On the asymptotic solutions of ordinary differential equations, with an application to the Bessel functions of large order

is possible, so that the resulting forms represent asymptotically a pair of solutions of the equation given. Moreover, each of these forms is then known to represent one and the same solution so long as p remains in a region of the complex plane in which its pure imaginary part is either invariably greater than or invariably less than some constant. The forms (2) are evidently of oscillatory or exponential type according as 42(x) is positive or negative on the x interval under account, and on different intervals in which +2(x) is of opposite sign the forms are respectively of the two opposing types. Between two such intervals a change in the character of any solution must, therefore, take place. It is evident, however, that the manner of this change is not deducible from the forms (2), for whatever the magnitude of p these forms fail to remain significant at a zero of 02(X).