# §9.12 Scorer Functions

## §9.12(i) Differential Equation

 9.12.1 $\frac{{\mathrm{d}}^{2}w}{{\mathrm{d}z}^{2}}-zw=\frac{1}{\pi}.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $z$: complex variable and $w$: function Source: Olver (1997b, (12.07), p. 430) Referenced by: (9.12.20), §9.12(i), §9.17(ii) Permalink: http://dlmf.nist.gov/9.12.E1 Encodings: TeX, pMML, png See also: Annotations for §9.12(i), §9.12 and Ch.9

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), §9.12 and Ch.9

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 $-\operatorname{Gi}\left(z\right)$, $\operatorname{Hi}\left(z\right)$, $e^{\mp 2\pi i/3}\operatorname{Hi}\left(ze^{\mp 2\pi i/3}\right)$,

where

 9.12.4 $\operatorname{Gi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{z}^{% \infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t+\operatorname{Ai}\left(z% \right)\int_{0}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t,$ ⓘ Defines: $\operatorname{Gi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function) Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable Source: Olver (1997b, (12.08), p. 430) Referenced by: (9.10.1) Permalink: http://dlmf.nist.gov/9.12.E4 Encodings: TeX, pMML, png See also: Annotations for §9.12(i), §9.12 and Ch.9
 9.12.5 $\operatorname{Hi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{-\infty}^% {z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-\operatorname{Ai}\left(z\right% )\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t.$ ⓘ Defines: $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function) Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable Source: Olver (1997b, (12.09), p. 430) Referenced by: (9.10.3) Permalink: http://dlmf.nist.gov/9.12.E5 Encodings: TeX, pMML, png See also: Annotations for §9.12(i), §9.12 and Ch.9

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

## §9.12(iv) Numerically Satisfactory Solutions

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

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

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

and

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

## §9.12(v) Connection Formulas

 9.12.11 $\operatorname{Gi}\left(z\right)+\operatorname{Hi}\left(z\right)=\operatorname{% Bi}\left(z\right),$ ⓘ Symbols: $\operatorname{Bi}\left(\NVar{z}\right)$: Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function) and $z$: complex variable Proof sketch: Verify with the aid of §§9.2(ii) and 9.12(iii). Referenced by: (9.10.2), (9.12.15), (9.12.19), §9.12(viii) Permalink: http://dlmf.nist.gov/9.12.E11 Encodings: TeX, pMML, png See also: Annotations for §9.12(v), §9.12 and Ch.9
 9.12.12 $\operatorname{Gi}\left(z\right)=\tfrac{1}{2}e^{\pi i/3}\operatorname{Hi}\left(% ze^{-2\pi i/3}\right)+\tfrac{1}{2}e^{-\pi i/3}\operatorname{Hi}\left(ze^{2\pi i% /3}\right),$
 9.12.13 $\operatorname{Gi}\left(z\right)=e^{\mp\pi i/3}\operatorname{Hi}\left(ze^{\pm 2% \pi i/3}\right)\pm i\operatorname{Ai}\left(z\right),$
 9.12.14 $\operatorname{Hi}\left(z\right)=e^{\pm 2\pi i/3}\operatorname{Hi}\left(ze^{\pm 2% \pi i/3}\right)+2e^{\mp\pi i/6}\operatorname{Ai}\left(ze^{\mp 2\pi i/3}\right).$ ⓘ Symbols: $\operatorname{Ai}\left(\NVar{z}\right)$: Airy function, $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $z$: complex variable Proof sketch: Verify with the aid of §§9.2(ii) and 9.12(iii). Referenced by: §9.12(viii) Permalink: http://dlmf.nist.gov/9.12.E14 Encodings: TeX, pMML, png See also: Annotations for §9.12(v), §9.12 and Ch.9

## §9.12(vi) Maclaurin Series

 9.12.15 $\operatorname{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos% \left(\frac{2k-1}{3}\pi\right)\Gamma\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)% ^{k}}{k!},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\operatorname{Gi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable Proof sketch: Substitute into (9.12.11) by means of (9.12.17), (9.4.3), (9.2.5), (9.2.6), and use (5.5.3). Referenced by: (9.12.16) Permalink: http://dlmf.nist.gov/9.12.E15 Encodings: TeX, pMML, png See also: Annotations for §9.12(vi), §9.12 and Ch.9
 9.12.16 $\operatorname{Gi}'\left(z\right)=\frac{3^{-1/3}}{\pi}\*\sum_{k=0}^{\infty}\cos% \left(\frac{2k+1}{3}\pi\right)\Gamma\left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)% ^{k}}{k!}.$
 9.12.17 $\operatorname{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi}\sum_{k=0}^{\infty}\Gamma% \left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)^{k}}{k!},$ ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $k$: nonnegative integer and $z$: complex variable Proof sketch: Expand the integral in (9.12.20) in powers of $zt$ and integrate term-by-term by means of (5.2.1). Referenced by: (9.12.15), (9.12.18) Permalink: http://dlmf.nist.gov/9.12.E17 Encodings: TeX, pMML, png See also: Annotations for §9.12(vi), §9.12 and Ch.9
 9.12.18 $\operatorname{Hi}'\left(z\right)=\frac{3^{-1/3}}{\pi}\sum_{k=0}^{\infty}\Gamma% \left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)^{k}}{k!}.$

## §9.12(vii) Integral Representations

 9.12.19 $\operatorname{Gi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\tfrac% {1}{3}t^{3}+xt\right)\,\mathrm{d}t,$ $x\in\mathbb{R}$.
 9.12.20 $\operatorname{Hi}\left(z\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-% \tfrac{1}{3}t^{3}+zt\right)\,\mathrm{d}t,$ ⓘ Symbols: $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\exp\NVar{z}$: exponential function, $\int$: integral and $z$: complex variable Proof sketch: Verify by showing that the right-hand side satisfies the differential equation (9.12.1) and the initial conditions given in §9.12(iii). Referenced by: (9.12.17), (9.12.31) Permalink: http://dlmf.nist.gov/9.12.E20 Encodings: TeX, pMML, png See also: Annotations for §9.12(vii), §9.12 and Ch.9
 9.12.21 $\operatorname{Gi}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-% \tfrac{1}{3}t^{3}-\tfrac{1}{2}zt\right)\cos\left(\tfrac{1}{2}\sqrt{3}zt+\tfrac% {2}{3}\pi\right)\,\mathrm{d}t.$

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

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

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

### Mellin–Barnes Type Integral

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

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

## §9.12(viii) Asymptotic Expansions

### Functions and Derivatives

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

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

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

### Integrals

 9.12.30 $\int_{0}^{z}\operatorname{Gi}\left(t\right)\,\mathrm{d}t\sim\frac{1}{\pi}\ln z% +\frac{2\gamma+\ln 3}{3\pi}-\frac{1}{\pi}\sum_{k=1}^{\infty}\frac{(3k-1)!}{k!(% 3z^{3})^{k}},$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi-\delta$. ⓘ Symbols: $\gamma$: Euler’s constant, $\operatorname{Gi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase, $k$: nonnegative integer, $z$: complex variable and $\delta$: small positive constant Proof sketch: Integrate (9.12.25) and obtain the constant term by combining (9.12.12) and (9.12.31).(this equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.) Permalink: http://dlmf.nist.gov/9.12.E30 Encodings: TeX, pMML, png See also: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9
 9.12.31 $\int_{0}^{z}\operatorname{Hi}\left(-t\right)\,\mathrm{d}t\sim\frac{1}{\pi}\ln z% +\frac{2\gamma+\ln 3}{3\pi}+\frac{1}{\pi}\sum_{k=1}^{\infty}(-1)^{k-1}\frac{(3% k-1)!}{k!(3z^{3})^{k}},$ $|\operatorname{ph}z|\leq\tfrac{2}{3}\pi-\delta$, ⓘ Symbols: $\gamma$: Euler’s constant, $\operatorname{Hi}\left(\NVar{z}\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase, $k$: nonnegative integer, $x$: real variable, $z$: complex variable and $\delta$: small positive constant Proof sketch: Except for the constant term, this be verified by termwise integration of (9.12.27). To evaluate the constant term replace $z$ by $-x$ ($\leq 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi\int_{0}^{x}\operatorname{Hi}\left(-t\right)\,\mathrm{d}t=\int_{0}^{\infty}% (1-e^{-xt})e^{-\frac{1}{3}t^{3}}t^{-1}\,\mathrm{d}t$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to\infty$ the asymptotic expansions of $\int_{0}^{\infty}xe^{-xt}e^{-\frac{1}{3}t^{3}}(\ln t)\,\mathrm{d}t$ and $\int_{0}^{\infty}e^{-xt}t^{2}e^{-\frac{1}{3}t^{3}}(\ln t)\,\mathrm{d}t$ follow from (2.3.9). Also, $\int_{0}^{\infty}t^{2}e^{-\frac{1}{3}t^{3}}(\ln t)\,\mathrm{d}t$ can be found by replacing $\frac{1}{3}t^{3}$ by $t$ and referring to the first of (5.9.18). (This equation first appeared in Rothman (1954a). As noted in this reference these results were derived by the author of the present DLMF chapter, but the proof was not included.) Referenced by: (9.12.30) Permalink: http://dlmf.nist.gov/9.12.E31 Encodings: TeX, pMML, png See also: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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

## §9.12(ix) Zeros

All zeros, real or complex, of $\operatorname{Gi}\left(z\right)$ and $\operatorname{Hi}\left(z\right)$ are simple.

Neither $\operatorname{Hi}\left(z\right)$ nor $\operatorname{Hi}'\left(z\right)$ has real zeros.

$\operatorname{Gi}\left(z\right)$ has no nonnegative real zeros and $\operatorname{Gi}'\left(z\right)$ has exactly one nonnegative real zero, given by $z=0.60907\;54170\;7\dotsc$. Both $\operatorname{Gi}\left(z\right)$ and $\operatorname{Gi}'\left(z\right)$ 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.