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Tag Archives: functional
Dilogarithm difference formula proof
$$\mathrm{Li}_2(z) + \mathrm{Li}_2 \left(\frac{z}{z1} \right) = – \frac{1}{2} \log^2 (1z) \,\,\,\, \, z<1$$ $$ \textit{proof} $$ Start by the following $$\mathrm{Li}_2 \left(\frac{z}{z1} \right) = \int^{\frac{z}{z1}}_0 \frac{ \log(1t)}{t}\, dt$$ Differentiate both sides with respect to $z$ $$\frac{d}{dz}\mathrm{Li}_2 \left(\frac{z}{z1} \right) = \frac{1}{(z1)^2}\left( \frac{ \log … Continue reading
Dilogarithm functional equation proof
$$\mathrm{Li}_2(z) + \mathrm{Li}_{2}(1z) = \frac{\pi^2}{6}\log(z) \log(1z) \,\,\,\, ,\,0<z<1$$ $$\textit{proof}$$ Start by the following $$\mathrm{Li}_2\left(z\right) = \int^{z}_0 \frac{\log(1t)}{t} \, dt $$ Now integrate by parts to obtain $$\mathrm{Li}_2\left(z\right)= \int^z_0 \frac{\log(t)}{1t} \, dt \log(z) \log(1z) $$ By the change of variable \(t=1x … Continue reading
Posted in Dilogarithm, Polylogarithm
Tagged difference, dilogarithm, formula, functional, proof
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