Parabolic cylinder functions revisited using the Laplace transform

Abstract

In this paper we gather and extend classical results for parabolic cylinder functions, namely solutions of the Weber differential equations, using a systematic approach by Borel-Laplace methods. We revisit the definition and construction of the standard solutions $U,V$ of the Weber differential equation \begin{equation*} w''(z)-\left(\frac{z^2}{4}+a\right)w(z)=0 \end{equation*} and provide representations by Laplace integrals extended to include all values of the complex parameter $a$; we find an integral integral representation for the function $V$; none was previously available. For the Weber equation in the form \begin{equation*} u''(x)+\left(\frac{x^2}{4}-a\right)u(x)=0, \end{equation*} we define a new fundamental system $E_\pm$ which is analytic in $a\in\mathbb{C}$, based on asymptotic behavior; they appropriately extend and modify the classical solutions $E,E^*$ of the real Weber equation to the complex domain. The techniques used are general and we include details and motivations for the approach.