Communications in Mathematical Sciences

Volume 12 (2014)

Number 1

Absence of zeros and asymptotic error estimates for airy and parabolic cylinder functions

Pages: 175 – 200



Felix Finster (Fakultät für Mathematik, Universität Regensburg, Germany)

Joel Smoller (Mathematics Department, University of Michigan, Ann Arbor, Mich., U.S.A.)


We derive WKB approximations for a class of Airy and parabolic cylinder functions in the complex plane, including quantitative error bounds. We prove that all zeros of the Airy function lie on a ray in the complex plane, and that the parabolic cylinder functions have no zeros. We also analyze the Airy and Airy-WKB limit of the parabolic cylinder functions.


airy functions, parabolic cylinder functions, asymptotic error estimates

2010 Mathematics Subject Classification

33C10, 33C15, 34D05, 34E05

Published 20 August 2013