Difference between revisions of "Legendre chi in terms of polylogarithm"
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<strong>[[Legendre chi in terms of polylogarithm|Proposition]]:</strong> The following formula holds: | <strong>[[Legendre chi in terms of polylogarithm|Proposition]]:</strong> 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 $\mathrm{Li}$ denotes the [[polylogarithm]]. | + | where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{Li}_{\nu}$ denotes the [[polylogarithm]]. |
<div class="mw-collapsible-content"> | <div class="mw-collapsible-content"> | ||
<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
</div> | </div> | ||
</div> | </div> | ||
Revision as of 01:27, 21 March 2015
Proposition: 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.
Proof: █
