Some Asymptotic Expansions for an Incomplete Probability Integral

For the D. Owen function
$$ T(h,a)=\frac{1}{2\pi}\int_0^a e^{-\frac{h^2}{2}(1+x^2)}\frac{dx}{1+x^2} $$
asymptotic expansions are derived in the cases 1) $h\to\infty$, $a\to 1$, 2) $h\to\infty$, $a\to 0$. Numerical computations by the formulas obtained are given. A correspondence between $T(h,a)$ and an incomplete probability integral is established.