Abstract
Asymptotic solutions of the differential equations d2w/dz2 = {uzn +f{z)} w 0, 1) for large positive values of u, have the formal expansions P(z) 1+ Z5=1 (f> l+ ^ M y us s 5=0 IIs where
P
is an exponential or Airy function for n 0 or 1 respectively. The coefficients
A
s
(z) and
B
s
(z) are given by recurrence relations. This paper proves that solutions of the differential equations exist whose asymptotic expansions in Poincaré’s sense are given by these series, and that the expansions are uniformly valid with respect to the complex variable z. The method of proof differs from those of earlier writers and fewer restrictions are made.
Reference23 articles.
1. British Association Mathematical Tables 1946 The Airy integral. Cambridge University Press.
2. California Institute of Technology 1953 Tech. rep. No. 1. Asymptotic solutions o f differential equations with turning points.
Cited by
110 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献