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Bessel functions of purely imaginary order, with an application to second-order linear differential equations having a large parameter. (English) Zbl 0703.33002

The main objective of the author is to give some formulas of practical importance connected with certain particular solutions of Bessel’s equation. The particular solutions considered in the paper are defined in terms of classical ones as follows: ${F}_{\mu }\left(z\right)=\frac{1}{2}\left\{{e}^{\mu \pi i/2}{H}_{\mu }^{\left(1\right)}\left(z\right)+{e}^{-\mu \pi i/2}{H}_{\mu }^{\left(2\right)}\left(z\right)\right\},{G}_{\mu }\left(z\right)=\frac{1}{2i}\left\{{e}^{\mu \pi i/2}{H}_{\mu }^{\left(1\right)}\left(z\right)-{e}^{-\mu \pi i/2}{H}_{\mu }^{\left(2\right)}\left(z\right)\right\},{L}_{\mu }\left(z\right)=\frac{\pi i}{2sin\mu \pi }\left\{{I}_{\mu }\left(z\right)+{I}_{-\mu }\left(z\right)\right\},\phantom{\rule{1.em}{0ex}}\left(\mu \ne 0\right)·$For these functions recurrence relations, analytic continuation formulas, power series and integral representations, asymptotic expressions for the zeros as well as some uniformly valid asymptotic expansions are derived. The derivations are achieved, in general, by a direct use of known results related to the classical solutions. A particular emphasis is given to the case of purely imaginary order where $\mu =i\nu$ with positive $\nu$. By using these latter a rather general equation of the form

${w}^{\text{'}\text{'}}=\left\{\frac{{u}^{2}}{4z}-\frac{{v}^{2}+1}{4{z}^{2}}+\frac{\psi \left(z\right)}{z}\right\}w$

is examined for large u.

Reviewer: M.Idemen
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 34E20 Asymptotic singular perturbations, turning point theory, WKB methods (ODE)
Bessel equation