Difference between revisions of "Polylogarithm"
From specialfunctionswiki
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$$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$ | $$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$ | ||
| − | + | <div align="center"> | |
| + | <gallery> | ||
| + | File:Polylog.png|Various polylogarithms plotted on $[-2,1]$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | {{:Lerch transcendent polylogarithm}} | ||
Revision as of 01:10, 21 March 2015
The polylogarithm $\mathrm{Li}_s$ is defined by the formula $$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$
Various polylogarithms plotted on $[-2,1]$.
Contents
Properties
Theorem
The following formula holds: $$\Phi(z,n,1)=\dfrac{\mathrm{Li}_n(z)}{z},$$ where $\Phi$ denotes the Lerch transcendent and $\mathrm{Li_n}$ denotes the polylogarithm.
