On the roots of certain transcendental equations, involving complex confluent hypergeometric functions and their application in the theory of waveguides

The positive purely imaginary roots of four equations, taking in complex confluent hypergeometric functions, are studied numerically. The first of them involves the Kummer function only, the second is stated through two Kummer and two Tricomi ones, the third is written by two complex Kummer and four real Bessel and Neumann functions, and the fourth - in terms of four Kummer and Tricomi and two Bessel ones. The first parameter of confluent functions is complex with a real part, equal the half of their second one - a positive integer and their independent variable is positive purely imaginary. A special relation links the latter with that of the cylindrical functions. The results are depicted in a graphical form. It is established that the roots of the first two equations grow monotonously from infinitesimal to unlimited values when the imaginary part of the first parameter of confluent functions increases from minus to plus infinity. Unlike them, the roots of the third and fourth equation possess a bell-like region in case the imaginary part mentioned varies in the vicinity of zero while when it gets extremely large (positive and negative), they become very small. The use of quantities sifted in analyzing the eigenvalue spectrum and transmission properties of the circular and coaxial waveguides, containing azimuthally magnetized ferrite only or concentric ferrite and dielectric layers which support normal TE_0n modes, is discussed.