论文标题
抛物线缸功能比率的均匀（非常）锋利的边界
Uniform (very) sharp bounds for ratios of Parabolic Cylinder functions
论文作者
论文摘要
抛物线缸功能（PCF）是经典的特殊功能，在许多不同的领域中具有应用。但是，几乎没有关于这些功能简单统一近似和边界的信息。我们获得了比率$φ_n（x）= u（n-1，x）/u（n，x）$的非常清晰的边界，并且在基本函数（代数或trigonometric）方面，双比$φ_n（x）/φ___________________________________________________________________n（x）/u（x）/u（x）/φ_n（x）/φ_{n+1}（x）$还提供了$ u（n，z）/u（n，y）$的范围。边界非常尖锐，如$ x \ rightarrow \ pm \ infty $和$ n \ rightarrow +\ infty $，并且在三个不同方向上的同时清晰度解释了它们出色的全球精度。获得上下基础界限，能够为中等大的$ | x | $和/或$ n $产生几位准确性。
Parabolic Cylinder functions (PCFs) are classical special functions with applications in many different fields. However, there is little information available regarding simple uniform approximations and bounds for these functions. We obtain very sharp bounds for the ratio $Φ_n(x)=U(n-1,x)/U(n,x)$ and the double ratio $Φ_n(x)/Φ_{n+1}(x)$ in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for $U(n,z)/U(n,y)$ are also made available. The bounds are very sharp as $x\rightarrow \pm \infty$ and $n\rightarrow +\infty$, and this simultaneous sharpness in three different directions explains their remarkable global accuracy. Upper and lower elementary bounds are obtained which are able to produce several digits of accuracy for moderately large $|x|$ and/or $n$.
