FUNDAMENTALNAYA I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

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

Zeroes of Schrödinger's radial function Rnl(r) and Kummer's function 1F1(-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 $a \le 4$ 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 ${}_1F_1(-4; c; z) = 0$ are based on \emph{one} (\emph{common} for all methods) equation of cubic resolvent of FEL-type. For greater geometrical clarity of (\emph{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)\sqrt{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 $2 \le c < \infty$ for the cases $a = 3$ and $a = 4$ are obtained. While calculating zeroes $x_{k}$ of the $R_{nl}(r)$ and ${}_1F_1$ functions, the ``singular'' cases $(a,c) = (4,6), (6,4), (8,14),\ldots$ are found.

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