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Zbl 0905.33001
MacDonald, D.A.
On the computation of zeroes of $J_n(z)-iJ_{n+1}(z)=0$. (English)
[J] Q. Appl. Math. 55, No.4, 623-633 (1997). ISSN 0033-569X; ISSN 1552-4485/e

This paper is devoted to the study of certain methods for finding the zeros of the equation $$J_n(z) -iJ_{n+1} (z)=0.$$ Three methods, one analytic and two numerical, are discussed. The analytic method utilizes known series expansion which in turn is used to obtain asymptotic formulas for the roots in question. Second method allows the user to approximate those zeros using the Newton-Raphson iteration. The last method is derived from the ascending series representation for the modified Bessel functions of the first kind. Numerical results are included.
[E.Neuman (Carbondale)]

MSC 2000:: *33C10 Cylinder functions, etc.
41A60 Asymptotic problems in approximation

Keywords: asymptotic expansions; Newton-Raphson iteration; Bessel functions

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