Ifantis, E. K.; Siafarikas, P. D. Upper bounds for the first zeros of Bessel functions. (English) Zbl 0616.65017 J. Comput. Appl. Math. 17, 355-358 (1987). Upper bounds for the first positive zero of the Bessel functions of first kind \(J_{\mu}(z)\) for \(\mu >-1\) are obtained. Ritz’s approximation method, applied to the eigenvalue problem of a compactself adjoint operator, defined on an abstract separable Hilbert space is used to establish the results. Reviewer: C.L.Koul Cited in 4 Documents MSC: 65D20 Computation of special functions and constants, construction of tables 65H05 Numerical computation of solutions to single equations 33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\) 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 65J10 Numerical solutions to equations with linear operators 47A10 Spectrum, resolvent Keywords:Ritz’s method; positive zero of the Bessel functions of first kind; eigenvalue problem; compactself adjoint operator; Hilbert space PDFBibTeX XMLCite \textit{E. K. Ifantis} and \textit{P. D. Siafarikas}, J. Comput. Appl. Math. 17, 355--358 (1987; Zbl 0616.65017) Full Text: DOI References: [1] Ahmed, S.; Calogero, F., On the zeros of Bessel functions IV, Lett. Nuovo Cimento, 21, 531-534 (1978) [2] Chambers, L. I., An upper bound for the first zero of Bessel functions, Math. Comput., 38, 589-591 (1982) · Zbl 0483.33005 [3] Dirschmid, H., Bemerkungen zu einer Arbeit von G. Polya zur Bestimmung der Nullstellen ganzer Funktionen, Numer. Math., 13, 344-348 (1969) · Zbl 0185.40405 [4] Dirschmid, H., Zur Einschliessung der Eigenwerte Vollstetiger Positiver Operatoren in separablen Hilbert-Räumen,, Computing, 5, 119-127 (1970), I and II · Zbl 0209.46502 [5] Elbert, A., Some inequalities concerning Bessel functions of first kind, Studia Sci. Math. Hungar, 6, 277-285 (1971) · Zbl 0239.33009 [6] Elbert, A.; Laforgia, A., Further results on the zeros of Bessel functions, Analysis, 5, 71-86 (1983) · Zbl 0564.33005 [7] Giordano, C.; Laforgia, A., Elementary approximations for zeros of Bessel functions, J. Comput. Appl. Math., 9, 221-228 (1983) · Zbl 0516.33007 [8] Grad, J.; Zakrajsek, E., Method for evaluation of zeros of Bessel functions, J. Inst. Math. Applic., 11, 57-72 (1973) · Zbl 0252.33007 [9] Hethcote, H. W., Bounds for zeros of some special functions, Proc. Amer. Math. Soc., 25, 72-74 (1970) · Zbl 0189.34404 [10] Hethcote, H. W., Error bounds for asymptotic approximations of zeros of transcendental functions, SIAM J. Math. Anal., 2, 147-152 (1970) · Zbl 0199.49902 [11] Ifantis, E. K.; Siafarikas, P. D.; Kouris, C. B., Conditions for solution of a linear first order differential equation in the Hardy-Lebesgue space and applications, J. Math. Anal. Appl., 104, 454-466 (1984) · Zbl 0558.34006 [12] Ifantis, E. K.; Siafarikas, P. D., An inequality related the zeros of two ordinary Bessel functions, Applicable Analysis, 19, 251-263 (1985) · Zbl 0569.33006 [13] Ifantis, E. K.; Siafarikas, P. D., A differential equation for the zeros of Bessel functions, Applicable Analysis, 20, 269-281 (1985) · Zbl 0553.33004 [14] Laforgia, A.; Muldoon, M. E., Inequalities and approximations for zeros of Bessel functions of small order, SIAM J. Math. Anal., 14, 383-388 (1983) · Zbl 0514.33006 [15] Nemat-Nasser, S.; Minagawa, S., Harmonic waves in layered composites: Comparison among several schemes, J. Appl. Mech., 42, 699-704 (1975) · Zbl 0359.73025 [16] Piessens, R., A series expansion for the first positive zero of the Bessel functions, Math. Comput., 42, 195-197 (1984) · Zbl 0536.33005 [17] Polya, G., Graeffe’s method for eigenvalues, Numer. Math., 11, 315-319 (1968) · Zbl 0191.15903 [18] Vulikh, B. Z., Introduction to Functional Analysis for Scientists and Technologists (1963), Pergamon Press: Pergamon Press New York · Zbl 0109.08103 [19] Watson, G. N., A Treatise on the Theory of Bessel Functions (1966), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0174.36202 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.