Error Bounds for Asymptotic Expansions in Turning-Point Problems
Error Bounds for Asymptotic Expansions in Turning-Point Problems
复制标题
转折点问题中渐近展开的误差界
DOI：
10.1137/0112019
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发表时间：
1964
期刊：
影响因子：
--
通讯作者：
F. Olver
中科院分区：
文献类型：
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作者：
F. Olver
1.01 dx {u，p（u，x）+ q（u，x）l，vo，其中u是一个大的正参数，x在一个真实的区间上变化.这个区间不包含函数p（u，x）和q（u，x）的奇点，只包含p（u，x）的一个零点;它不需要有界。在本文中，我们推广这些结果在两个方面。首先，我们考虑截断包含Airy函数及其一阶导数的微分方程的某些形式（一般发散）级数解所得到的高阶逼近。第二，我们考虑x和u的复值。我们正在研究的级数本身并不是新的;我们已经知道[2，3，4，5]，它们表示参数u的大值的解的一致渐近展开式，对p（u，x）和q（u，x）有一定的限制。下面给出的结果的新特点是，它们提供了精确的，现实的，与截断级数相关的余项的界限。
1.01 dx {u p (u,)+ q (u, x) l vo, in which u is a large positive parameter, andx ranges over a real interval. This interval contains no singularities of the functions p (u, x) and q (u, x) and just one zero of p (u, x); it need not be bounded. In the present paper, we extend these results in two ways. First, we consider higher approxi-mations which are obtained by truncating certain formal (generally di-vergent) series solutions of the differential equation involing the Airy functions and their first derivatives. Second, we consider complex alues of x and u.The series we are investigating are not themselves new; it is already known [2, 3, 4, 5] that they represent uniform asymptotic expansions of solutions for large alues of the parameteru, subject to certain restrictions on p (u, x) and q (u, x). The new feature of the results given below is that they supply precise, and realistic, bounds for the remainder terms associated with the truncated series.