具有最佳精度的变形贝塞尔函数比的简单界
经典分析与常微分方程
2023-04-17 v3
摘要
本文刻画了变形贝塞尔函数比的形如 B ( α , β , γ , x ) = ( α + β 2 + γ 2 x 2 ) / x B(\alpha,\beta,\gamma,x)=(\alpha+\sqrt{\beta^2+\gamma^2 x^2})/x B ( α , β , γ , x ) = ( α + β 2 + γ 2 x 2 ) / x α \alpha α β \beta β γ \gamma γ B ( α , β , γ , x ) B(\alpha,\beta,\gamma,x) B ( α , β , γ , x ) x → 0 + x\rightarrow 0^+ x → 0 + x → + ∞ x\rightarrow +\infty x → + ∞ Φ ν ( x ) = I ν − 1 ( x ) / I ν ( x ) \Phi_{\nu}(x)=I_{\nu-1} (x)/I_{\nu}(x) Φ ν ( x ) = I ν − 1 ( x ) / I ν ( x ) B ( α , β , γ , x ) B(\alpha,\beta,\gamma,x) B ( α , β , γ , x ) Φ ν ( x ) \Phi_{\nu}(x) Φ ν ( x ) x = x ∗ > 0 x=x_*>0 x = x ∗ > 0 x x x B ( α , β , γ , x ) B(\alpha,\beta,\gamma,x) B ( α , β , γ , x ) Φ ν ( x ) \Phi_{\nu}(x) Φ ν ( x ) x ∗ x_* x ∗ Φ ν ( x ) = K ν + 1 ( x ) / K ν ( x ) \Phi_{\nu}(x)=K_{\nu+1} (x)/K_{\nu}(x) Φ ν ( x ) = K ν + 1 ( x ) / K ν ( x ) ν \nu ν x ∗ → 0 + x_*\rightarrow 0^+ x ∗ → 0 + x ∗ → + ∞ x_*\rightarrow +\infty x ∗ → + ∞ 0 + 0^+ 0 + + ∞ +\infty + ∞ x ∗ x_* x ∗
引用
@article{arxiv.2207.02713,
title = {Simple bounds with best possible accuracy for ratios of modified Bessel functions},
author = {J. Segura},
journal= {arXiv preprint arXiv:2207.02713},
year = {2023}
}