Weber Differential Equations

Consider the differential equation satisfied by

$$w=z^{-1/2} W_{k,-1/4}({\textstyle{1\over 2}}z^2),$$ (1)

where $W$ is a Whittaker Function.

$${d\over z\,dz}\left[{d(wz^{1/2})\over z\,dz}\right]+\left({-{1\over 4}+{2k\over z^2} +{3\over 4z^4}}\right)wz^{1/2}=0$$ (2)

$${d^2w\over dz^2}+(2k-{\textstyle{1\over 4}}z^2)w=0.$$ (3)

This is usually rewritten

$${d^2D_n(z)\over dz^2}+(n+{\textstyle{1\over 2}}-{\textstyle{1\over 4}}z^2)D_n(z)=0.$$ (4)

The solutions are Parabolic Cylinder Functions.

The equations

$${d^2U\over du^2}-(c+k^2u^2)U=0$$ (5)

$${d^2V\over dv^2}+(c-k^2v^2)V=0,$$ (6)

which arise by separating variables in Laplace's Equation in Parabolic Cylindrical Coordinates, are also known as the Weber differential equations. As above, the solutions are known as Parabolic Cylinder Functions.