# Questions tagged [polylogarithm]

For questions about or related to polylogarithm functions.

264 questions
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### A peculiar Euler sum

I would like a hand in the computation of the following Euler sum (Why isn't here a tag for Euler sums?) $$S=\sum_{m,n\geq 0}\frac{(-1)^{m+n}}{(2m+1)(2n+1)^2(2m+2n+1)} \tag{1}$$ which arises from ...
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### Prove $\text{Li}_2(e^{-2 i x})+\text{Li}_2(e^{2 i x})=\frac{1}{3} (6 x^2-6 \pi x+\pi ^2)$ when $0<x<\pi$

This is an identity I deduced when playing with the initial-boundary value problem of heat conduction equation asked here. It's easy to verify numerically with Mathematica: ...
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### Polylogarithmic integrals

I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form \begin{equation} \int \frac{\prod_{i=1}^N \...
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### Evaluate$\int\limits_0^1 [\log(x)\log(1-x)+\operatorname{Li}_2(x)]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]\mathrm dx$

$$\mathfrak{I}=\int\limits_0^1 \left[\log(x)\log(1-x)+\operatorname{Li}_2(x)\right]\left[\frac{\operatorname{Li}_2(x)}{x(1-x)}-\frac{\zeta(2)}{1-x}\right]\mathrm dx=4\zeta(2)\zeta(3)-9\zeta(5)\tag1$$ ...
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### Proving $\Im\operatorname{Li}_2(\sqrt i(\sqrt 2-1))=\frac34G+\frac18\pi\ln(\sqrt2-1)$

$\newcommand{\Li}{\operatorname{Li}_2}$ I found, numerically, that $$\Im\Li(\sqrt i(\sqrt 2-1))=\frac34G+\frac18\pi\ln(\sqrt2-1).$$ How can we prove it? My attempt of proving this equation: ...
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