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user153012's user avatar
user153012
  • Member for 10 years
  • Last seen more than a week ago
24 votes
Accepted

Prove that $\int_{0}^{1}\sin{(\pi x)}x^x(1-x)^{1-x}\,dx =\frac{\pi e}{24} $

20 votes

Power of a matrix, given its jordan form

20 votes

Closed Form for the Imaginary Part of $\text{Li}_3\Big(\frac{1+i}2\Big)$

15 votes

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

15 votes

Integral ${\large\int}_0^\infty\frac{\ln x}{1+x}\sqrt{\frac{x+\sqrt{1+x^2}}{1+x^2}}\ \mathrm dx$

14 votes

Is there a power of 2 that, written backward, is a power of 5?

14 votes

Simpler closed form for $\sum_{n=1}^\infty\frac{\Gamma\left(n+\frac{1}{2}\right)}{(2n+1)^4\,4^n\,n!}$

13 votes

Conjectured closed form for $\operatorname{Li}_2\!\left(\sqrt{2-\sqrt3}\cdot e^{i\pi/12}\right)$

13 votes
Accepted

Closed form for $\int_1^\infty\frac{\operatorname dx}{\operatorname \Gamma(x)}$

12 votes

Prove that $\int_0^1 \frac{{\rm{Li}}_2(x)\ln(1-x)\ln^2(x)}{x} \,dx=-\frac{\zeta(6)}{3}$

11 votes

Is zero a prime number?

11 votes
Accepted

Approximating Logs and Antilogs by hand

10 votes
Accepted

Importance of relation between harmonic, geometric,arithmetic and quadratic mean

9 votes

Simplification of an expression containing $\operatorname{Li}_3(x)$ terms

8 votes

Beautiful, simple proofs worthy of writing on this beautiful glass door

8 votes

How to find ${\large\int}_0^1\frac{\ln^3(1+x)\ln x}x\mathrm dx$

8 votes

Is there a way to relate prime numbers and the Fourier transform?

8 votes

Priority vector and eigenvectors - AHP method

7 votes
Accepted

Prove or disprove $\int_{-\infty}^\infty \frac{dx}{\cos x+\cosh x}=\frac{1512835691 \pi}{1983703776}$

7 votes

Finding the value of ${_2F_1}\left(\frac12,\frac12;\ 1;\ 1/9\right)$

7 votes

Does a positive semidefinite matrix always have a non-negative trace?

6 votes
Accepted

Identity $i\int_0^{\pi}\left[\mathrm{Li}_2\left(-1-e^{ix}\right)-\mathrm{Li}_2\left(-1-e^{-ix}\right)\right]dx=\frac{7}{3}\zeta(3) $

6 votes
Accepted

A closed-form of product the gamma functions containing $\pi$ and $\phi$

6 votes

An equation that generates a beautiful or unique shape for motivating students in mathematics

6 votes
Accepted

Proof of a dilogarithm identity

5 votes
Accepted

Reference to proof of closed form for of $\zeta_{\text{Ai}}(2)$

5 votes

Calculating pi manually

5 votes

A conjectured result for $\sum_{n=1}^\infty\frac{(-1)^n\,H_{n/5}}n$

5 votes
Accepted

Prove ${_2F_1}\left({{\tfrac16,\tfrac23}\atop{\tfrac56}}\middle|\,\frac{80}{81}\right)=\frac 35 \cdot 5^{1/6} \cdot 3^{2/3}$

5 votes
Accepted

Closed form of $\int_{0}^{\pi/2}x\cot\left(x\right)\cos\left(x\right)\log\left(\sin\left(x\right)\right)dx$

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