Some asymptotics for the Bessel functions with an explicit error term

Abstract

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 $$c_1|{\nu }^2-1/4|<\underset{x\ge 0}{\sup }{x}^{3/2}|J_{\nu }(x)-\sqrt{\frac{2}{\pi x}}\cos (x-\frac{\pi \nu }{2}-\frac{\pi }{4})|<c_2|{\nu }^2-1/4|,$$ $\nu \ge -1/2,$ for some explicit numerical constants $c_1$ and $c_2.$

Typos corrected