GLOBALLY EXACT ASYMPTOTICS FOR INTEGRALS WITH ARBITRARY ORDER SADDLES

被引:13
作者
Bennett, T. [1 ]
Howls, C. J. [2 ]
Nemes, G. [3 ]
Daalhuis, A. B. Olde [3 ]
机构
[1] Univ Kent, SMSAS, Sibson Bldg,Parkwood Rd, Canterbury CT2 7FS, Kent, England
[2] Univ Southampton, Math Sci, Southampton SO17 1BJ, Hants, England
[3] Univ Edinburgh, Sch Math, James Clerk Maxwell Bldg,Kings Bldg, Edinburgh EH9 3FD, Midlothian, Scotland
基金
英国工程与自然科学研究理事会;
关键词
integral asymptotics; asymptotic expansions; hyperasymptotics; error bounds; saddle points; BESSEL HANKEL; ERROR-BOUNDS; LARGE VALUES; HYPERASYMPTOTICS; EXPANSIONS; ARGUMENT;
D O I
10.1137/17M1154217
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We derive the first exact, rigorous but practical, globally valid remainder terms for asymptotic expansions about saddles and contour endpoints of arbitrary order degeneracy derived from the method of steepest descents. The exact remainder terms lead naturally to sharper novel asymptotic bounds for truncated expansions that are a significant improvement over the previous best existing bounds for quadratic saddles derived two decades ago. We also develop a comprehensive hyperasymptotic theory, whereby the remainder terms are iteratively reexpanded about adjacent saddle points to achieve better-than-exponential accuracy. By necessity of the degeneracy, the form of the hyperasymptotic expansions is more complicated than in the case of quadratic endpoints and saddles and requires generalizations of the hyperterminants derived in those cases. However, we provide efficient methods to evaluate them, and we remove all possible ambiguities in their definition. We illustrate this approach for three different examples, providing all the necessary information for the practical implementation of the method.
引用
收藏
页码:2144 / 2177
页数:34
相关论文
共 48 条