Uniform asymptotic formulae for functions with transition points

in which R(x, w), Q(x, w) are regular at x=O, w=0 and f(x) has a simple zero at x=0; for (see ?2) this may be reduced to the form (1.1) by a change of variables, regular at x = z = 0. When all the symbols denote real numbers, the solutions of (1.2) are monotonic or oscillating according as OffQ +4-1R2+ 2-1dR/dx is positive or negative; and when v is large, this quantity changes sign at a point near x = 0, on account of the simple zero of f(x) at x = 0. Thus as x passes through 0 the solutions change from monotonic to oscillating, and we may call x = 0 a transition point('). It is this transition which distinguishes our problem from the simpler one in which the x-region includes no zero of f(x). The guiding idea of the investigation is familiar: "approximately identical differential equations have approximately identical solutions." A significant approximation to (1.1) will be an equation having the same features when z is near 0. Accordingly the project stated in the first paragraph is crystallized: the approximations to solutions of (1.1) are to be solutions of a differential equation of the sameform as (1.1). The simplest equation of this form is the Airy equation