Some asymptotics for the Bessel functions with an explicit error term

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 $J_ν(x)$ 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 $c_1 | ν^2-1/4\,| < \sup_{x \ge 0} x^{3/2}|J_ν(x)-\sqrt{\frac{2}{πx}} \, \cos (x-\frac{πν}{2}-\fracπ{4}\,)| <c_2 |ν^2-1/4\,|,$ $ν\ge -1/2 \, ,$ for some explicit numerical constants $c_1$ and $c_2.$

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