Geodesic
Asymptotic ratios of Bessel functions of purely imaginary argument
Applications of Mathematics, Tome 19 (1974) no. 1, pp. 1-6

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The Riccati equations as well as some interesting inequalities for the ratios of Bessel functions of purely imaginary argument $T_p(x;1)=\frac {K_p(x)}{xK_{p+1}(x)}$ and $T_p(x;-1)=\frac {I_p(x)}{xI_{p-1}(x)}$ are derived. Solutions of the Riccati equations are given in terms of power series in $p^{-1}$. In particular, the asymptotic formulae for $T_p(x;\pm 1)$ with a remainder of order $O(p^{-5})$ are obtained. For $p$ small, they represent an asymptotic expansion in $x$ up to the order $O(x^{-5})$.
The Riccati equations as well as some interesting inequalities for the ratios of Bessel functions of purely imaginary argument $T_p(x;1)=\frac {K_p(x)}{xK_{p+1}(x)}$ and $T_p(x;-1)=\frac {I_p(x)}{xI_{p-1}(x)}$ are derived. Solutions of the Riccati equations are given in terms of power series in $p^{-1}$. In particular, the asymptotic formulae for $T_p(x;\pm 1)$ with a remainder of order $O(p^{-5})$ are obtained. For $p$ small, they represent an asymptotic expansion in $x$ up to the order $O(x^{-5})$.
DOI : 10.21136/AM.1974.103508
Classification : 33A40, 33C10 
Trlifaj, Ladislav. Asymptotic ratios of Bessel functions of purely imaginary argument. Applications of Mathematics, Tome 19 (1974) no. 1, pp. 1-6. doi: 10.21136/AM.1974.103508
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