In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation

This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates.

The above equation may be brought into two distinct forms (A) and (B) by completing the square and rescaling z, called H. F. Weber's equations:

and

If $${\displaystyle f(a,z)}$$ is a solution, then so are $${\displaystyle f(a,-z),f(-a,iz){\text{ and }}f(-a,-iz).}$$

If $${\displaystyle f(a,z)\,}$$ is a solution of equation (A), then $${\displaystyle f(-ia,ze^{(1/4)\pi i})}$$ is a solution of (B), and, by symmetry, $${\displaystyle f(-ia,-ze^{(1/4)\pi i}),f(ia,-ze^{-(1/4)\pi i}){\text{ and }}f(ia,ze^{-(1/4)\pi i})}$$ are also solutions of (B).

There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun (1965)): $${\displaystyle y_{1}(a;z)=\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {1}{4}};\;{\tfrac {1}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,\,\,\,\,\,(\mathrm {even} )}$$ and $${\displaystyle y_{2}(a;z)=z\exp(-z^{2}/4)\;_{1}F_{1}\left({\tfrac {1}{2}}a+{\tfrac {3}{4}};\;{\tfrac {3}{2}}\;;\;{\frac {z^{2}}{2}}\right)\,\,\,\,\,\,(\mathrm {odd} )}$$ where ${\displaystyle \;_{1}F_{1}(a;b;z)=M(a;b;z)}$ is the confluent hypergeometric function.

Other pairs of independent solutions may be formed from linear combinations of the above solutions. One such pair is based upon their behavior at infinity: $${\displaystyle U(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}}}\left[\cos(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)-{\sqrt {2}}\sin(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}$$ $${\displaystyle V(a,z)={\frac {1}{2^{\xi }{\sqrt {\pi }}\Gamma [1/2-a]}}\left[\sin(\xi \pi )\Gamma (1/2-\xi )\,y_{1}(a,z)+{\sqrt {2}}\cos(\xi \pi )\Gamma (1-\xi )\,y_{2}(a,z)\right]}$$ where $${\displaystyle \xi ={\frac {1}{2}}a+{\frac {1}{4}}.}$$