Inequalities and Approximations for the Zeros of Bessel Functions

Abstract

We present some results obtained in the last years on the zeros $c_{\nu,k}$ of the general Bessel function
$$C_{\nu}(x)=\cos {\alpha} \, J_\nu (x) - \sin {\alpha} Y_\nu (x), \qquad 0 \le \alpha <\pi, $$ where  $J_\nu$ and $Y_\nu$ are the Bessel functions of the first and second kind, respectively. In particular we discuss inequalities for a) fixed $\nu$ and large $k$; b) fixed $k$ and large $\nu$. In some cases a numerical comparison with old results is also made.

References

1. Elbert A. and A. Laforgia, An upper bound for the zeros of cylider functions, MIA, Mathematical Inequalities and Applications, 1 (1998), 105-111.

2. Elbert A. and A. Laforgia, Further results on McMahon asymptotic approximations, to appear.

3. Gatteschi L. and C. Giordano, Error bounds for McMahon's asymptotic approximations of the zeros of the Bessel functions, Integral Transforms and Special Functions, to appear.

See latex file of the abstract.