Smoothing of Stokes discontinuities

When the asymptotics of an analytic function f(z) are given in terms of two asymptotic forms S1(z), S2(z), alternately dominant in alternating sectors of the z-plane, Stokes observed that the coefficients of S1, S2, while constant in any one sector, can change from sector to sector, and that to a first approximation the change appears to be discontinuous. Recently, Berry has given an interesting formal argument which shows in more detail how the change takes place, and the present paper gives a rigorous treatment of Berry’s result.