On the zeros of Jn(z)±iJn+1(z) and [Jn+1(z)]2−Jn(z)Jn+2(z)

摘要

The zeros of Jn(z)±iJn+1(z) and [Jn+1(z)]2−Jn(z)Jn+2(z) play an important role in certain physical applications. At the origin these functions have a zero of multiplicity n (if n⩾1) and 2n+2, respectively. We prove that all the zeros that lie in C0 are simple. zebec (Kravanja et al., Comput. Phys. Commun. 113(2–3) (1998) 220–238) is a reliable software package for calculating zeros of Bessel functions of the first, the second, or the third kind, or their first derivatives. It can be easily extended to calculate zeros of any analytic function, provided that the zeros are known to be simple. Thus, zebec is the package of choice to calculate the zeros of Jn(z)±iJn+1(z) or [Jn+1(z)]2−Jn(z)Jn+2(z). We tabulate the first 30 zeros of J5(z)−iJ6(z) and J10(z)−iJ11(z) that lie in the fourth quadrant as computed by zebec.