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Solutions of general homogeneous linear second-order differential equations having an irregular singularity, of unit rank, at infinity are examined. The classical Poincare expansions for such solutions generally are valid in a sector for which the argument of the independent variable z has the range STT — b where b denotes an arbitrary positive constant. In this paper, n terms, say, are taken in the Poincare expansion; the remainder is expressed in the form hn(z) + en(z)i and an explicit expansion involving the generalised exponential integral then is derived for hn(z). By the method of successive approximations, explicit error bounds are derived for Isn^)!Then it is shown from the error bounds that when n = |2| + a where a is bounded and |;z| —> oo, Sn(z) is asymptotically smaller than hn(z) in a sector with a range of arg(2;) which is larger than 47r. In addition to being valid in a larger sector than the corresponding Poincare expansions, the expansions provide greater accuracy, and give a smooth interpretation of the Stokes phenomenon. The present results provide explicit and realistic error bounds for general exponentially-improved asymptotic solutions of differential equations.

Error bounds for exponentially improved asymptotic solutions of ordinary differential equations having irregular singularities of rank one