A NOTE ON A MONOTONICITY PROPERTY OF BESSEL FUNCTIONS

STAMATIS KOUMANDOS

《International Journal of Mathematics and Mathematical Sciences》 1997卷 (1997/12) Pp. 561-566

https://doi.org/10.1155/S0161171297000756

摘要

A theorem of Lorch, Muldoon and Szegö states that the sequence {∫_(jα,k)^(jα,k+1) t^(-α)|J_α(t)|dt}_(k=1)^∞ is decreasing for α＞ -1/2, where J_α(t) the Bessel function of the first kind order α and j_(α,k) its kth positive root. This monotonicity property implies Szegö's inequality ∫_0^x t^(-α)J_α(t)dt ≥ 0, when α ≥ α' and α' is the unique solution of ∫_0^(jα,2) t^(-α)J_α(t)dt=0. We give a new and simpler proof of these classical results by expressing the above Bessel function integral as an integral involving elementary functions.

關鍵字

Bessel functions ； positive integral of Besśel functions ； monotonicity property ofBessel functions