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

Abstract

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 ${\lim }_{\nu \to n}\{I_{\nu -1/2}(x)-{\mathbf{L}}_{1/2-\nu }(x)\}/\sin (\pi \nu )$, for $n\in {\mathbb{Z}}^{+}$, $x>0$.

7 pages