Days on Diffraction 

Abstract: In the paper we deal with the Heun functions — solutions of the Heun equation. Despite the increasing interest to the equation and numerous applications of the functions in a wide variety of physical problems, it is only Maple amidst known software packages which is able to evaluate the functions numerically. But the Maple routine is known to be imperfect. The purpose of the work is to develop alternative algorithms for numerical evaluation of the Heun functions. A procedure based on power series expansions and analytic continuation is suggested.

Abstract: The known problem of resonances in the semiclassical approximation theory is analyzed. It is shown how a specific ?quantum behavior? near the resonance motion appears. In the classical limit, we observe a Poisson micro-geometry concomitant with the resonance. We discuss what the corresponding quantum geometry is and construct irreducible representations of resonance algebras and their coherent states. This results in solving the resonance problems in question.

Abstract: In this paper we consider the confluent Heun equation, which is a linear differential equation of second order with three singular points (two regular and one irregular). A procedure for numerical evaluation of the equation’s solutions (confluent Heun functions) is proposed. The scheme is based on power series, asymptotic expansions and analytic continuation. Results of numerical tests are given.

Abstract: The positive real numbers Z(c,n) are defined for c - arbitrary, real and n = 1, 2, 3, … The cases c - diferent from zero or a negative integer, c - zero or an even negative integer and c - an odd negative integer, are considered. If a = c/2 - jk - complex, c - real, (c ≠ l, l = 0,−1,−2, …), k - real, x = jz, z - real, positive and n = 1, 2, 3, …, Z(c,n) numbers are called the positive purely imaginary zeros ζ(c) k,n in x of the complex Kummer confluent hypergeometric function Φ(a, c; x), on condition that k = 0. A theorem is formulated and proved numerically that extends the definition of numbers also for c = l in which Φ(a, c; x) has simple poles and reveals some of their properties. It is composed of three lemmas. Lemmas 1 and 2 substantiate the existence of quantities and determine them when l = 2p, p = 0,−1,−2, … (l = 0,−2,−4, …, c - zero or an even negative integer) and l = 2p − 1 (l = −1,−3,−5, …, c - an odd negative integer), as the common limits of the couples of infinite sequences of positive real numbers {Z(l - e,n)} and {Z(l+e,n)}, and {Z(l-e,n)} and {Z(l+e,n+1)}), resp. for e → 0 (e - infinitesimal positive real number). Lemma 3 demonstrates the possibility to express the numbers Z(l,n), (l - an odd negative integer) through the ones Z(2−l,n), (the symmetry of both quantities with respect to the point c = 1) and presents the relation of Z(0,n) and Z(2,n) with the Ludolphian number π. Tables and graphs visualize the effect of parameters c and n on Z(c,n). The role of the latter is shown in the theory of azimuthally magnetized circular ferrite waveguides that support normal TE 0n modes.