Zaid Alyafeai's user avatar
Zaid Alyafeai's user avatar
Zaid Alyafeai's user avatar
Zaid Alyafeai
  • Member for 10 years, 8 months
  • Last seen more than 3 years ago
86 votes
9 answers
10k views

Generalized Euler sum $\sum_{n=1}^\infty \frac{H_n}{n^q}$

43 votes
7 answers
2k views

Triple Euler sum result $\sum_{k\geq 1}\frac{H_k^{(2)}H_k }{k^2}=\zeta(2)\zeta(3)+\zeta(5)$

34 votes
8 answers
2k views

Evaluating $\int^1_0 \frac{\log(1+x)\log(1-x) \log(x)}{x}\, \mathrm dx$

29 votes
4 answers
9k views

Calculating alternating Euler sums of odd powers

26 votes
7 answers
3k views

Alternating harmonic sum $\sum_{k\geq 1}\frac{(-1)^k}{k^3}H_k$

21 votes
1 answer
10k views

Real and imaginary part of Gamma function

21 votes
10 answers
1k views

A logarithmic integral $\int^1_0 \frac{\log\left(\frac{1+x}{1-x}\right)}{x\sqrt{1-x^2}}\,dx$

19 votes
1 answer
2k views

An integral of a rational function of logarithm and nonlinear arguments

19 votes
4 answers
670 views

A closed form for a triple integral with sines and cosines

18 votes
3 answers
906 views

Show that $\int_0^\infty \frac{x\log(1+x^2)}{e^{2\pi x}+1}dx=\frac{19}{24} - \frac{23}{24}\log 2 - \frac12\log A$

16 votes
5 answers
1k views

Double Euler sum $ \sum_{k\geq 1} \frac{H_k^{(2)} H_k}{k^3} $

16 votes
2 answers
692 views

prove $\sum\limits_{n\geq 1} (-1)^{n+1}\frac{H_{\lfloor n/2\rfloor}}{n^3} = \zeta^2(2)/2-\frac{7}{4}\zeta(3)\log(2)$

15 votes
2 answers
1k views

Elliptic integral $\int^1_0 \frac{K(k)}{\sqrt{1-k^2}}\,dk$

14 votes
2 answers
2k views

Definite Dilogarithm integral $\int^1_0 \frac{\operatorname{Li}_2^2(x)}{x}\, dx $

13 votes
3 answers
4k views

Integrating around a dog bone contour

12 votes
5 answers
752 views

Dilogarithm integral $\int^x_0 \frac{\operatorname{Li}_2(1-t)\log(1-t)}{t}\, dt$

11 votes
2 answers
414 views

An integral of rational function with third power of cosine hyperbolic function

10 votes
2 answers
814 views

Logarithm integral with golden ratio $\int^1_0 \frac{\log(1+\phi x^2)}{1+x}\, dx$

10 votes
1 answer
578 views

A solution for $\int^{2\pi}_0e^{\cos \theta}\cos(a\theta -\sin \theta)\,d \theta $

9 votes
4 answers
577 views

Quadirlogarithm value $\operatorname{Li}_4 \left( \frac{1}{2}\right)$

9 votes
1 answer
542 views

Intruiging Symmetric harmonic sum $\sum_{n\geq 1} \frac{H^{(k)}_n}{n^k}\, = \frac{\zeta{(2k)}+\zeta^{2}(k)}{2}$

9 votes
3 answers
302 views

Evaluate the series $\sum_{k\geq 1} \frac{1}{2^k k^2}$

9 votes
3 answers
910 views

Limit involving the Sine integral function

8 votes
0 answers
1k views

Integrals that can't be solved by real methods [closed]

8 votes
2 answers
407 views

Solving higher order logarithms integrals without the beta function

8 votes
7 answers
885 views

Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$

8 votes
2 answers
3k views

contour integration of a function with two branch points: $\int^{\infty}_0 \frac{\log(1+x)}{x^p(1+x)}\, dx \,\,\, 0<\Re(p)<1$

8 votes
1 answer
2k views

General case integral $\int^1_0 f(x) \, f'(1-x) \, dx $

7 votes
4 answers
811 views

Double harmonic sum $\sum_{n\geq 1}\frac{H^{(p)}_nH_n}{n^q}$

7 votes
1 answer
401 views

Find the integral $\int_0^\infty \frac{\log(1+x^2)}{e^{2\pi x}+1}dx$