Difference between revisions of "Polylogarithm"
From specialfunctionswiki
(→Properties) |
|||
| (8 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
| − | The polylogarithm $\mathrm{Li}_s$ is defined by the formula | + | __NOTOC__ |
| + | The polylogarithm $\mathrm{Li}_s$ is defined by the formula for $|z|<1$ by | ||
$$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$ | $$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$ | ||
| + | A special case of the polylogarithm with $s=2$ is called a [[dilogarithm]]. | ||
<div align="center"> | <div align="center"> | ||
| Line 9: | Line 11: | ||
=Properties= | =Properties= | ||
| − | + | [[Lerch transcendent polylogarithm]]<br /> | |
| − | + | [[Legendre chi in terms of polylogarithm]]<br /> | |
| + | |||
| + | =Videos= | ||
| + | [https://www.youtube.com/watch?v=6v60ivoC2z8 polylogarithm function] | ||
| + | |||
| + | =References= | ||
| + | |||
| + | {{:Logarithm and friends footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 20:28, 25 June 2017
The polylogarithm $\mathrm{Li}_s$ is defined by the formula for $|z|<1$ by $$\mathrm{Li}_s(z) = \sum_{k=1}^{\infty} \dfrac{z^k}{k^s} = z + \dfrac{z^2}{2^s} + \dfrac{z^3}{3^s} + \ldots$$ A special case of the polylogarithm with $s=2$ is called a dilogarithm.
Various polylogarithms plotted on $[-2,1]$.
Properties
Lerch transcendent polylogarithm
Legendre chi in terms of polylogarithm
