Tag Archives: Multiple

Multiple integral related to Euler sums

Posted by Cornel Ioan Valean on Facebook Show that $$\small \int^1_0 \cdots \int^1_0 \frac{\mathrm{d}\varphi_1\cdots \mathrm{d}\varphi_5}{(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)} = \frac{59}{32}\zeta(5)-\frac{1}{2}\zeta(2)\zeta(3)$$ $$\textit{proof}$$ Consider the integral $$I = \int^1_0 \cdots \int^1_0 \frac{\mathrm{d}\varphi_1\cdots \mathrm{d}\varphi_5}{(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)(\varphi_1+\varphi_2+\varphi_3+\varphi_4+\varphi_5+1)}$$ Use the series expansion $$I = \sum_{n=0}^\infty \sum_{k=0}^\infty \frac{(-1)^{n+k}}{n+1} \int^1_0\cdots \int^1_0 (\varphi_1\cdots … Continue reading

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