FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA
(FUNDAMENTAL AND APPLIED MATHEMATICS)
2000, VOLUME 6, NUMBER 1, PAGES 143-162
I. B. Kozhukhov
N. I. Platonov
Abstract
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Two methods of the polynomial approximation of zeros of
the Bessel function $J_\nu(x)$ of the first genus and its
derivative $J_\nu'(x)$ are considered, where they are assumed
to be implicit functions on $\nu$ defined by equations
$J_\nu(x)=0$ and $J_\nu'(x)=0$ . The first method is based on
the approximation of zeros by Taylor polynomials and uses the common
algorithm for calculating the high derivative of an implicit
function. The asymptotic expressions for zeros of $J_\nu'(x)$
are obtained as well as the numeric values of some first
coefficients of decomposition. The region of applications
of the formulas is investigated. The second method uses
the approximation of the Bessel function by a polynomial of 4th
degree and reduces to the solution of a certain system of
algebraic equations. An analysis of the accuracy of these
methods is carried out.
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