Difference between revisions of "Legendre chi in terms of polylogarithm"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\chi_{\nu}(z)=\dfrac{1}{2}[\mathrm{Li}_{\nu}(z)-\mathrm{Li}_{\nu}(-z)] = \mathrm{Li}_{\nu}(z)-2^{-\nu}\mathrm{Li}_{\nu}(z^2),$$ | $$\chi_{\nu}(z)=\dfrac{1}{2}[\mathrm{Li}_{\nu}(z)-\mathrm{Li}_{\nu}(-z)] = \mathrm{Li}_{\nu}(z)-2^{-\nu}\mathrm{Li}_{\nu}(z^2),$$ | ||
where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{Li}_{\nu}$ denotes the [[polylogarithm]]. | where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{Li}_{\nu}$ denotes the [[polylogarithm]]. | ||
| − | + | ||
| − | + | ==Proof== | |
| − | + | ||
| − | + | ==References== | |
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 07:59, 8 June 2016
Theorem
The following formula holds: $$\chi_{\nu}(z)=\dfrac{1}{2}[\mathrm{Li}_{\nu}(z)-\mathrm{Li}_{\nu}(-z)] = \mathrm{Li}_{\nu}(z)-2^{-\nu}\mathrm{Li}_{\nu}(z^2),$$ where $\chi$ denotes the Legendre chi function and $\mathrm{Li}_{\nu}$ denotes the polylogarithm.
