Today's integral is so called Abel's integral, which is given by
Problem. \begin{equation} \label{eq:prob} \int_{0}^{\infty} \frac{t \; dt}{(e^{\pi t} - e^{-\pi t})(t^2 + 1)} \end{equation}
Now our aim is to evaluate \eqref{eq:prob} in closed form. We preliminarily introduce a function that will play a key role throughout the calculation. Let $k$ be a nonnegative integer. Then we define polygamma of degree $k$ by
where $\ln \Gamma(s)$ is the logarithmic gamma function. Then it is not hard to confirm that
Also, Euler's reflexion formula yields
Now we begin the calculation.
Solution. let $\alpha$, $\beta$ be positive real numbers and we define
Then we find that
Especially, if we plug $\beta = 1/2$, then we obtain the representation of $I\left(\alpha, \frac{1}{2}\right)$ in terms of polygamma functions:
Finally, we can put $\alpha = 2 \pi$ to obtain the value of Abel's integral.
Posted by aficionado

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