Advances in the Theory of A, B, C Numbers and Its Application to Waveguide Propagation

摘要

Harnessing a recently elaborated iterative technique, the dependence of the real positive numbers A_1, B_1, C_1, defined earlier by the positive purely imaginary zeros ζ_(k,n)~((c)) in x (n = 1,2,3,...) of the Kummer function Φ(a, c; x), (a = c/2 - jk - complex, c = 3, x = jz, k, z - real, -∞ < k < +∞, z > 0), on the real positive parameters |α| and fo (0 < |α| < 1, r/-_0 > ζ_(0,n)~((c))/2), satisfying certain criterion, is analyzed numerically, stipulating that n = 1 and r/-_0 ≥ 10. The outcomes are pictured in a graphical form. A second limit of the domain of existence of A_1, B_1 and C_1 in the |α| - A_1- and r/-_0 - (A_1-), |α| -B_1- and fo - B_1-, and |α| - C_1- and r/-_0 - C_1-planes, resp. is specified, linked with a representative of the L_1(c, n) numbers, akin to the zeros ζ_(k,n)~((c)) for k → -∞, in addition to the previously found first one, related to ζ_(0,n)~((c))/2. It is established also that as for ζ_(0,n)~((c)) < r/-_0 ≤ 10, so for r/-_0 ≥ 10, the impact of |α| on A_1, of r/-_0 on B_1 and of both α and fo on C_1, is slight. Interpreting physically fo, and n in a suitable way, the employment of the quantities studied in the computation of the differential phase shift produced by the circular waveguide, wholly filled with ferrite of azimuthal magnetization and propagating normal TE_(0n) mode, is demonstrated. The investigation is a contribution to the theory of A, B, C numbers of which the A_1, B_1, C_1 ones constitute one of the two sub-classes, introduced by now.