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Zbl 0948.33001
Landau, L.J.
Bessel functions: monotonicity and bounds. (English)
[J] J. Lond. Math. Soc., II. Ser. 61, No.1, 197-215 (2000). ISSN 0024-6107; ISSN 1469-7750/e

The aim of this paper is the derivation of monotonicity properties of Bessel functions, leading to precise bounds which are uniform in order or argument. Monotonicity with respect to the order $\nu$ of the magnitude of general Bessel functions. ${\cal C}_\nu(x)= aJ_\nu(x)+ bY_\nu(x)$ at positive stationary points of associated functions is derived. In particular, the magnitude of ${\cal C}_\nu$ at its positive stationary points is strictly decreasing in $\nu$ for all positive $\nu$. It follows that $\sup_x|J_\nu (x)|$ strictly decreases from 1 to 0 as $\nu$ increases from 0 to $\infty$. The magnitude of $x^{1\over 2}{\cal C}_\nu$ at its positive stationary points is strictly increasing in $\nu$. It follows that $\sup_x|x^{1\over 2}J_\nu (x)|$ equals $\sqrt{2\over \pi}$ for $0\le\nu \le{1 \over 2}$ and strictly increases to $\infty$ as $\nu$ increases from ${1\over 2}$ to $\infty$. It is also shown that $\nu^{1\over 3}\sup|J_\nu(x)|$ strictly increases from 0 to $b=0.674885\dots$ as $\nu$ increases from 0 to $\infty$. Hence for all positive $\nu$ and real $x$, $|J_\nu(x)|<b\nu^{- {1\over 3}}$ where $b$ is the best possible such constant. Additionally, errors in the work by Abramowitz and Stegun by Watson are pointed out.
[R.S.Dahiya (Ames)]

MSC 2000:: *33C10 Cylinder functions, etc.

Keywords: Bessel functions

Cited in: Zbl 1108.33004

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