On the characteristic function of the asymmetric Student's <i>t</i>-distribution and an integral involving the sine function

We obtain a new closed-form formula for the characteristic function of the asymmetric Student's <i>t-</i>distribution. As part of our analysis, we derive a new closed-form formula for the integral<i> ∫</i>₀^∞sin(<i>ax</i>)/(<i>b²</i>+ <i>x</i>²)ⁿd<i>x</i>, for a, b &gt; 0,<i> n</i> € 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_{<i> v→n</i>} {<i style="">I_v-</i>_1/2(x) -<b>L</b> _1/2-V(x)} / sin(Πν), for n ε Z⁺, x &gt; 0.<i style="height: 1px; width: 1px; opacity: 0.001;"/>