论文标题
在圆柱和通风函数的真实零上的两个猜想的证明
Proofs of two conjectures on the real zeros of the cylinder and Airy functions
论文作者
论文摘要
我们证明了圆柱体和通风函数的大型真实零零的已知分歧渐近扩展的包裹特性，从而在Elbert和Laforgia以及Fabijonas和Olver提出的肯定性两个猜想中回答。证明的本质是构建分析函数，这些函数沿着某些离散的实数集进行评估时返回零。通过操纵这些函数的轮廓积分，我们得出了有限数量的项以及可以有效估计的剩余数量后截断的大零的渐近膨胀。然后将这些猜想推导为这些估计的推论。还讨论了相关的相位函数的类似结果。
We prove the enveloping property of the known divergent asymptotic expansions of the large real zeros of the cylinder and Airy functions, and thereby answering in the affirmative two conjectures posed by Elbert and Laforgia and by Fabijonas and Olver, respectively. The essence of the proof is the construction of analytic functions that return the zeros when evaluated along certain discrete sets of real numbers. By manipulating contour integrals of these functions, we derive the asymptotic expansions of the large zeros truncated after a finite number of terms plus remainders that can be estimated efficiently. The conjectures are then deduced as corollaries of these estimates. An analogous result for the associated phase function is also discussed.
