when the parameter K is a positive integer, has been widely discussed. The forms of its solutions, with respect to their asymptotic dependence upon K, are of importance, and have been determined under certain restrictions upon the variables z and K. These restrictions, when heaviest, have confined z to real and K to positive integral values; when lightest, they have permitted z to vary in a strip of the complex plane of finite length and width, and K over the real axis. In the present paper it is purposed to remove these restrictions: to derive asymptotic forms of the solutions of the equation (1) valid in the entire z plane for large values of K, real or complex. It may be recalled that the polynomials (2) were introduced into analysis by Hermitet in 1864. Five years later, Weberl noted that the harmonic functions applicable to the parabolic cylinder satisfy an ordinary differential equation of the form