014 - 7:49

score 44 · Answer 6 · 2014-08-18 13:30:54

44

Views

Approximate $_2F_1(a,b;c;x)$ for large (maybe negative) values of $a, b, c$?

Published on 18 Aug 2014 - 13:30

score 344 · Answer 7 · 2014-08-24 06:47:53

344

Views

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

Published on 24 Aug 2014 - 6:47

score 229 · Answer 8 · 2014-08-25 18:19:22

229

Views

Relations between the Eisenstein series and the hypergeometric series

Published on 25 Aug 2014 - 18:19

score 1.5k · Answer 9 · 2014-08-31 16:26:48

1.5k

Views

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

Published on 31 Aug 2014 - 16:26

score 300 · Answer 10 · 2014-09-01 09:33:15

300

Views

How to express a hypergeometric confluent function in the pFq form?

Published on 01 Sep 2014 - 9:33

score 88 · Answer 11 · 2014-09-03 18:03:11

88

Views

Hypergeometric Distrbution issue

Published on 03 Sep 2014 - 18:03

score 528 · Answer 12 · 2014-09-03 18:30:45

528

Views

MLE of Urn Without Replacement Problem

Published on 03 Sep 2014 - 18:30

score 7k · Answer 13 · 2014-09-04 02:15:19

7k

Views

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

Published on 04 Sep 2014 - 2:15

score 33 · Answer 14 · 2014-09-05 02:36:07

33

Views

Evaluation of hypergeometric function when the argument is sum of two matrices

Published on 05 Sep 2014 - 2:36

score 153 · Answer 15 · 2014-09-07 02:03:43

153

Views

A generalization of Clausen's formula

Published on 07 Sep 2014 - 2:03

List Question

Prove ${\large\int}_0^1\frac{\ln(1+8x)}{x^{2/3}\,(1-x)^{2/3}\,(1+8x)^{1/3}}dx=\frac{\ln3}{\pi\sqrt3}\Gamma^3\!\left(\tfrac13\right)$

Hypergeometric function ratios: $\frac{_{2}F_{1}(a+1,b;c;x)}{_{2}F_{1}(a,b;c;x)}$?

Continued Fraction Expansion

How to evaluate the integral $\int_{1}^{\infty} x^{-5/3} \cos\left((x-1) \tau\right) dx$

Integral's Closed-form expression in terms of hypergeometric function

Approximate $_2F_1(a,b;c;x)$ for large (maybe negative) values of $a, b, c$?

Prove: $\int_{0}^{1}\frac{\ln{x}\,\mathrm{d}x}{\sqrt[3]{x(1-x^2)^2}}\stackrel{?}{=}-\frac18\left[\Gamma{\left(\frac13\right)}\right]^3$

Relations between the Eisenstein series and the hypergeometric series

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

How to express a hypergeometric confluent function in the pFq form?

Hypergeometric Distrbution issue

MLE of Urn Without Replacement Problem

Closed form for ${\large\int}_0^1\frac{\ln^2x}{\sqrt{1-x+x^2}}dx$

Evaluation of hypergeometric function when the argument is sum of two matrices

A generalization of Clausen's formula

Trending Questions

Popular # Hahtags

Popular Questions