오늘의 계산 39 - Abel's Integral

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.