# Parabolic cylinder function

Weber function, Weber–Hermite function

A solution of the differential equation

$$\tag{* } \frac{d ^ {2} y }{dz ^ {2} } + \left ( \nu + \frac{1}{2} - \frac{z ^ {2} }{4} \right ) y = 0,$$

which is obtained as a result of separating the variables in the wave equation $\Delta u = k ^ {2} u$ in parabolic cylindrical coordinates (cf. Parabolic coordinates). Ordinarily one uses the solution

$$D _ \nu ( z) \equiv U \left ( - \nu - \frac{1}{2} , z \right ) = \ 2 ^ {( \nu - 1)/2 } e ^ {- z ^ {2} /4 } \Psi \left ( \frac{1 - \nu }{2} , \frac{3}{2} ; \ \frac{z ^ {2} }{2} \right ) ,$$

where $\Psi ( a, b; z)$ is the confluent hypergeometric function. The functions $D _ \nu (- z)$ and $D _ {- \nu - 1 } (\pm iz)$ also satisfy equation (*). The functions $D _ \nu ( z)$ and $D _ {- \nu - 1 } (\pm iz)$ are linearly independent for arbitrary $\nu$, and $D _ \nu ( z)$ and $D _ \nu (- z)$ are linearly independent for $\nu \neq 0, \pm 1 , \dots$. 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, $n = 1, 2 ,\dots$:

$$\frac{d ^ {n} }{dz ^ {n} } \left [ e ^ {z ^ {2} /4 } D _ \nu ( z) \right ] = \ (- 1) ^ {n} (- \nu ) _ {n} e ^ {z ^ {2} /4 } D _ {\nu - n } ( z),$$

$$\frac{d ^ {n} }{dz ^ {n} } \left [ e ^ {- z ^ {2} /4 } D _ \nu ( z) \right ] = (- 1) ^ {n} e ^ {- z ^ {2} /4 } D _ {\nu + n } ( z) .$$

The recurrence formulas are:

$$D _ {\nu + 1 } ( z) - zD _ \nu ( z) + \nu D _ {\nu - 1 } ( z) = 0,$$

$$D _ \nu ^ \prime ( z) + \frac{z}{2} D _ \nu ( z) - \nu D _ {\nu - 1 } ( z) = 0,$$

$$D _ \nu ^ \prime ( z) - \frac{z}{2} D _ \nu ( z) + D _ {\nu + 1 } ( z) = 0.$$

Asymptotically, for fixed $\nu$ and $| \mathop{\rm arg} z | < 3 \pi /4$, as $z \rightarrow \infty$, one has

$$D _ \nu ( z) = \ z ^ \nu e ^ {- z ^ {2} /4 } \left [ \sum _ { k= 0} ^ { N } \frac{(- \nu /2) _ {k} ( 1/2 - \nu /2) _ {k} }{k!} \left ( \frac{z ^ {2} }{- 2} \right ) ^ {- k}\right .+ \left . O ( | z | ^ {- 2N- 2 } ) \right ] ,$$

and, for bounded $| z |$ and $| \mathop{\rm arg} ( - \nu ) | \leq \pi /2$, as $| \nu | \rightarrow \infty$, one has

$$D _ \nu ( z) = \ \frac{1}{\sqrt 2 } \mathop{\rm exp} \left [ \frac \nu {2} \mathop{\rm ln} (- \nu ) - \frac \nu {2} - \sqrt {- \nu } z \right ] \left [ 1 + O \left ( \frac{1}{\sqrt {| \nu | } } \right ) \right ] .$$

Parabolic cylinder functions are related to other functions as follows ($n = 0, 1 , \dots$): To the Hermite polynomials by

$$D _ {n} ( z) = \ 2 ^ {- n/2} e ^ {- z ^ {2} /4 } H _ {n} \left ( \frac{z}{\sqrt 2} \right ) ,$$

to the probability integral (error function) by

$$D _ {-} n- 1 ( z) = \ \frac{(- 1) ^ {n} \sqrt 2 }{n!} e ^ {- z ^ {2} /4 } \frac{d ^ {n} }{dz ^ {n} } \left ( e ^ {z ^ {2} /3 } \mathop{\rm erfc} \frac{z}{\sqrt z } \right ) ,$$

and to the Bessel functions by

$$D _ {- 1/2} ( z) = \ \sqrt { \frac{\pi z }{2} } K _ {1/4} \left ( \frac{z ^ {2} }{4} \right ) .$$

#### References

 [1] H. Bateman (ed.) A. Erdélyi (ed.) et al. (ed.) , Higher transcendental functions , 2. Bessel functions, parabolic cylinder functions, orthogonal polynomials , McGraw-Hill (1953) [2] J.C.P. Miller, "Giving solutions of the differential equation , tables of Weber parabolic cylinder functions" , H.M. Stationary Office (1955)
How to Cite This Entry:
Parabolic cylinder function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Parabolic_cylinder_function&oldid=52277
This article was adapted from an original article by Yu.A. BrychkovA.P. Prudnikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article