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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 (Weber 1869) harv error: no target: CITEREFWeber1869 (help):
(A)
and
(B)
If
is a solution, then so are
If
is a solution of equation (A), then
is a solution of (B), and, by symmetry,
are also solutions of (B).
Solutions
There are independent even and odd solutions of the form (A). These are given by (following the notation of Abramowitz and Stegun (1965)):
Other pairs of independent solutions may be formed from linear combinations of the above solutions (see Abramowitz and Stegun). One such pair is based upon their behavior at infinity:
where
The function U(a, z) approaches zero for large values of z and |arg(z)| < π/2, while V(a, z) diverges for large values of positive real z .
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902)) that are themselves sometimes called parabolic cylinder functions (see Abramowitz and Stegun (1965)):
Function Da(z) was introduced by Whittaker and Watson as a solution of eq.~(1) with bounded at . It can be expressed in terms of confluent hypergeometric functions as
Weber, H.F. (1869) "Ueber die Integration der partiellen Differentialgleichung ". Math. Ann., 1, 1–36
Whittaker, E.T. (1902) "On the functions associated with the parabolic cylinder in harmonic analysis" Proc. London Math. Soc.35, 417–427.
Whittaker, E. T. and Watson, G. N. "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348, 1990.