The asymptotic solutions of a linear differential equation of the second order with two turning points

Abstract

in which (i) X2 is a large complex parameter; (ii) s is the real variable over some interval a < s < b; (iii) 02(s) is a real function which has precisely two zeros, both simple, on the interval (a, b); (iv) X(s, X) is bounded in s and X, and is integrable in s. The zeros of 02(s) are referred to as turning points of the differential equation. The solution forms of a differential Equation (1.1) over an interval that includes just one turning point are known, and can be expressed asymptotically with respect to X through the use of Bessel functions. That being so, it could well seem that forms having validity over the entire interval with two turning points might be obtainable by dealing with the interval by halves, and, in the familiar way, patching together the forms appropriate to the respective halves, along with their derivatives. That procedure, however, is not practicable when the forms in hand are not actual representations but only the leading terms of asymptotic ones. For such a form is explicit only to a certain degree, and because of that ordinarily represents not just a single particular solution over an appropriate x-interval, but a whole infinity of them. In other sub-intervals these solutions may be wholly dissimilar. To fit together two such forms, along with their derivatives, at some chosen point between the two turninlg points, therefore does not identify the solutions to which that fit applies, and accordingly cannot be taken as a basis for inferring the continuations of those solutions in intervals beyond the turning points. In the present paper the interval is not treated by halves. It is shown that forms which are valid over the whole interval can be given in terms of ap-