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This integral plays a central role in a physics problem (Casimir effect)${}^\ast$ $$\Omega(\phi,L)=-\frac{1}{\pi}\operatorname{Re}\int_0^\infty \ln\bigl[1+\beta(\omega)^2 e^{i\phi-2\omega L}\bigr]\,d\...

In a quantum mechanical problem I encountered the integral $$I_k=\int_0^\infty x^{2(l+1)-k}j_k(\sigma x)e^{-x}[L_{n-l-1}^{2l+1}(x)]^2 dx,$$ where $j_k(x)$ is a spherical Bessel function, and $\sigma$ ...

I've encountered the following product, \begin{equation} \, _2F_1\left(3 d+2,3 d+2;6 d+4;1-\frac{1}{t^2}\right) \, _3F_2\left(-\frac{d}{2},\frac{d}{2},d;\frac{d}{2}+1,\frac{3 d}{2}+1;t^2\right) \...

In a quantum-information-theoretic context, I've encountered the problem of integrating over $r \in [0,1]$, the function \begin{equation} r^{2 d-1} \, _2F_1\left(-\frac{d}{2},\frac{d}{2};\frac{d+2}{2}...

Consider the following series: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + m^2}/2)}{\sinh( \pi \sqrt{n^2 + m^2})} $$ From the physical ...