Absence of Zeros and Asymptotic Error Estimates for Airy and Parabolic Cylinder Functions

Abstract

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.

25 pages, LaTeX, 7 figures (published version)

References in corpus (2)

• Spectral Estimates and Non-Selfadjoint Perturbations of Spheroidal Wave Operators
• Parabolic Cylinder Functions: Examples of Error Bounds For Asymptotic Expansions

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• Refined Error Estimates for the Riccati Equation with Applications to the Angular Teukolsky Equation
• Linear Stability of Rotating Black Holes: Outline of the Proof
• Lectures on Linear Stability of Rotating Black Holes