On the characteristic function of the asymmetric Student's $t$-distribution and an integral involving the sine function

We obtain a new closed-form formula for the characteristic function of the asymmetric Student's $t$-distribution. As part of our analysis, we derive a new closed-form formula for the integral $\int_0^\infty \sin(ax)/(b^2+x^2)^n\,\mathrm{d}x$, for $a,b>0$, $n\in\mathbb{Z}^+$, expressed in terms of the exponential integral function. As a consequence of our integral formula, we deduce a closed-form formula for the limit $\lim_{ν\rightarrow n} \{I_{ν-1/2}(x)-\mathbf{L}_{1/2-ν}(x)\}/\sin(πν)$, for $n\in\mathbb{Z}^+$, $x>0$.