Coordinate surfaces of parabolic cylindrical coordinates. Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinatesPlot of the parabolic cylinder function D v(z) with v=5 in the complex plane from -2-2i to 2+2i with colors created with Mathematica 13.1 function ComplexPlot3D
In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation
Other pairs of independent solutions may be formed from linear combinations of the above solutions.[2] 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 .
and
The functions U and V can also be related to the functions Dp(x) (a notation dating back to Whittaker (1902))[3] that are themselves sometimes called parabolic cylinder functions:[2]
Function Da(z) was introduced by Whittaker and Watson as a solution of eq.~(1) with bounded at .[4] It can be expressed in terms of confluent hypergeometric functions as
Power series for this function have been obtained by Abadir (1993).[5]
References
^Weber, H.F. (1869), "Ueber die Integration der partiellen Differentialgleichung ", Math. Ann., vol. 1, pp. 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. (1990) "The Parabolic Cylinder Function." §16.5 in A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 347-348.
^ Abadir, K. M. (1993) "Expansions for some confluent hypergeometric functions." Journal of Physics A, 26, 4059-4066.