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In an answer to a question I needed the following integral: $$ f(z):=\int\limits_0^\infty t\coth(zt)e^{-t^2}dt; $$ it represented deviation from modularity of some other function. However I noticed ...

This question comes from physics... We have the following functional (or function dose not matter): $$S\left[x\right]:=\intop_{-\frac{t_{0}}{2}}^{+\frac{t_{0}}{2}}dt\,\mathcal{L}\left[x\right]\::\:{\...