# §9.12 Scorer Functions

## §9.12(i) Differential Equation

Solutions of this equation are the Scorer functions and can be found by the method of variation of parameters (§1.13(iii)). The general solution is given by

where and are arbitrary constants, and are any two linearly independent solutions of Airy’s equation (9.2.1), and is any particular solution of (9.12.1). Standard particular solutions are

where

and are entire functions of .

## §9.12(ii) Graphs

See Figures 9.12.1 and 9.12.2.

 Figure 9.12.1: , . Symbols: : Scorer function (inhomogeneous Airy function) and : real variable Referenced by: §9.12(ii), §9.12(ix) Permalink: http://dlmf.nist.gov/9.12.F1 Encodings: pdf, png Figure 9.12.2: , . Symbols: : Scorer function (inhomogeneous Airy function) and : real variable Referenced by: §9.12(ii), §9.12(ix) Permalink: http://dlmf.nist.gov/9.12.F2 Encodings: pdf, png

## §9.12(iv) Numerically Satisfactory Solutions

is a numerically satisfactory companion to the complementary functions and on the interval . is a numerically satisfactory companion to and on the interval .

## §9.12(vii) Integral Representations

If or , and is the modified Bessel function (§10.25(ii)), then

where the last integral is a Cauchy principal value (§1.4(v)).

### ¶ Mellin–Barnes Type Integral

where the integration contour separates the poles of from those of .

## §9.12(viii) Asymptotic Expansions

### ¶ Functions and Derivatives

For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. For example, with the notation of §9.7(i).

9.12.29.

## §9.12(ix) Zeros

All zeros, real or complex, of and are simple.

Neither nor has real zeros.

has no nonnegative real zeros and has exactly one nonnegative real zero, given by . Both and have an infinity of negative real zeros, and they are interlaced.

For the above properties and further results, including the distribution of complex zeros, asymptotic approximations for the numerically large real or complex zeros, and numerical tables see Gil et al. (2003c).

For graphical illustration of the real zeros see Figures 9.12.1 and 9.12.2.