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
$$_{1}F_{1}(-a;c;z)
($n\; <\; 10$, $l\; <\; 4$)

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 $$_{1}F_{1}(-4;c;z) = 0
are based on *one* (*common* for all methods) equation of
cubic resolvent of FEL-type.
For greater geometrical clarity of (*nonuniform* for $a\; >\; 3$) distribution of
zeroes $x$_{k} = z_{k}-(c+a-1)
on the axis $y\; =\; 0$
the "circular" diagrams with the radius $R$_{a} = (a-1)√(c+a-1)
are introduced for the first time.
It allows to notice some singularities of distribution of these zeroes
and their "images", i. e.
the points $T$_{k} on the circle.
Exact "angle" asymptotics of the points $T$_{k} for
$2$£
c < ¥ for the
cases $a\; =\; 3$ and
$a\; =\; 4$ are
obtained.
While calculating zeroes $x$_{k} of the
$R$_{nl}(r) and
$$_{1}F_{1}
functions, the "singular" cases $(a,c)\; =\; (4,6),\; (6,4),\; (8,14),$¼ are found.

All articles are
published in Russian.

Location: http://mech.math.msu.su/~fpm/eng/k02/k024/k02415h.htm

Last modified: April 10, 2003