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. The purpose of this paper is to put forward a ring-ideal theoretic basis for the theory of uniform asymptotic expansions, and to study a new elementary technique for the asymptotic solution of differential equations containing one or more paramcters. In Part 1 we formulate a new notion of asymptotic expansion and ancillary concepts of formal convergence and formal series. Because we consider this formulation to have significance beyond the uses exhibited here, we make it at a level of generality which seems to us to best reveal its essential nature. In Part 2, we use the notions of Part 1 to frame an investigation of some turning point problems for linear equations of arbitrary order containing a single parameter. Our analysis is based on a new classification of such problems depending on a refinement of the notion of a turning point. In Part 3, we study a second order equation containing two parameters. This analysis contains, as a special case, results for Bessel's equation as both the independent variable and the parameter tend to infinity. We obtain uniform asymptotic results under weak relative growth restrictions on the two parameters which differ qualitatively from any previously known to us. This investigation is an outgrowth of a previous study [1] of turning point problems for second order linear differential equations. We are indebted to Professor Wolfgang Wasow for reading preliminary versions of Part 2 and to the late Professor R. E. Langer for calling the third order problem of ?2.6 to our attention.

A new basis for uniform asymptotic solution of differential equations containing one or several parameters