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Bessel Functions: Monotonicity and Bounds
Author(s) -
Landau L. J.
Publication year - 2000
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610799008352
Subject(s) - monotonic function , bessel function , mathematics , magnitude (astronomy) , constant (computer programming) , combinatorics , order (exchange) , mathematical analysis , stationary point , physics , finance , astronomy , computer science , economics , programming language
Monotonicity with respect to the order v of the magnitude of general Bessel functions C v ( x ) = aJ v ( x )+ bY v ( x ) at positive stationary points of associated functions is derived. In particular, the magnitude of C v at its positive stationary points is strictly decreasing in v for all positive v . It follows that sup x ∣ J v ( x )∣ strictly decreases from 1 to 0 as v increases from 0 to ∞. The magnitude of x 1/2 C v ( x ) at its positive stationary points is strictly increasing in v . It follows that sup x ∣ x 1/2 J v ( x )∣ equals √2/π for 0 ⩽ v ⩽ 1/2 and strictly increases to ∞ as v increases from 1/2 to ∞. It is shown that v 1/3 sup x ∣ J v ( x )∣ strictly increases from 0 to b = 0.674885… as v increases from 0 to ∞. Hence for all positive v and real x ,| J v ( x ) | < b v‐ 1 / 3where b is the best possible such constant. Furthermore, for all positive v and real x ,| J v ( x ) | ⩽ c| x |‐ 1 / 3where c = 0.7857468704… is the best possible such constant. Additionally, errors in work by Abramowitz and Stegun and by Watson are pointed out.