Contribution to the theory of A1 numbers

A numerical modeling of the real positive numbers A₁ is performed, determined by means of the relation A₁=A_1--A₁₊ in which A_1± = A_1±(c,|α|, r̅₀, n) are also such numbers, defined through the positive purely imaginary zeros ζ_k,n^(c) of the complex Kummer confluent hypergeometric function Φ(a,c;x) with a = c/2-jk-complex, c - positive integer, x = jz - positive purely imaginary, k, z - real, -∞<;k<;+∞, z>;0, (|α|, r₀-real, positive, n = 1, 2, 3, ... - number of the zero). The parameters |α|<;1 and r̅₀ >; ζ_0,n^(c)/2 are subject to the condition ζ_0,n^(c)/2 <; r̅₀√1-α² <; L₁(c, n)/|α| in which the quantities L₁(c, n) are the common real positive limits of the sequences of numbers {|k|ζ_k,n^(c)} and {|a|ζ_k,n^(c)}, attained at k→-∞. It is assumed that c = 3 and n = 1 for which ζ_k,n^(c) = 7.66341 19404 and L₁(c, n)= 6.59365 41068. The computations are made for a set of equidistant numerical equivalents of the {|α|, r̅₀} - pairs, harnessing a specially developed iterative scheme. The discussion is focused on the values of r̅₀ >; 2 L₁(c, n). The results are presented in a tabular form. They complement previous ones, conforming to the interval ζ_{0, n}^(c)/2 <;2L₁(c, n). The analysis discloses that in the case considered the area of existence of the A₁ numbers splits into two parts, relevant to |α|<;1/√2 and |α|>;1/√2. The joint consideration of the ne- and earlier outcomes reveals that for any fixed ζ_{0, n}^(c)/2 <; r̅₀ <; + ∞, wherever they are available, the quantities A₁ remain practically constant with regard to |α|. Simultaneously, for all |α|´s, irrespective of their particular values, A₁ diminish when r̅₀ grows, following approximately functions of the kind f(x) = F/r̅₀ with almost identical factors F. Interpreting r̅₀ as the normalized in an appropriate way radius of the circular ferrite waveguide, magnetized in azimuthal direction to remanence and |α| as the modulus of off-diagonal element of the permeability tensor of its anisotropic filling, it is shown that the formula Δβ̅=A₁|α| makes possible to figure up the pertinent to a given {|α|, r̅₀}-pair normalized differential phase shift, provided by the structure for the normal TE₀₁ mode, using as a coefficient the A₁ number, relevant to the same pair.