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Approximations for the Bessel and Airy functions with an explicit error term
Part of:
Approximations and expansions
Published online by Cambridge University Press: 01 May 2014
Abstract
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We show how one can obtain an asymptotic expression for some special functions 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 ${\rm Ai}(x).$ In particular, we answer a question raised by Olenko and find a sharp bound on the difference between $J_\nu (x)$ and its standard asymptotics. We also give a very simple and surprisingly precise approximation for the zeros ${\rm Ai}(x).$
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- © The Author 2014
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