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polylogarithm [2016/05/30 16:39] nikolaj |
polylogarithm [2016/05/30 16:41] nikolaj |
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| ===== Polylogarithm ===== | ===== Polylogarithm ===== | ||
| ==== Function ==== | ==== Function ==== | ||
| - | >todo | + | >Context: s |
| - | | @#FFBB00: definiendum | @#FFBB00: $ \mathrm{Li}_s: ?? \to ??$ | | + | | @#FFBB00: definiendum | @#FFBB00: $ \mathrm{Li}_s: {\mathbb C} \to {\mathbb C}$ | |
| | @#FFBB00: definiendum | @#FFBB00: $ \mathrm{Li}_s(z) := \begin{cases} \sum_{n=0}^\infty\, n^{-s} z^n&\hspace{.5cm} \mathrm{if}\hspace{.5cm} |z|<1,\hspace{.5cm} \\\\ \text{analytic continuation}\hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$ | | | @#FFBB00: definiendum | @#FFBB00: $ \mathrm{Li}_s(z) := \begin{cases} \sum_{n=0}^\infty\, n^{-s} z^n&\hspace{.5cm} \mathrm{if}\hspace{.5cm} |z|<1,\hspace{.5cm} \\\\ \text{analytic continuation}\hspace{.5cm} &\hspace{.5cm} \mathrm{else} \end{cases}$ | | ||
| - | "$\text{analytic continuation}$" | + | >todo "$\text{analytic continuation}$" |
| ----- | ----- | ||
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| This relates to the Bose-Einstein distribution where $z$ is the [[Fugacity]]. | This relates to the Bose-Einstein distribution where $z$ is the [[Fugacity]]. | ||
| - | == Special values == | + | == Relation to other functions == |
| $\zeta(s)=\lim_{z\to{1}}\mathrm{Li}_s(z)$ | $\zeta(s)=\lim_{z\to{1}}\mathrm{Li}_s(z)$ | ||
