On the zeros of Kummer function: Basic properties and their application in the theory of the azimuthally magnetized circular ferrite waveguide

A numerical evaluation of the first several purely imaginary (real) zeros ζ_k,n^(c) in x, resp. in z (ζ̂_k̂,n̂^(c) in x̂) of the complex (real) Kummer confluent hypergeometric function Φ(a,c;x), [Φ(â,ĉ;x̂)] is performed, assuming a = c / 2 - jk, c = 1 and c = 3, x = jz, k, z - real, -∞ <; k <; +∞, z > 0 (â = ĉ/2 + k̂, ĉ=1 and ĉ = 3, k̂, x̂ - real, -∞ <; k̂ <; -ĉ / 2, x̂ > 0). The results of analysis are presented in a tabular form, depending on the imaginary part k of parameter a of Φ̂̂(a,c; x) [on the addition k̂ in parameter â of Φ̂(â,ĉ;x̂)]. Momentous characteristics of quantities studied, concerning the number n (n̂) of zeros of the complex (real) function, their specification and their behaviour in case k → -∞ (k̂→-∞), are established. Some related numbers, called L₁(c,n) [L̂₁(ĉ,n̂)] ones and their features, are also considered. The employment of zeros ζ_k,n^(c) and ζ̂_k̂,n̂^(ĉ), and of the derivative of them quantities L₁(c,n) and L̂₁(ĉ,n̂) in the theory of waveguides, is discussed, as well.