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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Infinite sums of roots for a class of transcendental equations and Bessel functions of order one-half
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by N. Liron PDF
Math. Comp. 25 (1971), 769-781 Request permission

Abstract:

The roots of Bessel functions of order one-half are special cases of roots of transcendental equations of the form $\tan z = A(z)/B(z)$, where $A(z),B(z)$ are polynomials and $A(z)/B(z)$ is odd. We prove that the function $f(z) = B(z)\sin z - A(z)\cos z,f(z)$ even or odd, satisfies the conditions of Hadamard’s factorization theorem, and derive recurrences for sums of the form ${S_l} = \sum \nolimits _{k = 1}^\infty {\alpha _k^{ - 2l},l = 1,2, \cdots }$, where the ${\alpha _k}$’s are the nonzero roots of $f(z)$. We also prove under what conditions on $A(z)$ and $B(z)$ is ${S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)$ or ${S_l} = {\pi ^{ - 2l - 2}}\zeta (2l + 2)({2^{2l + 2}} - 1)$, where $\zeta$ is the Riemann zeta function. We prove that, although Bessel functions of positive half-order, ${J_{l + 1/2}}$, have only real roots, perturbation of any one of its coefficients introduces nonreal roots for $l > 2$.
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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Math. Comp. 25 (1971), 769-781
  • MSC: Primary 33A40
  • DOI: https://doi.org/10.1090/S0025-5718-1971-0304726-X
  • MathSciNet review: 0304726