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

propriate Weber functions w(z, k), namely of appropriate solutions of Weber's differential equation d2w . .(1.2) -+ {2K + 1 -z2}w = 0. dz2 These Weber functions are thus shown to fill, in the instance of two turning points, a role that is analogous to that which the Bessel functions fill when there is only one.A study of the solution forms of the Weber Equation (1.2) with both the variable and the parameter complex, and the latter large, was made by Schwid, [l](2), and more recently by Erdelyi, Kennedy and McGregor,[2].References to the literature are given by these authors, who also remark that differential equations of the type (1.1) present themselves in quantum mechanics, and in problems of wave motion, of diffraction, of vibrations, etc.No use of these other studies is, however, made in the present paper.The reasons are that the solutions singled out and set forth in them are not optimal ones for the present investigation, and that the solution forms given by them, applying as they do to domains of the complex variable, are of greater intricacy than is here needed.The particular Weber functions that assume a basic role here, are ones that have monotonic absolute values, either on each side of a turning point, or over the given interval as a whole.The cases in which the parameter X2 is real are special ones.There are two of them, and these are quite distinct.If ~K262(s) is negative between the turning points, the solutions of the equation are of an exponential character there, and make transitions to oscillatory forms when the turning points are traversed.An earlier study of the Equation (1.1) in this case was made, under(2) Numerals within brackets will be used to refer to the bibliography attached.