Newbetuts

Two powerful alternating sums $\sum_{n=1}^\infty\frac{(-1)^nH_nH_n^{(2)}}{n^2}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n^3}{n^2}$

How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$

Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?

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

Definite integral involving polylogarithm

Prove that $\int_0^1\frac{\operatorname{Li}_3(1-z)}{\sqrt{z(1-z)}}\mathrm dz=-\frac{\pi^3}{3}\log 2+\frac{4\pi}3\log^3 2+2\pi\zeta(3)$

Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$

How can you approach $\int_0^{\pi/2} x\frac{\ln(\cos x)}{\sin x}dx$

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Closed-form of $\int\limits_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

Dilogarithm identity containing the tribonacci constant

Known exact values of the $\operatorname{Li}_3$ function

Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Proof of a dilogarithm identity

Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

A logarithmic integral, generalization of a result of Shalev

New posts in polylogarithm

Two powerful alternating sums $\sum_{n=1}^\infty\frac{(-1)^nH_nH_n^{(2)}}{n^2}$ and $\sum_{n=1}^\infty\frac{(-1)^nH_n^3}{n^2}$

Inverse of the polylogarithm

Strategies for evaluating sums $\sum_{n=1}^\infty \frac{H_n^{(m)}z^n}{n}$

Branch Points of the Polylog function

How to evaluate$J(k) = \int_{0}^{1} \frac{\ln^2x\ln\left ( \frac{1-x}{1+x} \right ) }{(x-1)^2-k^2(x+1)^2}\text{d}x$

Closed forms of Nielsen polylogarithms $\int_0^1\frac{(\ln t)^{n-1}(\ln(1-z\,t))^p}{t}dt$?

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

Definite integral involving polylogarithm

Prove that $\int_0^1\frac{\operatorname{Li}_3(1-z)}{\sqrt{z(1-z)}}\mathrm dz=-\frac{\pi^3}{3}\log 2+\frac{4\pi}3\log^3 2+2\pi\zeta(3)$

Different ways to evaluate $\sum_{n=1}^\infty\frac{H_nH_n^{(2)}}{(n+1)(n+2)(n+3)}$

How can you approach $\int_0^{\pi/2} x\frac{\ln(\cos x)}{\sin x}dx$

Is there a closed form for $\int_0^1\frac{\ln^4(1-x)\operatorname{Li}_4(x)}{x}dx\ ?$

Closed-form of $\int\limits_0^1\left(\frac{\left(x^2+1\right)\arcsin(x)}{\sqrt{1-x^2}}+2x\ln\left(x^2+1\right)\right)\frac{\ln x}{x^3+x}\,dx$

Dilogarithm identity containing the tribonacci constant

Known exact values of the $\operatorname{Li}_3$ function

Compute this following integral without Fourier series : $\int_0^{\pi/4}x\ln(\tan x)dx$

Proof of a dilogarithm identity

Compute $\int_0^\infty \frac{\operatorname{Li}_3(x)}{1+x^2}\ dx$

More on the integral $\int_0^1\int_0^1\int_0^1\int_0^1\frac{1}{(1+x) (1+y) (1+z)(1+w) (1+ x y z w)} \ dx \ dy \ dz \ dw$

A logarithmic integral, generalization of a result of Shalev