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prove-that-0-0-log-1-e-x-yLi-2-e-x-y-Li-3-e-x-y-1-e-x-y-e-x-y-dxdy-21-8-6-2-3-by-MATH-AMIN-




Question Number 154208 by amin96 last updated on 15/Sep/21
∵∴∵∴prove that  ∫_0 ^∞ ∫_0 ^∞ ((log(1−e^(−x) )(yLi_2 (e^(−x−y) )+Li_3 (e^(−x−y) ))/(1−e^(x+y) ))e^(x+y) dxdy=((21)/8)ζ(6)+ζ^2 (3)  ∵∴∵by MATH.AMIN∴∵∴
$$\because\therefore\because\therefore{prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{{log}\left(\mathrm{1}−{e}^{−{x}} \right)\left({yLi}_{\mathrm{2}} \left({e}^{−{x}−{y}} \right)+{Li}_{\mathrm{3}} \left({e}^{−{x}−{y}} \right)\right.}{\mathrm{1}−{e}^{{x}+{y}} }{e}^{{x}+{y}} {dxdy}=\frac{\mathrm{21}}{\mathrm{8}}\zeta\left(\mathrm{6}\right)+\zeta^{\mathrm{2}} \left(\mathrm{3}\right) \\ $$$$\because\therefore\because\boldsymbol{\mathrm{by}}\:\boldsymbol{\mathrm{MATH}}.\boldsymbol{\mathrm{AMIN}}\therefore\because\therefore \\ $$

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