Some asymptotics for the Bessel functions with an explicit error term

Details Some asymptotics for the Bessel functions with an explicit error term Some asymptotics for the Bessel functions with an explicit error term

2011-07-11

arxiv.org/abs/1107.2007v2

Mathematics

Classical Analysis and ODEs

Typos corrected

Scientific paper

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}\,)|

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