thumb|right|350px|[[Coordinate system#Coordinate surface|Coordinate surfaces of parabolic cylindrical coordinates. Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinates]]
alt=Plot of the parabolic cylinder function D<sub>ν</sub>(z) with ν = 5 in the complex plane from -2-2i to 2+2i|thumb|Plot of the parabolic cylinder function D<sub>ν</sub>(z) with in the complex plane from to
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 , called H. F. Weber's equations:
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