Question

Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

score 1.2k · Answer 1 · 2014-12-07 18:21:17

1.2k

Views

Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

Published on 07 Dec 2014 - 18:21

score 291 · Answer 2 · 2026-03-25 14:03:25

291

Views

Why does the tribonacci constant have a trilogarithm ladder?

Published on 25 Mar 2026 - 14:03

score 379 · Answer 3 · 2014-12-25 00:35:57

379

Views

How to calculate $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}h_n}{n^3}$ where $h_n=\sum_{k=1}^{n}\frac{1}{2k-1}$?

Published on 25 Dec 2014 - 0:35

score 403 · Answer 4 · 2019-10-08 15:49:59

403

Views

Integral related to the softplus function

Published on 08 Oct 2019 - 15:49

score 44 · Answer 5 · 2019-10-12 14:22:41

44

Views

The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$.

Published on 12 Oct 2019 - 14:22

score 280 · Answer 6 · 2019-10-16 06:43:10

280

Views

Computing $\sum\limits_{n=0}^\infty\frac{(-1)^nH_{n/2}}{(2n+1)^2}$

Published on 16 Oct 2019 - 6:43

score 306 · Answer 7 · 2019-10-17 23:22:54

306

Views

Find the series expansion of $\frac{\ln^4(1-x)}{1-x}$

Published on 17 Oct 2019 - 23:22

score 262 · Answer 8 · 2019-10-19 03:39:11

262

Views

Compute $\sum_{n=1}^\infty\frac{H_n^3}{n^4}-3\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^4}$

Published on 19 Oct 2019 - 3:39

score 257 · Answer 9 · 2019-10-23 20:55:07

257

Views

How to prove $\int_0^\infty e^{-a q} \Gamma (0,q)^2 \, dq = -\frac{1}{a}\left(2 \operatorname{Li}_2(-a-1)+\frac{\pi ^2}{6}\right)$

Published on 23 Oct 2019 - 20:55

score 393 · Answer 10 · 2019-10-25 21:26:51

393

Views

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Published on 25 Oct 2019 - 21:26

score 1.5k · Answer 11 · 2026-03-26 01:11:56

1.5k

Views

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

Published on 26 Mar 2026 - 1:11

score 82 · Answer 12 · 2019-11-05 14:41:31

82

Views

Prove that $\mathrm{Li}_s(x)=\sum_{k=1}^{\infty}\frac{x^k}{k^s}$ is a rational function.

Published on 05 Nov 2019 - 14:41

score 438 · Answer 13 · 2019-11-07 20:24:22

438

Views

Compute $\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

Published on 07 Nov 2019 - 20:24

score 584 · Answer 14 · 2019-11-09 19:38:56

584

Views

Compute $\int_0^{\pi/2} x^2\left(\sum_{n=1}^\infty (-1)^{n-1} \cos^n(x)\cos(nx)\right)dx$

Published on 09 Nov 2019 - 19:38

score 305 · Answer 15 · 2019-11-11 22:54:04

305

Views

Is there a closed form for $\int_0^1\frac{\ln(x) \sin^{-1}(x)}{x\sqrt{1-x^2}}dx\ ?$

Published on 11 Nov 2019 - 22:54

List Question

Proof of dilogarithm reflection formula $\zeta(2)-\log(x)\log(1-x)=\operatorname{Li}_2(x)+\operatorname{Li}_2(1-x)$

Why does the tribonacci constant have a trilogarithm ladder?

How to calculate $\sum_{n=1}^{\infty}\frac{(-1)^{n-1}h_n}{n^3}$ where $h_n=\sum_{k=1}^{n}\frac{1}{2k-1}$?

Integral related to the softplus function

The equation $100\log(5x)\log(2x)+1 = 0$ has two distinct real roots $\alpha$ and $\beta$. Find the value of $\alpha\beta$.

Computing $\sum\limits_{n=0}^\infty\frac{(-1)^nH_{n/2}}{(2n+1)^2}$

Find the series expansion of $\frac{\ln^4(1-x)}{1-x}$

Compute $\sum_{n=1}^\infty\frac{H_n^3}{n^4}-3\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{n^4}$

How to prove $\int_0^\infty e^{-a q} \Gamma (0,q)^2 \, dq = -\frac{1}{a}\left(2 \operatorname{Li}_2(-a-1)+\frac{\pi ^2}{6}\right)$

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Conjectural closed-form of $\int_0^1 \frac{\log^n (1-x) \log^{n-1} (1+x)}{1+x} dx$

Prove that $\mathrm{Li}_s(x)=\sum_{k=1}^{\infty}\frac{x^k}{k^s}$ is a rational function.

Compute $\int_0^1\frac{\ln x\operatorname{Li}_2(x^2)}{\sqrt{1-x^2}}dx$

Compute $\int_0^{\pi/2} x^2\left(\sum_{n=1}^\infty (-1)^{n-1} \cos^n(x)\cos(nx)\right)dx$

Is there a closed form for $\int_0^1\frac{\ln(x) \sin^{-1}(x)}{x\sqrt{1-x^2}}dx\ ?$

Trending Questions

Popular # Hahtags

Popular Questions