Difference between revisions of "Legendre chi in terms of polylogarithm"

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==Theorem==
<strong>[[Legendre chi in terms of polylogarithm|Proposition]]:</strong> The following formula holds:
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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]].
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where $\chi$ denotes the [[Legendre chi]] function and $\mathrm{Li}_{\nu}$ denotes the [[polylogarithm]].
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<strong>Proof:</strong> █
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==Proof==
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==References==
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[[Category:Theorem]]
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[[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.

Proof

References