FUNDAMENTALNAYA
I PRIKLADNAYA MATEMATIKA

(FUNDAMENTAL AND APPLIED MATHEMATICS)

2000, VOLUME 6, NUMBER 1, PAGES 143-162

**The polynomial approximation of zeros of the Bessel functions**

I. B. Kozhukhov

N. I. Platonov

Abstract

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Two methods of the polynomial approximation of zeros of
the Bessel function $J$_{n}(x) of
the first genus and its derivative $J$_{n}'(x) are
considered, where they are assumed to be implicit functions
on $$n defined by equations
$J$_{n}(x)=0 and
$J$_{n}'(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 $J$_{n}'(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.

All articles are
published in Russian.

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Last modified: April 11, 2000