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Asymptotic approximations for modified Bessel functions

The behavior of a q ν(x) =I ν(x)/I ν a (x), where I ν is a modified Bessel function with integral or half‐integral index ν and I ν a the leading term of its asymptotic series, is investigated for x≫1. It is shown that q ν(x) may be approximated by e ν(x) =exp(−ν2/2x), the difference r ν(x) =q ν(x)−e ν(x) being of order x −1/4. Bounds for r ν(x) depending only on x are derived for each of the two classes of ν’s and an application of these results in scattering theory is indicated.