# Questions tagged [polylogarithm]

For questions about or related to polylogarithm functions.

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### Generalised Polylogarithm Polynomials and Related Integer Sequences

Consider the generalised infinite summation $$S_{n,m}=m^{n+1} \sum_{k=1}^\infty \frac{k^n}{(m+1)^k}=m^{n+1}\,\mathrm{Li}_{(-n)} \left(\frac{1}{m+1}\right)$$ where $m$ and $n$ are positive integers, ...
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### $\int\frac{\ln^kx\ln(1-x)}x dx$ Vs. $\int\frac{\ln x\ln^k(1-x)}x dx$

By using WolframAlpha, experimentely I observed that $$\frac1{k!}\int\frac{\ln^kx\ln(1-x)}x dx=(-1)^{k+1}\text{Li}_{k+2}(x)+\sum_{i=1}^k \frac{(-1)^{k+1-i}}{i!} \text{Li}_{k+2-i}(x)\ln^{i}x+c.$$ On ...
1 vote
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### asymptotic behaviour of polylogarithmic function

I would like to understand the asymptotic behaviour as $a \rightarrow 0$ of the function $$f(a) := \sum\limits_{k=2}^{\infty} e^{ - a^2 k}{k^{-3/2}}$$ More precisely, I would like to obtain an ...
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As you can see, here I performed a derivation of a quite simple formula, not much differing from the standard integral representation of the Polylogarithm. Seeking to make it fancier, I arrived at ...
63 views

### Evaluating an integral from 0 to 1 with a parameter, (and a dilogarithm)

So I need to evaluate the following integral (in terms of a): $$\int_{0}^{1} \frac{\ln{|1-\frac{y}{a}|}}{y} dy$$ Till now I have tried u-sub ($u = \ln{|1-\frac{y}{a}|}$, $u=\frac{y}{a}$) and ...
193 views

### A cool integral: $\int^{\ln{\phi}}_{0}\ln\left(e^{x}-e^{-x}\right)dx=-\frac{\pi^2}{20}$

I was looking at the equation $\ln{e^{x}-e^{-x}}$ and found that the zero was at $x=\ln{\phi}$ where $\phi$ is the golden ratio. I thought that was pretty cool so I attempted to find the integral. I ...
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### Calculate $\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$

this integral got posted on a mathematics group by a friend $$I=\int _0^1\frac{\arcsin ^2\left(x\right)\ln \left(x\right)\ln \left(1-x\right)}{x}\:\mathrm{d}x$$ I tried seeing what I'd get from ...
1 vote