Parabolic cylinder function

Summary

Parabolic cylinder functions are solutions of Weber's equation $y'' + \left( \nu + \frac12 - \frac{z^2}{4} \right) y = 0$. The standard solution $D_\nu(z)$ decays like $z^\nu e^{-z^2/4}$ as $z \to +\infty$. They arise when the Helmholtz equation or Laplace's equation is solved by separation of variables in parabolic cylindrical coordinates. For integer $\nu \ge 0$ they reduce to Hermite functions. They are confluent hypergeometric functions in the variable $z^2/2$.