Question

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

score 210 · Answer 1 · 2022-04-24 16:41:02

210

Views

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

Published on 24 Apr 2022 - 16:41

score 51 · Answer 2 · 2022-05-25 17:53:17

Proving $-\text{Li}_2(x^{-1}-1)+\text{Li}_2(1-x^{-1})-\text{Li}_2(2 x)+\text{Li}_2(2-2 x)+\text{Li}_2(2) = i \pi \log(x)$ for $x>1/2$.

score 433 · Answer 3 · 2022-06-03 16:59:00

433

Views

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Published on 03 Jun 2022 - 16:59

score 311 · Answer 4 · 2022-06-11 03:34:06

311

Views

Find close form for $\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$

Published on 11 Jun 2022 - 3:34

score 311 · Answer 5 · 2022-06-14 10:13:43

311

Views

Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$

Published on 14 Jun 2022 - 10:13

score 91 · Answer 6 · 2026-02-23 08:39:31

91

Views

Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?

Published on 23 Feb 2026 - 8:39

score 209 · Answer 7 · 2022-07-05 20:19:59

209

Views

Is there an analytic solution to $\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$?

Published on 05 Jul 2022 - 20:19

score 253 · Answer 8 · 2022-07-15 10:22:48

253

Views

Find closed-form of: $\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$

Published on 15 Jul 2022 - 10:22

score 312 · Answer 9 · 2022-07-20 14:23:50

312

Views

Two tough integrals with logarithms and polylogarithms

Published on 20 Jul 2022 - 14:23

score 74 · Answer 10 · 2022-07-26 07:24:11

74

Views

The summation $\sum_{k=1}^{\infty} \frac{\phi^{-2k}}{k^2}=\text{Li}_2(1/\phi^2)=\frac{\pi^2}{15}-\ln^2 \phi$

Published on 26 Jul 2022 - 7:24

score 70 · Answer 11 · 2026-02-23 22:50:12

70

Views

Bounding fractional moments of geometric random variable

Published on 23 Feb 2026 - 22:50

score 116 · Answer 12 · 2022-09-05 12:49:48

116

Views

Polylog integral $\int_{0}^{1}\frac{x\log x\operatorname{Li}_2(x)}{x^2+1}dx$

Published on 05 Sep 2022 - 12:49

score 27 · Answer 13 · 2022-10-08 19:32:27

27

Views

Can $\text{Li}_2(x/y)$ be expressed as a sum of other $\text{Li}_2$?

Published on 08 Oct 2022 - 19:32

score 2.2k · Answer 14 · 2022-11-04 06:06:51

2.2k

Views

How to Evaluate the Integral? $\int_{0}^{1}\frac{\ln\left( \frac{x+1}{2x^2} \right)}{\sqrt{x^2+2x}}dx=\frac{\pi^2}{2}$

Published on 04 Nov 2022 - 6:06

score 144 · Answer 15 · 2022-11-07 13:34:35

144

Views

I am stuck in this integral: $\int_{\sqrt{3}}^{\infty} \frac{\ln(x-1)}{x^2-1}dx$

Published on 07 Nov 2022 - 13:34

List Question

How can we prove a closed form for $\frac{1}{8} \text{Li}_2\left(\frac{2+\sqrt{3}}{4} \right)+\text{Li}_2\left(2+\sqrt{3}\right)$?

Proving $-\text{Li}_2(x^{-1}-1)+\text{Li}_2(1-x^{-1})-\text{Li}_2(2 x)+\text{Li}_2(2-2 x)+\text{Li}_2(2) = i \pi \log(x)$ for $x>1/2$.

Is the closed form of $\int_0^1\frac{\text{Li}_{2a+1}(x)}{1+x^2}dx$ known in the literature?

Find close form for $\int_0^1 \frac{\log(x)\log(1+x^2)}{1+x^2}dx$

Evaluating $\int_0^1\frac{\ln^2(1+x)+2\ln(x)\ln(1+x^2)}{1+x^2}dx$

Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?

Is there an analytic solution to $\int_a^b \frac{\arctan(A+Bt)}{C^2 + (t-Z)^2}dt$?

Find closed-form of: $\int_{0}^{1}\frac{x\log^{3}{(x+1)}}{x^2+1}dx$

Two tough integrals with logarithms and polylogarithms

The summation $\sum_{k=1}^{\infty} \frac{\phi^{-2k}}{k^2}=\text{Li}_2(1/\phi^2)=\frac{\pi^2}{15}-\ln^2 \phi$

Bounding fractional moments of geometric random variable

Polylog integral $\int_{0}^{1}\frac{x\log x\operatorname{Li}_2(x)}{x^2+1}dx$

Can $\text{Li}_2(x/y)$ be expressed as a sum of other $\text{Li}_2$?

How to Evaluate the Integral? $\int_{0}^{1}\frac{\ln\left( \frac{x+1}{2x^2} \right)}{\sqrt{x^2+2x}}dx=\frac{\pi^2}{2}$

I am stuck in this integral: $\int_{\sqrt{3}}^{\infty} \frac{\ln(x-1)}{x^2-1}dx$

Trending Questions

Popular # Hahtags

Popular Questions