# Questions tagged [polylogarithm]

For questions about or related to polylogarithm functions.

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### Why is $B_{2n}(\frac12+ix)\in\mathbb R$ whenever $x\in\mathbb R$?

I just noticed that $B_{2n}(\frac12+ix)\in\mathbb R$, where: $x\in\mathbb R$, $n\in\mathbb N$, and $B_n(x)$ is the $n$th Bernoulli Polynomial. Why? Is there a simple, slick proof? Does it follow from ...
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### Canonical reference for algebraic theory of polylogs?

I've noticed that there are lots of similar-looking (at least to my untrained eye) questions on Math Stackexchange involving polylogarithms, all equally delicious, in which the OP asks about how to ...
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### Evaluating improper integral $\int_0^1 \frac{\log(x)}{x+\alpha}\; dx$ for small positive $\alpha$

Let $\alpha$ be a small positive real number. How do I obtain $$I = \int_0^1 \frac{\log(x)}{x+\alpha}\; dx = -\frac{1}{2}(\log\alpha)^2 - \frac{\pi^2}{6} - \operatorname{Li}_2(-\alpha)$$? Maxima told ...
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### Can this formula for $\zeta(3)$ be proven or simplified further?

This question is related to the equivalence of formulas (1) and (2) below where formula (1) is from a post on the Harmonic Series Facebook group and formula (2) is based on evaluation of the integral ...
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### Proving $\int_0^{1/2}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3)$

By comparing some results, I found that $$\int_0^{\frac12}\frac{\text{Li}_2(-x)}{1-x}dx=-\text{Li}_3\left(-\frac12\right)-\frac{13}{24}\zeta(3).\tag{1}$$ I tried to prove it starting with applying IBP:...
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### When are $\sqrt{-\log x}^{1+\varepsilon}\operatorname{Li}_{1/2}(x)$ and $\sqrt{-\log x}\operatorname{Li}_\nu(x)$ strictly increasing?

[Cross-posted on MathOverflow] This problem (posed by a Discord user) originates from the study of the fire-diffuse-fire model of calcium dynamics. Briefly, I was told that the goal was to prove there ...
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### Is the closed form of $\int_0^1 \frac{x\ln^a(1+x)}{1+x^2}dx$ known in the literature?

We know how hard these integrals $$\int_0^1 \frac{x\ln(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^2(1+x)}{1+x^2}dx; \int_0^1 \frac{x\ln^3(1+x)}{1+x^2}dx; ...$$ can be. So I decided to come up with a ...
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### Evaluating $\int_0^\infty\frac{\tan^{-1}av\cot^{-1}av}{1+v^2}\,dv$

The Weierstrass substitution stuck in my head after I used it to prove the rigidity of the braced hendecagon (and tridecagon). Thus I had another look at this question which I eventually answered in a ...
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### Prove $\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$

I managed here to prove $$\int_0^1\frac{\text{Li}_2(-x^2)}{\sqrt{1-x^2}}\,dx=\pi\int_0^1\frac{\ln\left(\frac{2}{1+\sqrt{1+x}}\right)}{x}\,dx$$ but what I did was converting the LHS integral to a ...
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### Closed form evaluation of a trigonometric integral in terms of polylogarithms

Define the function $\mathcal{K}:\mathbb{R}\times\mathbb{R}\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\times\left(-\frac{\pi}{2},\frac{\pi}{2}\right)\rightarrow\mathbb{R}$ via the definite ...
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### evaluate $\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$

I came across this integral: $$\int_{0}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx=\frac{1}{2}\int_{-1}^{1}\frac{\text{Li}_2(\frac{x^2-1}{4})}{1-x^2}dx$$ One way to evaluate is to start with the ...
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