Abstract
In order to perform the rigorous computation of the zeros of Bessel functions, we exploit classical approximants going back to Euler and recently generalized in the light of the Trefftz-Fichera orthogonal invariants method. Working in multiple precision (up to 140 significant figures in decimal system), we compute explicit approximants obtaining more than 100 exact figures for the first positive zero. By inversion of this zero as a function of the order, we evaluate the first eigenvalue of the exponential potential Schrödinger operator with the same accuracy.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. I. Akhiezer,The Classical Moment Problem (1965), Oliver and Body, Edinburgh.
G. Fichera,Numerical and Quantitative Analysis (1978), Pitman, London.
V. Grecchi,Zeros of Bessel functions by means of the Trefftz-Fichera orthogonal invariants method through the Schrödinger equation, Rend., Cl. Sci. Fis. Mat. Natur. (1–2)63 (1977), 59–70.
E. H. Lieb, B. Simon, A. S. Wightman,Studies in Mathematical Physics. Essays in Honor of Valentine Bargmann (1976), Princeton University Press, Princeton.
R. Osserman,A note on Hayman's theorem on bass note of a drum, Comment. Math. Helvetici,52 (1977), 545–555.
Waring,Meditationes Analyticae (1776), Cambridge.
G. N. Watson,A Treatise on the Theory of Bessel Functions (1944), Cambridge University Press, London.
A. Weinstein, W. Stenger,Methods of Intermediate Problems for Eigenalues. Theory and Ramifications (1972), Academic Press, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Zironi, F. Multiple precision computation of some zeros of bessel functions by rigorous explicit formulae. Calcolo 18, 321–335 (1981). https://doi.org/10.1007/BF02576434
Issue Date:
DOI: https://doi.org/10.1007/BF02576434