Uniform asymptotic expansions for the zeros of parabolic cylinder functions
T. M. Dunster, Amparo Gil, Diego Ruiz‐Antolín, Javier Segura · arXiv (Cornell University) · 2024
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.