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Zbl 0603.65029
Kerimov, M.K.; Skorokhodov, S.L.
On computation of complex zeros of Hankel functions and their derivatives. (Russian)
[J] Zh. Vychisl. Mat. Mat. Fiz. 25, No.11, 1628-1643 (1985). ISSN 0044-4669

As a continuation of their previous papers [ibid. 24, No.5, 650-664 (1984; Zbl 0568.65010); No.10, 1497-1513 (1984; Zbl 0561.65011)], the authors discuss the computation of complex zeros of the Hankel functions $H\sb{\nu}\sp{(1)}(z)$ and $H\sb{\nu}\sp{(2)}(z)$ with real index $\nu$. The method is due to Newton's iteration, starting from a suitable asymptotic approximate value. First they discuss on the complex zeros of $K\sb{\nu}(z)$, and give several formulas concerning these functions. In the computation of their derivatives, they use asymptotic formulas containing Ai-function, and they remark the relation $Ai(z)=\pi\sp{- 1}(z/3)\sp{1/2}$ $K\sb{1/3}(2z\sp{3/2}/3)$. They give tables of first 50 zeros of $H\sb 0\sp{(1)}(z)$ and $H\sb 1\sp{(1)}(z)$.
[S.Hitotumatu]

MSC 2000:: *65H05 Single nonlinear equations (numerical methods)
65D20 Computation of special functions
33C10 Cylinder functions, etc.

Keywords: complex zeros; Hankel functions; Newton's iteration; asymptotic formulas; Ai-function

Citations: Zbl 0568.65010; Zbl 0561.65011

Cited in: Zbl 0614.65055

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