# §12.14 The Function

## §12.14(i) Introduction

In this section solutions of equation (12.2.3) are considered. This equation is important when and are real, and we shall assume this to be the case. In other cases the general theory of (12.2.2) is available. and form a numerically satisfactory pair of solutions when .

12.14.1
12.14.2

## §12.14(iii) Graphs

For the modulus functions and see §12.14(x).

 Figure 12.14.1: , , , . Symbols: : parabolic cylinder function, : real variable and Permalink: http://dlmf.nist.gov/12.14.F1 Encodings: pdf, png Figure 12.14.2: , , , . Symbols: : parabolic cylinder function, : real variable and Permalink: http://dlmf.nist.gov/12.14.F2 Encodings: pdf, png
 Figure 12.14.3: , , , . Symbols: : parabolic cylinder function, : real variable and Permalink: http://dlmf.nist.gov/12.14.F3 Encodings: pdf, png Figure 12.14.4: , , . Symbols: : parabolic cylinder function, : real variable and Permalink: http://dlmf.nist.gov/12.14.F4 Encodings: pdf, png

## §12.14(iv) Connection Formula

where

12.14.5
12.14.6
12.14.7

the branch of being zero when and defined by continuity elsewhere.

## §12.14(v) Power-Series Expansions

12.14.8

Here and are the even and odd solutions of (12.2.3):

12.14.9
12.14.10

where and satisfy the recursion relations

12.14.11

with

12.14.12

Other expansions, involving and , can be obtained from (12.4.3) to (12.4.6) by replacing by and by ; see Miller (1955, p. 80), and also (12.14.15) and (12.14.16).

## §12.14(vi) Integral Representations

These follow from the contour integrals of §12.5(ii), which are valid for general complex values of the argument and parameter . See Miller (1955, p. 26).

## §12.14(vii) Relations to Other Functions

### ¶ Bessel Functions

For the notation see §10.2(ii). When

12.14.13

### ¶ Confluent Hypergeometric Functions

For the notation see §13.2(i).

The even and odd solutions of (12.2.3) (see §12.14(v)) are given by

## §12.14(viii) Asymptotic Expansions for Large Variable

Write

where

with given by (12.14.7). Then as

The coefficients and are obtainable by equating real and imaginary parts in

12.14.22

Equivalently,

## §12.14(ix) Uniform Asymptotic Expansions for Large Parameter

The differential equation

follows from (12.2.3), and has solutions . For real and oscillations occur outside the -interval . Airy-type uniform asymptotic expansions can be used to include either one of the turning points . In the following expansions, obtained from Olver (1959), is large and positive, and is again an arbitrary small positive constant.

### ¶ Positive ,

uniformly for . Here is as in §12.10(ii), is defined by

12.14.27

with given by (12.10.7), and

12.14.28

with as in §12.10(ii). The function has the asymptotic expansion

with

12.14.30

### ¶ Positive ,

uniformly for , with given by (12.10.23) and given by (12.10.24).

The expansions for the derivatives corresponding to (12.14.25), (12.14.26), and (12.14.31) may be obtained by formal term-by-term differentiation with respect to ; compare the analogous results in §§12.10(ii)12.10(v).

### ¶ Negative ,

In this case there are no real turning points, and the solutions of (12.2.3), with replaced by , oscillate on the entire real -axis.

uniformly for , where

12.14.36

and and the coefficients and as in §12.10(v).

## §12.14(x) Modulus and Phase Functions

As noted in §12.14(ix), when is negative the solutions of (12.2.3), with replaced by , are oscillatory on the whole real line; also, when is positive there is a central interval in which the solutions are exponential in character. In the oscillatory intervals we write

where is defined in (12.14.5), and (0), , (0), and are real. or is the modulus and or is the corresponding phase. Compare §12.2(vi).

For properties of the modulus and phase functions, including differential equations and asymptotic expansions for large , see Miller (1955, pp. 87–88). For graphs of the modulus functions see §12.14(iii).

## §12.14(xi) Zeros of ,

For asymptotic expansions of the zeros of and , see Olver (1959).