组合数学
贝塞尔函数
数学
上下界
简单(哲学)
BETA(编程语言)
物理
数学分析
哲学
认识论
计算机科学
程序设计语言
标识
DOI:10.1016/j.jmaa.2023.127211
摘要
The best bounds of the form B(α,β,γ,x)=(α+β2+γ2x2)/x for ratios of modified Bessel functions are characterized: if α, β and γ are chosen in such a way that B(α,β,γ,x) is a sharp approximation for Φν(x)=Iν−1(x)/Iν(x) as x→0+ (respectively x→+∞) and the graphs of the functions B(α,β,γ,x) and Φν(x) are tangent at some x=x⁎>0, then B(α,β,γ,x) is an upper (respectively lower) bound for Φν(x) for any positive x, and it is the best possible at x⁎. The same is true for the ratio Φν(x)=Kν+1(x)/Kν(x) but interchanging lower and upper bounds (and with a slightly more restricted range for ν). Bounds with maximal accuracy at 0+ and +∞ are recovered in the limits x⁎→0+ and x⁎→+∞, and for these cases the coefficients have simple expressions. For the case of finite and positive x⁎ we provide uniparametric families of bounds which are close to the optimal bounds and retain their confluence properties.
科研通智能强力驱动
Strongly Powered by AbleSci AI