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and po(x), p1(x), and the qj(x)'s are suitable restricted bounded functions. Specifically, formulas are sought which represent solutions of (1.1) in a certain asymptotic sense when lXI > N(2). Existing asymptotic theory of the equation (1.1) is known to depend upon the nature of the vanishing of po(x) and p1(x), these being coefficients of positive powers of X. A zero of po(x) has been called a turning point of the equation, the order of the turning point being the order of the zero. In an interval where po(x) -dO, complete asymptotic expansions of a pair of solutions have long been known; this case was the subject of the classical work of Birkhoff [1 ]. A recent advance, due to R. E. Langer [3 ], has brought the theory of the simple turning point to a similar degree of completion. For higher order turning points, on the other hand, the theory is still fragmentary, being most highly developed for a turning point of order two. In this case expansions have in fact been given [2; 4], but only the leading term of an expansion has been expressed in any simple way. For terms after the first, the known expression are extremely complicated, each successive term a degree more involved than the one before. It is the object of the present paper to obtain expansions, valid in the presence of a second order turning point, whose terms to any order are simply expressed. An algorithm will be given which involves only simple power series

The solutions of second order linear ordinary differential equations about a turning point of order two