Parabolic cylinder functions occur when separation of variables is used on Laplace's equation in these coordinates. In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation.

2025年1月31日 · Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions D_nu (z) as solutions to the Weber differential equation y^ ('') (z)+ (nu+1/2-1/4z^2)y (z)=0. (1) The two independent solutions are given by y=D_nu (z) and y=D_ (-nu-1) (iz), where D_nu (z) = 2^ (nu/2+1/4)z^ (-1/2)W_ (nu/2+1/4,-1/4) (1/2z^2) (2) =...

2025年1月31日 · A quadratic surface given by the equation x^2+2rz=0.

This chapter is based in part on Abramowitz and Stegun (1964, Chapter 19) by J. C. P. Miller. The main references used in writing this chapter are Erdélyi et al. (1953b, v. 2), Miller (1955), and Olver (1959). For additional bibliographic reading see Buchholz (1969), Lebedev (1965), Magnus et al. (1966), Olver (1997b), and Temme (1996b).

ParabolicCylinderD [\ [Nu], z] gives the parabolic cylinder function \ [Nu].

2022年4月22日 · The parabolic cylinder functions are entire functions of $ z $. The function $ D _ \nu ( z) $ is real for real $ \nu $ and $ z $. The differentiation formulas are, $ n = 1, 2 ,\dots $:

Cylinders are given more specific names based on the shape of their cross-sections. For example, the function z = x 2 has no y dependence and is parabolic. It should have the same parabolic cross section for any value of y. Thus it is called a parabolic cylinder. The graph of x 2 + y 2 = 1 is the same for any value of z and is shaped like a ...

Compute the parabolic cylinder functions or the derivatives of the parabolic cylinder functions. Description: The parabolic cylinder functions are defined as:

textual description of parabolic cylinder; finite cone; infinite cone; ruled surfaces; parabolic cylinder vs lip surface

1999年5月26日 · Whittaker and Watson (1990, p. 347) define the parabolic cylinder functions as solutions to the Weber Differential Equation