Complex zeros of the modified Bessel function 𝐾_{𝑛}(𝑍)

Abstract

Abstract.The complex zeros of Kn(Z) are computed for integer orders n = 2(1)10, to 9D figures, using an iterative interpolation scheme. Introduction.The investigation of wave propagation and scattering in elastic media is often performed by means of integral transform methods. The analysis of such problems in cylindrical coordinates often leads to waves whose transformed potential functions are expressed in terms of modified Bessel functions. In particular, the potentials for outgoing radiating waves which decay with increasing distance from the source are expressed in terms of modified Bessel functions of the second kind, Kn(Z). In the course of a recent study using the Laplace transform, it was necessary to determine complex zeros of K£Z) in order to locate poles of the solution required for the inversion. It is believed that the tabulated zeros given below will permit the evaluation of several significant scattering problems.Several methods for the evaluation of zeros of Bessel functions, notably by means of asymptotic expansions, have been given by Olver [1] and Luke [2]. The methods developed by Olver, however, are not entirely applicable in the present case, since the convergence only improves with large orders of n. On the other hand, the rational approximations given by Luke have been proved, under appropriate restrictions of the parameters, to converge in the first quadrant; convergence for |arg Z\ < it has recently been established by Fields [3]. It should be observed that Cochran and Hoffspiegel [4] have extended the work of Olver for determining the positive real or purely imaginary zeros of Hankel functions with respect to noninteger orders when the variable is held fixed. However, their methods are inapplicable, since the zeros of the modified Bessel functions are seen to be always complex. A method for the evaluation of complex zeros of cylinder functions was given by Döring [5] based on MacMahon and Olver expansions. Tabulated values were given for some zeros of Hankel functions Hn(Z) which can be related to those of Kn(Z).The zeros presented below to 9D significant figures were calculated according to a method described in the following section.

Characteristic approach to the soliton resolution

“…We remark that in this case the analysis based on recurrence relation is purely academic because the eigenvalue equation can be solved exactly and the outgoing solution is given by v out = y − 1 2 e s/y K 2 (s/y), where K 2 (z) is the modified Bessel function of the second kind. Thus, the quantization condition for the quasinormal modes is K 2 (s) = 0, which has exactly one pair of complex conjugate zeros on the principal branch s = −1.281 373 ± 0.429 4849i [19]. We verified that the roots of C + (s) are the same, which provides a reassuring benchmark test for Leaver's method.…”

An Application of the Finite Element Approximation Method to Find the Complex Zeros of the Modified Bessel Function K n (z)

“…Other methods have also been employed successfully [12]. Olver [10] evaluated zeros of Bessel functions of large order using uniform asymptotic expansions.…”

“…By tracing the argument of Kn(z) along the four sides of a square, and using a double linear interpolation, Parnes [12] has given an iterative scheme to obtain the complex z zeros of Kn(z) for fixed integer orders n = 2, 3, . .…”