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On the computation of zeroes of ${J}_{n}\left(z\right)-i{J}_{n+1}\left(z\right)=0$. (English) Zbl 0905.33001

This paper is devoted to the study of certain methods for finding the zeros of the equation

${J}_{n}\left(z\right)-i{J}_{n+1}\left(z\right)=0·$

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.

##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$ 41A60 Asymptotic approximations, asymptotic expansions (steepest descent, etc.)