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I am interested in computing the integral of this function: \begin{align} \int_0^\infty\frac{2du(u^2+1)}{(1-e^{2\pi u})}, \end{align} which of course at first sight, does not converge. But in QFT it's usually possible to regulate such a function. Thus, the question is, does anyone know how to regulate this function?

The question is unanswerable, because there is no physical context. In quantum field theory, you regulate so as to define the continuum quantum field theory as a limit, say of lattices. Here it is not clear what the regulating structure is supposed to be, because you didn't specify the physical system it is describing. An obvious answer is simply to change the lower limit to epsilon, and subtract $\log(\epsilon)/\pi$, but without a proper context, you can't say whether this makes sense.

is there an i missing in the exponential term?

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