The L(c, n) numbers and their application in the theory of waveguides

The theorem for existence and for the basic properties of the L(c, n) numbers (positive real numbers, coherent with the positive purely imaginary zeros zeta
<sub>k,

n</sub>

^(c)
in x of the complex Kummer confluent hypergeometric function Phi(a, c; x) with a = c/2 - jk - complex, c - real, (c ne l, l = 0, -1, -2, -3,...), k - real, x = jz, z - real, positive and n = 1, 2, 3,...), is formulated and proved numerically. It is composed of three lemmas. Lemmas 1 and 2 demonstrate the existence of quantities and determine them in case c ne l and c = l (in which Phi(a, c; x) is not defined) as the common limits of the couples of infinite sequences of positive real numbers {K_(c, n, k)_)} and {M_(c, n, k_)}, (K_(c, n, k_) = |k_|zeta
<sub>k_,

n</sub>

^(c)
, M_(c, n, k_) = |a_|zeta
<sub>k_,

n</sub>

^(c)
for k_ rarr -infin, and {L(l - epsiv, n)} and {L(l + epsiv, n + 1)}, (epsiv - infinitesimal positive real number) for epsiv rarr 0, resp. (The subscript ldquo-rdquo indicates quantities, corresponding to negative k.) Lemma 3 reveals the main features of the numbers (recurrence relation, formula for symmetry and bond of some of them with the Ludolphian number). Tables and graphs illustrate the influence of parameters c and n on L(c, n). The benefit of numbers is manifested in the theory of azimuthally magnetized circular ferrite waveguides, propagating normal TE
_0n
modes.