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I'm a physicist looking at the Fredholm inverse of some integral equation. In attempting to solve the equation I stumbled upon a type of integral of the form \begin{equation} \int \frac{\prod_{i=1}^N \operatorname{Li}_{s_i}(x)}{x+\alpha} \mathrm dx. \end{equation}

At first glance the above may be integrated by elementary methods, using the relation \begin{equation} x\frac{\mathrm d\operatorname{Li}_s(x)}{\mathrm dx} = \operatorname{Li}_{s-1}(x). \end{equation}

However, in the above it spawns expressions with products of $N+1$ (!!) polylogarithms under the integral sign, after many partial integrations.

I ask this question hoping someone could point me to some reading material that would allow me to solve the above integral.

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  • Note that $$\mathrm{Li}{s}(z)=\int_0^z\mathrm{Li}{s-1}(t)\frac{\mathrm dt}t$$ which may help you to turn the product into the form $$\mathrm{Li}^N_{K}(z)$$ – clathratus Jan 08 '19 at 22:20

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