Question

Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$

score 269 · Answer 1 · 2023-04-26 06:11:09

269

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Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$

Published on 26 Apr 2023 - 6:11

score 91 · Answer 2 · 2023-04-26 18:55:19

91

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Function upper-bounding polylogarithm function when $0 < z < 1$

Published on 26 Apr 2023 - 18:55

score 257 · Answer 3 · 2023-04-28 02:03:10

257

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How to find the exact value of $\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{n^2 \cdot 2^{\frac{n}{2}}} $?

Published on 28 Apr 2023 - 2:03

score 237 · Answer 4 · 2026-02-23 03:03:13

237

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Generalized formula for the polylogarithm

Published on 23 Feb 2026 - 3:03

score 77 · Answer 5 · 2026-02-23 02:55:24

77

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How to derive this polylogarithm identity (involving Bernoulli polynomials)?

Published on 23 Feb 2026 - 2:55

score 122 · Answer 6 · 2023-05-21 13:24:48

122

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Show that $\int_0^1 \frac{Li_{1 - 2m}(1 - 1/x)}{x} dx = 0$.

Published on 21 May 2023 - 13:24

score 32 · Answer 7 · 2023-05-25 00:25:51

32

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Proving $2^{\log^{1-\varepsilon}n}\in \omega(\operatorname{polylog}(n))$

Published on 25 May 2023 - 0:25

score 321 · Answer 8 · 2023-06-10 14:32:08

321

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What's a better time complexity, $O(\log^2(n))$, or $O(n)$?

Published on 10 Jun 2023 - 14:32

score 280 · Answer 9 · 2023-06-13 15:28:54

280

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Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $

Published on 13 Jun 2023 - 15:28

score 63 · Answer 10 · 2026-03-26 12:33:51

63

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Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function

Published on 26 Mar 2026 - 12:33

score 139 · Answer 11 · 2023-06-24 14:03:48

139

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Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$

Published on 24 Jun 2023 - 14:03

score 77 · Answer 12 · 2026-03-25 22:24:51

77

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Efficient calculation for Lerch Transcendent Expression

Published on 25 Mar 2026 - 22:24

score 315 · Answer 13 · 2023-07-22 02:19:08

315

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How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$

Published on 22 Jul 2023 - 2:19

score 197 · Answer 14 · 2026-02-23 04:53:24

197

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Verification of the generalized polylogarithm formula

Published on 23 Feb 2026 - 4:53

score 67 · Answer 15 · 2023-08-12 14:59:12

67

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Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms

Published on 12 Aug 2023 - 14:59

List Question

Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$

Function upper-bounding polylogarithm function when $0 < z < 1$

How to find the exact value of $\sum_{n=1}^{\infty} \frac{\sin \left(\frac{n \pi}{4}\right)}{n^2 \cdot 2^{\frac{n}{2}}} $?

Generalized formula for the polylogarithm

How to derive this polylogarithm identity (involving Bernoulli polynomials)?

Show that $\int_0^1 \frac{Li_{1 - 2m}(1 - 1/x)}{x} dx = 0$.

Proving $2^{\log^{1-\varepsilon}n}\in \omega(\operatorname{polylog}(n))$

What's a better time complexity, $O(\log^2(n))$, or $O(n)$?

Closed forms of the integral $ \int_0^1 \frac{\mathrm{Li}_n(x)}{(1+x)^n} d x $

Evaluate $\sum\limits_{n=0}^\infty\operatorname W(e^{e^{an}})x^n$ with Lambert W function

Prove that $-\int_{0}^{1}\frac{{\rm Li}_2(-x(1-x))}{x}\ dx=\frac{2}{5}\zeta(3)$

Efficient calculation for Lerch Transcendent Expression

How to show $\int_0^1\frac{\operatorname{Li}_2\left(\frac{1+x^2}{2}\right)}{1+x^2}dx=\ln(2)G$

Verification of the generalized polylogarithm formula

Evaluating $\int\frac{\log(x+a)}{x}\,dx$ in terms of dilogarithms

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