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We show how one can obtain an asymptotic expression for some specialfunctions satisfying a second order differential equation with a very expliciterror term starting from appropriate upper bounds.  We will work out the details for the Bessel function $J_\nu (x)$ and the Airyfunction $Ai(x)$ 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 $c_2.$

Some asymptotics for the Bessel functions with an explicit error term