On the L4 numbers and their application

The F₄ set of complex functions: F^a₄=F^a₄ (a,c;x,ε^a,ρ,α), F̅₄ = F̅₄(a,c;x,̅μ,ρ,α), F̃₄ = F̃₄(a,c;x,ε̃,ρ,α), equation and ̃F̃_4lim = F̃_4lim(a,c;x,ε̃,ρ,α) is defined by means of the complex Kummer and Tricomi confluent hypergeometric functions Φ(a,c;x) and Ψ(a,c;x), and the real ordinary and modified Bessel ones, J_v(y) and I_v(u), (a=c/2-jk-complex; c=3; x=jz; k, z, equation,ε^a, ε̅, ρ, α y, u - real; -∞ <; k <; +∞ in case of F̅₄ and F̅₄; and |k| <; |K_lim|, |k| >; |K_lim|, and |k| = |K_lim| = (|α|/2)√ε̃ /[(1-α²)-ε̃] in case of F̃₄,F̃₄ and F̃_4lim, resp.; z >; 0; ε^a >; 1,ε̅ = 1, 0 <; ε̃ <; 1; 0 <; ρ <; 1, if F^a₄ and F̅₄; |α| >; |α_lim|, if̃ F̃₄; |α| <; |α_lim|, if F̃₄ and |α| = |α_lim|, if F̃_4lim are (is) concerned, (|α_lim|= √1 - ε̃); α_- <; 0, α₊ >; 0; y >; 0, u >; 0; v=0.1). The infinite sequences of positive real numbers K_4±(c,ε,ρ,α_±,n,k_±=|k_±η^(c)_k±,n(ε,- - ρ,α_±) and M_4±(c,ε,ρ,α_±,n,k_±)= |a_±|η^(c)_k±,n(ε,ρ,α_±) are constructed in which η_k±,^(c)n(ε,ρ,α_±) are the n th positive purely imaginary zeros of the F₄ functions in x, n=1,2,3, sgn k= sgn α and the parameters acquire the values pointed out. The class of L₄ numbers, involving the subclasses L^a_4± = L^a_4±(c,ε^a,ρ,α_±,n), L̅_4± = L̅_4±(c,̅ε,ρ,α_±,n), L̃_4± = L̃_4±(c,̅ε,ρ,α_±,n), L̃_4± = L̃_4±(c,̅ε,ρ,α_±,n) and L̃_4lim± = L̃_4lim± (a,c;x,ε̃,ρ,α) is advanced. Each of its elements equals the common limits of the sequences mentioned, put together of the zeros of the respective F₄ function, attained, provided k₊ → +∞ or K_- → -∞ (the subscripts “+”, “-” correspond to the relevant sign of k). Some values of the quantities L₄ are computed and presented in a tabular form. The significance of the numbers considered for the theory of waveguides is debated.