# §9.12 Scorer Functions

## §9.12(i) Differential Equation

 9.12.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-zw=\frac{1}{\pi}.$

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

 9.12.2 $w(z)=Aw_{1}(z)+Bw_{2}(z)+p(z),$ Defines: $w$: function (locally) Symbols: $z$: complex variable, $A$: constant, $B$: constant, $w_{1}$: function, $w_{2}$: function and $p$: particular solution Permalink: http://dlmf.nist.gov/9.12.E2 Encodings: TeX, pMML, png See also: Annotations for 9.12(i)

where $A$ and $B$ are arbitrary constants, $w_{1}(z)$ and $w_{2}(z)$ are any two linearly independent solutions of Airy’s equation (9.2.1), and $p(z)$ is any particular solution of (9.12.1). Standard particular solutions are

 9.12.3 $-\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$, $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$, $e^{\mp 2\pi i/3}\mathop{\mathrm{Hi}\/}\nolimits\!\left(ze^{\mp 2\pi i/3}\right)$,

where

 9.12.4 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Bi}\/}% \nolimits\!\left(z\right)\int_{z}^{\infty}\mathop{\mathrm{Ai}\/}\nolimits\!% \left(t\right)\mathrm{d}t+\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\int_% {0}^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t,$
 9.12.5 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Bi}\/}% \nolimits\!\left(z\right)\int_{-\infty}^{z}\mathop{\mathrm{Ai}\/}\nolimits\!% \left(t\right)\mathrm{d}t-\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right)\int_% {-\infty}^{z}\mathop{\mathrm{Bi}\/}\nolimits\!\left(t\right)\mathrm{d}t.$

$\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ are entire functions of $z$.

## §9.12(ii) Graphs

See Figures 9.12.1 and 9.12.2.

## §9.12(iii) Initial Values

 9.12.6 $\displaystyle\mathop{\mathrm{Gi}\/}\nolimits\!\left(0\right)$ $\displaystyle=\tfrac{1}{2}\mathop{\mathrm{Hi}\/}\nolimits\!\left(0\right)=% \tfrac{1}{3}\mathop{\mathrm{Bi}\/}\nolimits\!\left(0\right)=1/\!\left(3^{7/6}% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{2}{3}\right)\right)=0.20497\;55424\ldots,$ 9.12.7 $\displaystyle\mathop{\mathrm{Gi}\/}\nolimits'\!\left(0\right)$ $\displaystyle=\tfrac{1}{2}\mathop{\mathrm{Hi}\/}\nolimits'\!\left(0\right)=% \tfrac{1}{3}\mathop{\mathrm{Bi}\/}\nolimits'\!\left(0\right)=1/\left(3^{5/6}% \mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}\right)\right)=0.14942\;94524\ldots.$

## §9.12(iv) Numerically Satisfactory Solutions

$-\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ on the interval $0\leq x<\infty$. $\mathop{\mathrm{Hi}\/}\nolimits\!\left(x\right)$ is a numerically satisfactory companion to $\mathop{\mathrm{Ai}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{Bi}\/}\nolimits\!\left(x\right)$ on the interval $-\infty.

In $\mathbb{C}$, numerically satisfactory sets of solutions are given by

 9.12.8 $-\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Ai}\/}% \nolimits\!\left(z\right),\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right),$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi$,
 9.12.9 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Ai}\/}% \nolimits\!\left(ze^{-2\pi i/3}\right),\mathop{\mathrm{Ai}\/}\nolimits\!\left(% ze^{2\pi i/3}\right),$ $|\mathop{\mathrm{ph}\/}\nolimits(-z)|\leq\tfrac{2}{3}\pi$,

and

 9.12.10 $e^{\mp 2\pi i/3}\mathop{\mathrm{Hi}\/}\nolimits\!\left(ze^{\mp 2\pi i/3}\right% ),\mathop{\mathrm{Ai}\/}\nolimits\!\left(z\right),\mathop{\mathrm{Ai}\/}% \nolimits\!\left(ze^{\pm 2\pi i/3}\right),$ $-\pi\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{1}{3}\pi$.

## §9.12(v) Connection Formulas

 9.12.11 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)+\mathop{\mathrm{Hi}\/}% \nolimits\!\left(z\right)=\mathop{\mathrm{Bi}\/}\nolimits\!\left(z\right),$
 9.12.12 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}e^{\pi i/3}\mathop% {\mathrm{Hi}\/}\nolimits\!\left(ze^{-2\pi i/3}\right)+\tfrac{1}{2}e^{-\pi i/3}% \mathop{\mathrm{Hi}\/}\nolimits\!\left(ze^{2\pi i/3}\right),$
 9.12.13 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)=e^{\mp\pi i/3}\mathop{\mathrm{% Hi}\/}\nolimits\!\left(ze^{\pm 2\pi i/3}\right)\pm i\mathop{\mathrm{Ai}\/}% \nolimits\!\left(z\right),$
 9.12.14 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)=e^{\pm 2\pi i/3}\mathop{% \mathrm{Hi}\/}\nolimits\!\left(ze^{\pm 2\pi i/3}\right)+2e^{\mp\pi i/6}\mathop% {\mathrm{Ai}\/}\nolimits\!\left(ze^{\mp 2\pi i/3}\right).$

## §9.12(vi) Maclaurin Series

 9.12.15 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)=\frac{3^{-2/3}}{\pi}\*\sum_{k=% 0}^{\infty}\mathop{\cos\/}\nolimits\!\left(\frac{2k-1}{3}\pi\right)\mathop{% \Gamma\/}\nolimits\!\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)^{k}}{k!},$
 9.12.16 $\mathop{\mathrm{Gi}\/}\nolimits'\!\left(z\right)=\frac{3^{-1/3}}{\pi}\*\sum_{k% =0}^{\infty}\mathop{\cos\/}\nolimits\!\left(\frac{2k+1}{3}\pi\right)\mathop{% \Gamma\/}\nolimits\!\left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)^{k}}{k!}.$
 9.12.17 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)=\frac{3^{-2/3}}{\pi}\sum_{k=0}% ^{\infty}\mathop{\Gamma\/}\nolimits\!\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z% )^{k}}{k!},$
 9.12.18 $\mathop{\mathrm{Hi}\/}\nolimits'\!\left(z\right)=\frac{3^{-1/3}}{\pi}\sum_{k=0% }^{\infty}\mathop{\Gamma\/}\nolimits\!\left(\frac{k+2}{3}\right)\frac{(3^{1/3}% z)^{k}}{k!}.$

## §9.12(vii) Integral Representations

 9.12.19 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\sin\/}\nolimits\!\left(\tfrac{1}{3}t^{3}+xt\right)\mathrm{d}t,$ $x\in\mathbb{R}$.
 9.12.20 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)=\frac{1}{\pi}\int_{0}^{\infty}% \mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{3}t^{3}+zt\right)\mathrm{d}t,$
 9.12.21 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)=-\frac{1}{\pi}\int_{0}^{\infty% }\mathop{\exp\/}\nolimits\!\left(-\tfrac{1}{3}t^{3}-\tfrac{1}{2}zt\right)% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\sqrt{3}zt+\tfrac{2}{3}\pi\right)% \mathrm{d}t.$

If $\zeta=\tfrac{2}{3}z^{3/2}$ or $\tfrac{2}{3}x^{3/2}$, and $\mathop{K_{1/3}\/}\nolimits$ is the modified Bessel function (§10.25(ii)), then

 9.12.22 $\displaystyle\mathop{\mathrm{Hi}\/}\nolimits\!\left(-z\right)$ $\displaystyle=\frac{4z^{2}}{3^{3/2}\pi^{2}}\int_{0}^{\infty}\frac{\mathop{K_{1% /3}\/}\nolimits\!\left(t\right)}{\zeta^{2}+t^{2}}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\tfrac{1}{3}\pi$, 9.12.23 $\displaystyle\mathop{\mathrm{Gi}\/}\nolimits\!\left(x\right)$ $\displaystyle=\frac{4x^{2}}{3^{3/2}\pi^{2}}\pvint_{0}^{\infty}\frac{\mathop{K_% {1/3}\/}\nolimits\!\left(t\right)}{\zeta^{2}-t^{2}}\mathrm{d}t,$ $x>0$,

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

### Mellin–Barnes Type Integral

 9.12.24 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)=\frac{3^{-2/3}}{2\pi^{2}i}\int% _{-i\infty}^{i\infty}\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}+\tfrac{1}{% 3}t\right)\mathop{\Gamma\/}\nolimits\!\left(-t\right)(3^{1/3}e^{\pi i}z)^{t}% \mathrm{d}t,$

where the integration contour separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(\tfrac{1}{3}+\tfrac{1}{3}t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(-t\right)$.

## §9.12(viii) Asymptotic Expansions

### Functions and Derivatives

As $z\to\infty$, and with $\delta$ denoting an arbitrary small positive constant,

 9.12.25 $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)\sim\frac{1}{\pi z}\sum_{k=0}^{% \infty}\frac{(3k)!}{k!(3z^{3})^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi-\delta$,
 9.12.26 $\mathop{\mathrm{Gi}\/}\nolimits'\!\left(z\right)\sim-\frac{1}{\pi z^{2}}\sum_{% k=0}^{\infty}\frac{(3k+1)!}{k!(3z^{3})^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi-\delta$.
 9.12.27 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\sim-\frac{1}{\pi z}\sum_{k=0}^% {\infty}\frac{(3k)!}{k!(3z^{3})^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits(-z)|\leq\tfrac{2}{3}\pi-\delta$,
 9.12.28 $\mathop{\mathrm{Hi}\/}\nolimits'\!\left(z\right)\sim\frac{1}{\pi z^{2}}\sum_{k% =0}^{\infty}\frac{(3k+1)!}{k!(3z^{3})^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits(-z)|\leq\tfrac{2}{3}\pi-\delta$.

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 $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)\sim-\frac{1}{\pi z}\sum_{k=0}^% {\infty}\frac{(3k)!}{k!(3z^{3})^{k}}+\frac{e^{\zeta}}{\sqrt{\pi}z^{1/4}}\sum_{% k=0}^{\infty}\frac{u_{k}}{\zeta^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta$.

### Integrals

 9.12.30 $\int_{0}^{z}\mathop{\mathrm{Gi}\/}\nolimits\!\left(t\right)\mathrm{d}t\sim% \frac{1}{\pi}\mathop{\ln\/}\nolimits z+\frac{2\gamma+\mathop{\ln\/}\nolimits 3% }{3\pi}-\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(3k-1)!}{k!(3z^{3})^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{3}\pi-\delta$.
 9.12.31 $\int_{0}^{z}\mathop{\mathrm{Hi}\/}\nolimits\!\left(-t\right)\mathrm{d}t\sim% \frac{1}{\pi}\mathop{\ln\/}\nolimits z+\frac{2\gamma+\mathop{\ln\/}\nolimits 3% }{3\pi}+\frac{1}{\pi}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(3k-1)!}{k!(3z^{3})^{k% }},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{2}{3}\pi-\delta$,

where $\gamma$ is Euler’s constant (§5.2(ii)).

## §9.12(ix) Zeros

All zeros, real or complex, of $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ are simple.

Neither $\mathop{\mathrm{Hi}\/}\nolimits\!\left(z\right)$ nor $\mathop{\mathrm{Hi}\/}\nolimits'\!\left(z\right)$ has real zeros.

$\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ has no nonnegative real zeros and $\mathop{\mathrm{Gi}\/}\nolimits'\!\left(z\right)$ has exactly one nonnegative real zero, given by $z=0.60907\;54170\;7\ldots$. Both $\mathop{\mathrm{Gi}\/}\nolimits\!\left(z\right)$ and $\mathop{\mathrm{Gi}\/}\nolimits'\!\left(z\right)$ 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.