Consider the differential equation satisfied by
![\begin{displaymath}
w=z^{-1/2} W_{k,-1/4}({\textstyle{1\over 2}}z^2),
\end{displaymath}](w_375.gif) |
(1) |
where
is a Whittaker Function.
![\begin{displaymath}
{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
\end{displaymath}](w_377.gif) |
(2) |
![\begin{displaymath}
{d^2w\over dz^2}+(2k-{\textstyle{1\over 4}}z^2)w=0.
\end{displaymath}](w_378.gif) |
(3) |
This is usually rewritten
![\begin{displaymath}
{d^2D_n(z)\over dz^2}+(n+{\textstyle{1\over 2}}-{\textstyle{1\over 4}}z^2)D_n(z)=0.
\end{displaymath}](w_379.gif) |
(4) |
The solutions are Parabolic Cylinder Functions.
The equations
![\begin{displaymath}
{d^2U\over du^2}-(c+k^2u^2)U=0
\end{displaymath}](w_380.gif) |
(5) |
![\begin{displaymath}
{d^2V\over dv^2}+(c-k^2v^2)V=0,
\end{displaymath}](w_381.gif) |
(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.
© 1996-9 Eric W. Weisstein
1999-05-26