Question

Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$

score 134 · Answer 1 · 2020-06-23 03:28:54

134

Views

Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$

Published on 23 Jun 2020 - 3:28

score 412 · Answer 2 · 2020-06-23 04:01:21

412

Views

Evaluate $\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$

Published on 23 Jun 2020 - 4:01

score 99 · Answer 3 · 2020-06-27 17:12:37

99

Views

What's this partial sum $ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

Published on 27 Jun 2020 - 17:12

score 221 · Answer 4 · 2020-07-11 10:22:12

221

Views

Evaluation of a log-trig integral in terms of the Clausen function (or other functions related to the dilogarithm)

Published on 11 Jul 2020 - 10:22

score 82 · Answer 5 · 2020-07-16 21:10:01

82

Views

Closed-form expressions for the zeros of $\text{Li}_{-n}(x)$?

Published on 16 Jul 2020 - 21:10

score 3.3k · Answer 6 · 2020-07-20 12:50:30

3.3k

Views

Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$

Published on 20 Jul 2020 - 12:50

score 74 · Answer 7 · 2016-07-26 22:24:27

74

Views

How can I prove that the following function is increasing in $x \in [0,1]$?

Published on 26 Jul 2016 - 22:24

score 313 · Answer 8 · 2016-08-10 21:17:03

313

Views

Justify $\zeta(3)=2\int_0^1 \left(Li_2(e^{-2\pi i x})+Li_2(e^{2\pi i x}\right))\log \Gamma(x)dx$

Published on 10 Aug 2016 - 21:17

score 668 · Answer 9 · 2016-08-15 15:49:01

668

Views

What's about $\int_0^\infty x^{-z}Li_{z+2}(e^{-xz})dx$ as $-\zeta(3)z^2\Gamma(-z)$?

Published on 15 Aug 2016 - 15:49

score 316 · Answer 10 · 2016-08-20 17:37:33

316

Views

Find $A =\int \frac{x \, dx}{e^x +1 }$

Published on 20 Aug 2016 - 17:37

score 183 · Answer 11 · 2016-08-24 23:24:57

183

Views

Simple way to evaluate $f(n)= \sum_{r=0}^{\infty}r^n x^r$

Published on 24 Aug 2016 - 23:24

score 610 · Answer 12 · 2016-08-25 18:27:21

610

Views

Sum of Dilogarithm and its Complex Conjugate

Published on 25 Aug 2016 - 18:27

score 77 · Answer 13 · 2016-09-30 11:15:45

77

Views

Natural proof of identity for $\text{Li}_2(x)+\text{Li}_2(1-x)$

Published on 30 Sep 2016 - 11:15

score 1.1k · Answer 14 · 2016-10-08 16:19:27

1.1k

Views

Proof of a dilogarithm identity

Published on 08 Oct 2016 - 16:19

score 161 · Answer 15 · 2016-10-14 01:34:28

161

Views

Verifying closed form evaluation of an Ising-class multiple integral

Published on 14 Oct 2016 - 1:34

List Question

Evaluating $\int_0^1\frac{\ln(1+x^2)\text{Li}_2(x)}{x}dx$ without using $\sum_{n=1}^\infty\frac{H_n}{n^3}x^n$

Evaluate $\int_0^1 \log (1-x)\ _3F_2\left(1,1,1;\frac{3}{2},\frac{3}{2};x\right) \, dx$

What's this partial sum $ \sum_{k=0}^{n-1} \dfrac{\log(k!)}{2^{k+1}}$ equal?

Evaluation of a log-trig integral in terms of the Clausen function (or other functions related to the dilogarithm)

Closed-form expressions for the zeros of $\text{Li}_{-n}(x)$?

Remarkable logarithmic integral $\int_0^1 \frac{\log^2 (1-x) \log^2 x \log^3(1+x)}{x}dx$

How can I prove that the following function is increasing in $x \in [0,1]$?

Justify $\zeta(3)=2\int_0^1 \left(Li_2(e^{-2\pi i x})+Li_2(e^{2\pi i x}\right))\log \Gamma(x)dx$

What's about $\int_0^\infty x^{-z}Li_{z+2}(e^{-xz})dx$ as $-\zeta(3)z^2\Gamma(-z)$?

Find $A =\int \frac{x \, dx}{e^x +1 }$

Simple way to evaluate $f(n)= \sum_{r=0}^{\infty}r^n x^r$

Sum of Dilogarithm and its Complex Conjugate

Natural proof of identity for $\text{Li}_2(x)+\text{Li}_2(1-x)$

Proof of a dilogarithm identity

Verifying closed form evaluation of an Ising-class multiple integral

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