FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2002, VOLUME 8, NUMBER 4, PAGES 1159-1178

Zeroes of Schrödinger's radial function $R$_nl(r) and Kummer's function $$₁F₁(-a;c;z) (, )

V. F. Tarasov

Abstract

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Exact formulae for calculation of zeroes of Kummer's polynomials at $a$£ 4 are given; in other cases ($a\; >\; 4$) their numerical values (to within $10$-15) are given. It is shown that the methods of L. Ferrari, L. Euler and J.-L. Lagrange that are used for solving the equation $$₁F₁(-4;c;z) = 0 are based on one (common for all methods) equation of cubic resolvent of FEL-type. For greater geometrical clarity of (nonuniform for $a\; >\; 3$) distribution of zeroes $x$_k = z_k-(c+a-1) on the axis $y\; =\; 0$ the "circular" diagrams with the radius $R$_a = (a-1)√(c+a-1) are introduced for the first time. It allows to notice some singularities of distribution of these zeroes and their "images", i. e. the points $T$_k on the circle. Exact "angle" asymptotics of the points $T$_k for $2$£ c < ¥ for the cases $a\; =\; 3$ and $a\; =\; 4$ are obtained. While calculating zeroes $x$_k of the $R$_nl(r) and $$₁F₁ functions, the "singular" cases $(a,c)\; =\; (4,6),\; (6,4),\; (8,14),$¼ are found.

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Last modified: April 10, 2003