The Harmonic Functions Associated with the Parabolic Cylinder

Abstract

PARABOLIC CYLINDER.117 equation, when n is unrestricted, is taken by Whittaker to be ) c -*-i(W (-t)-^ dt, and other solutions are D a {-z), Dn -\{+iz).When n is a positive integer, thenThe moat remarkable discovery which has been made concerning the function D n (z) is contained in Adamoff's* approximation, namely that, 'when n is a large integer and x is positive,and | S«(aO| < 3*35 ..., when n > 2 ; and ic,,(0) = 0 or £&/» (where -1 < 9 < 1), according as n is odd or even.In my former paper, I shewed that Adamoffs investigation would be extended to B lh {z), when z is complex, provided that \l(z)\ is not too large : and this result made it easy to give a formal proof of a theorem enunciated by Hermite and Whittaker, concerning the expansibility of an arbitrary analytic function in the form 2 a n D n (,?) :-a result analogous to Neumann's expansions in series of Legendre polynomials and Bessel functions.In this paper, I investigate various types of asymptotic formulae and expansions of D n (z), hy the method of steepest descents ; the results contained in the paper are therefore to be associated with those previously obtained by Debyet and by myself I for Bessel functions.It should be stated here that the method of steepest descents^ is a mode of discussing the integral f e" Jif) <p (s) ds by choosing a contour which passes through a zero of /' (s), and which * Ann.del'Inst.Poly technique de St. Petersbourg, t. 5 (1906), pp.127-143.More recently, Perron, Archiv der Math.u.Phys., Ser. 3, Vol.22 (1914), pp.329-340, has given some approximate formulae for the confluent hypergeometric function of which D n {z) is a special form, but the limits of error in the formulae are not investigated with the preciseness of the •earlier work of Adamoff.t Math.