Parabolic cylinder functions 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 and the double ratio in terms of elementary functions (algebraic or trigonometric) and prove the monotonicity of these ratios; bounds for are also made available. The bounds are very sharp as and , 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 and/or n.

中文翻译：

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抛物线圆柱函数比率的统一（非常）锐界

抛物线柱面函数是经典的特殊函数，在许多不同领域都有应用。然而，关于这些函数的简单均匀近似和边界的可用信息很少。我们根据初等函数（代数或三角函数）获得了比率和双比率的非常清晰的界限，并证明了这些比率的单调性；也提供了边界。边界非常清晰，因为和，并且在三个不同方向上的这种同时清晰解释了它们非凡的全局准确性。获得了能够为中等大和/或n产生几位数精度的基本上界和下界。