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On the complex zeros of H μ (z), J μ ' (z), J μ '' (z) for real or complex order. (English) Zbl 0756.33001

The authors establish some propositions about the nonexistence of complex zeros of the functions H μ (z), J μ ' (z) and J μ '' (z), for μ in general complex. Some bounds for the purely imaginary zeros of the above functions are also obtained assuming their existence. These bounds for the purely imaginary zeros of H μ (z) are as follows:

ρ 2 >-2(μ 1 +1)|α+μ| 2 /(μ 1 +α 1 ),-1<μ 1 <-α 1 ,|ρ 2 |>(|μ+α||μ+α| 2 +2μ 2 (μ 2 +α 2 )-|μ+α| 2 )/|μ 2 +α 2 |andμ 2 >max{0,-α 2 }orμ 2 <min{0,-α 2 },

where ρ 1 , ρ 2 , μ 1 , μ 2 , α 1 , α 2 are the real and imaginary parts of ρ, μ and α. The authors use these bounds to measure the purely imaginary zeros of J μ '' (z).

The results proved by the authors generalize some of the results given earlier by E. K. Ifantis, P. D. Siafarikas and C. B. Kouris [J. Math. Anal. Appl. 104, 454-466 (1984; Zbl 0558.34006)].

MSC:
33C10Bessel and Airy functions, cylinder functions, 0 F 1