Question

Prove $\text{Li}_2(e^{-2 i x})+\text{Li}_2(e^{2 i x})=\frac{1}{3} (6 x^2-6 \pi x+\pi ^2)$ when $0<x<\pi$

score 208 · Answer 1 · 2026-04-17 04:29:39

208

Views

Prove $\text{Li}_2(e^{-2 i x})+\text{Li}_2(e^{2 i x})=\frac{1}{3} (6 x^2-6 \pi x+\pi ^2)$ when $0<x<\pi$

Published on 17 Apr 2026 - 4:29

score 217 · Answer 2 · 2026-04-16 10:58:15

217

Views

closed form for $\int_0^1\frac{\mathrm{Li}_s(x-x^2)}{x-x^2}\mathrm dx$

Published on 16 Apr 2026 - 10:58

score 167 · Answer 3 · 2026-04-14 20:52:57

167

Views

I need help computing $\int {\ln x\over 2-x}\, dx$

Published on 14 Apr 2026 - 20:52

score 191 · Answer 4 · 2026-05-05 04:34:12

191

Views

Series power function over exponential function

Published on 05 May 2026 - 4:34

score 149 · Answer 5 · 2026-04-16 01:05:07

149

Views

Polylogarithm inequality: $(s+1)\frac{-\operatorname{Li}_{s+1} (-x)}{-\operatorname{Li}_s(-x)} > \log(x)$

Published on 16 Apr 2026 - 1:05

score 418 · Answer 6 · 2026-04-12 01:48:47

418

Views

An integral involving a Gaussian, error functions and the Owen's T function.

Published on 12 Apr 2026 - 1:48

score 182 · Answer 7 · 2026-04-14 12:11:40

182

Views

Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?

Published on 14 Apr 2026 - 12:11

score 76 · Answer 8 · 2026-04-15 10:05:03

76

Views

Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?

Published on 15 Apr 2026 - 10:05

score 199 · Answer 9 · 2026-05-10 15:14:12

199

Views

Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

Published on 10 May 2026 - 15:14

score 484 · Answer 10 · 2026-05-10 14:45:25

484

Views

A peculiar Euler sum

Published on 10 May 2026 - 14:45

score 131 · Answer 11 · 2026-04-12 23:20:52

131

Views

On $\sum_{k=1}^\infty1/(k!k^s)$.

Published on 12 Apr 2026 - 23:20

score 458 · Answer 12 · 2026-04-12 01:40:20

458

Views

A logarithmic integral, generalization of a result of Shalev

Published on 12 Apr 2026 - 1:40

score 472 · Answer 13 · 2026-04-10 05:11:25

472

Views

Yet another difficult logarithmic integral

Published on 10 Apr 2026 - 5:11

score 257 · Answer 14 · 2026-04-15 19:36:38

257

Views

On $\int_{-\pi/2}^{\pi/2}\operatorname{Li}_3(\sin x)dx$ and its derivative

Published on 15 Apr 2026 - 19:36

score 82 · Answer 15 · 2026-04-16 14:02:33

82

Views

An anti-derivative involving an arc tangent, a square root and a rational function.

Published on 16 Apr 2026 - 14:02

List Question

Prove $\text{Li}_2(e^{-2 i x})+\text{Li}_2(e^{2 i x})=\frac{1}{3} (6 x^2-6 \pi x+\pi ^2)$ when $0<x<\pi$

closed form for $\int_0^1\frac{\mathrm{Li}_s(x-x^2)}{x-x^2}\mathrm dx$

I need help computing $\int {\ln x\over 2-x}\, dx$

Series power function over exponential function

Polylogarithm inequality: $(s+1)\frac{-\operatorname{Li}_{s+1} (-x)}{-\operatorname{Li}_s(-x)} > \log(x)$

An integral involving a Gaussian, error functions and the Owen's T function.

Is a closed form possible for $\int\frac{\text{Li}_2(x)^2}{x}dx$?

Can every Gaussian integral be reduced to elementary functions and poly-logarithms only?

Evaluating $\int_{0}^{1-x} \frac{1}{1-x_{n}} \cdots \int_{0}^{1-x_3} \frac{1}{1-x_2} \int_{0}^{1-x_2} \frac{1}{1-x_1} dx_1 dx_2 \cdots dx_n$

A peculiar Euler sum

On $\sum_{k=1}^\infty1/(k!k^s)$.

A logarithmic integral, generalization of a result of Shalev

Yet another difficult logarithmic integral

On $\int_{-\pi/2}^{\pi/2}\operatorname{Li}_3(\sin x)dx$ and its derivative

An anti-derivative involving an arc tangent, a square root and a rational function.

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