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BESSEL FUNCTIONS: MONOTONICITY AND BOUNDS

Published online by Cambridge University Press:  01 February 2000

L. J. LANDAU
L. J. LANDAU
Affiliation:
Mathematics Department, King's College London, Strand, London WC2R 2LS; larry.landau@kcl.ac.uk
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Abstract

Monotonicity with respect to the order v of the magnitude of general Bessel functions [Cscr ]v(x) = aJv(x)+bYv(x) at positive stationary points of associated functions is derived. In particular, the magnitude of [Cscr ]v at its positive stationary points is strictly decreasing in v for all positive v. It follows that supx[mid ]Jv(x)[mid ] strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x1/2[Cscr ]v(x) at its positive stationary points is strictly increasing in v. It follows that supx[mid ]x1/2[Cscr ]v(x)[mid ] equals √2/π for 0 [les ] v [les ] 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞.

It is shown that v1/3supx[mid ]Jv(x)[mid ] strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x,

formula here

where b is the best possible such constant. Furthermore, for all positive v and real x,

formula here

where c = 0.7857468704… is the best possible such constant.

Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 197 - 215
Copyright
The London Mathematical Society 2000

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