Questions tagged [polylogarithm]
For questions about or related to polylogarithm functions.
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Contour Integral involving Dilogarithmic functions
I am considering the contour integral:
$\int Li_2\left( \frac{1-z}{2}\right)Li_2\left( \frac{z-1}{2z}\right) \frac{dz}{z}$.
The contour of integration is the unit circle excluding the pole $z = 0$. $...
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A dilogarithm identity?
I'm wondering whether there any nice identities (or relationships) that can simplify or possibly compact the following expressions:
$$\operatorname{Li}_2(\beta e^{\alpha x}) - \operatorname{Li}_2(\...
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Evaluate a certain one-dimensional integral involving inverse trigonometric functions
Demonstrate that the integral of
\begin{equation}
\cos (y) \left(\sqrt{4-\sin ^2(y)} \cos ^{-1}(\sin (y))+4 \cos (y) \csc ^{-1}(2 \csc
(y))\right)
\end{equation}
over $y \in [0,\frac{\pi}{2}]$ ...
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Definition of polylog for all complex $\text{z}$
What are the different definitions of:
$$\text{Li}_\text{n}\left(\text{z}\right)$$
For all possible $\text{z}$???
I know that when $\left|\text{z}\right|<1$ it is defined by:
$$\text{Li}_\text{...
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Evaluating Fermi Dirac integrals of order j<0
The complete Fermi Dirac integral (I'm purposely leaving off the gamma prefactor)...
$$F_j(x) =\int\limits_{0}^{\infty} \frac{t^j}{e^{t-x}+1} \: dt$$
is generally defined for j>-1. Is there a way to ...
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Understanding limits for the polylogarithmic function
Wikipedia lists 7 different limits for the polylogarithmic function which can be found here. For example, the fourth and sixth listed limits state that
$$\lim_{\mathrm{Re}(z) \rightarrow \infty} \...
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Do all complex zeros of $Li_s(z)\,- \, Li_{1-s}(z)$ get the shape $s=\dfrac12 + \dfrac{k \, \pi }{\,\ln(2)}\,i$ when $z \rightarrow 0^{-}$?
This is a subquestion of this question on MO.
Numerical evidence strongly suggests that when $z \rightarrow 0^{-}$ the complex zeros that lie in the critical strip $0 \lt \Re(s) < 1$ of:
$$Li_s(z)...
- 3,191
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intersection between exponential and polylogarithmic functions
It's possible to solve this equation without using Lambert function or any numerical method, but only with ordinary algebra?
$n^{k}lg_2(n) \le k^n$ with $k,n>0, k \in \mathbb{R}$
For $k=\frac{4}{...
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