We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function $J_\nu (x)$ and the Airy function $Ai(x)$ and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that $$c_1 | \nu^2-1/4\,| < \sup_{ x \ge 0} x^{ 3/2}|J_\nu(x)-\sqrt{ \frac{ 2}{ \pi x}} \, \cos (x-\frac{ \pi \nu}{ 2}-\frac{ \pi}{ 4}\,)| <c_2 |\nu^2-1/4\,|, $$ $ \nu \ge -1/2 \, ,$ for some explicit numerical constants $c_1$ and $c_2.$
Categories: math.CA