  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.
