Uniform asymptotic expansions for the zeros of parabolic cylinder functions

Abstract

The real and complex zeros of the parabolic cylinder function U (a,z) are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for a positive or negative and large in absolute value, uniformly for unbounded z (real or complex). The accuracy of the approximations of the complex zeros is then demonstrated with some comparative tests using a highly precise numerical algorithm for finding the complex zeros of the function.