Parabolic cylinder function
is a solution, then so are
is a solution of equation (A), then
is a solution of (B), and, by symmetry,
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)):
where is the confluent hypergeometric function.
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:
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)):
||This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (December 2010)|
- Abramowitz, Milton; Stegun, Irene A., eds. (1965), "Chapter 19", Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, p. 686, ISBN 978-0486612720, MR 0167642.
- Rozov, N.Kh. (2001), "Weber equation", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Temme, N. M. (2010), "Parabolic cylinder function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W., NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0521192255, MR 2723248
- 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.