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


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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 xk = zk-(c+a-1) on the axis y = 0 the "circular" diagrams with the radius Ra = (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 Tk on the circle. Exact "angle" asymptotics of the points Tk for 2 £ c < ¥ for the cases a = 3 and a = 4 are obtained. While calculating zeroes xk of the Rnl(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