Uniform Asymptotic Expansions for the Zeros of Parabolic Cylinder Functions

ABSTRACT The real and complex zeros of the parabolic cylinder function are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and these are valid for positive or negative and large in absolute value, uniformly for unbounded (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.