§9.12 Scorer Functions

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Keywords:: Scorer functions
Referenced by:: §2.4(v), §3.5(ix), §9.10(i)
Permalink:: http://dlmf.nist.gov/9.12
See also:: Annotations for Ch.9

§9.12(i) Differential Equation
§9.12(ii) Graphs
§9.12(iii) Initial Values
§9.12(iv) Numerically Satisfactory Solutions
§9.12(v) Connection Formulas
§9.12(vi) Maclaurin Series
§9.12(vii) Integral Representations
§9.12(viii) Asymptotic Expansions
§9.12(ix) Zeros

§9.12(i) Differential Equation

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Keywords:: Scorer functions, analytic properties, definition, differential equation, integral representations, standard solutions
Notes:: See Olver (1997b, pp. 430–431).
Referenced by:: §11.11(iii)
Permalink:: http://dlmf.nist.gov/9.12.i
See also:: Annotations for §9.12 and Ch.9

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

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : derivative of with respect to , : complex variable and : 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 $w(z)=Aw_{1}(z)+Bw_{2}(z)+p(z),$

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Defines:: : function (locally)
Symbols:: : complex variable, : constant, : constant, $w_{1}$ : function, $w_{2}$ : function and : 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 and are arbitrary constants, $w_{1}(z)$ and $w_{2}(z)$ 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

9.12.3

$-\operatorname{Gi}\left(z\right)$ ,

$\operatorname{Hi}\left(z\right)$ ,

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

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous 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 : complex variable
Source:: Olver (1997b, p. 430)
Permalink:: http://dlmf.nist.gov/9.12.E3
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png
See also:: Annotations for §9.12(i), §9.12 and Ch.9

where

9.12.4 $\operatorname{Gi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{z}^{% \infty}\operatorname{Ai}\left(t\right)\,\mathrm{d}t+\operatorname{Ai}\left(z% \right)\int_{0}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t,$

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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 , $\int$ : integral and : 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 $\operatorname{Hi}\left(z\right)=\operatorname{Bi}\left(z\right)\int_{-\infty}^% {z}\operatorname{Ai}\left(t\right)\,\mathrm{d}t-\operatorname{Ai}\left(z\right% )\int_{-\infty}^{z}\operatorname{Bi}\left(t\right)\,\mathrm{d}t.$

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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 , $\int$ : integral and : 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 .

§9.12(ii) Graphs

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Keywords:: Scorer functions, graphs
Notes:: These graphs were produced by NIST.
Permalink:: http://dlmf.nist.gov/9.12.ii
See also:: Annotations for §9.12 and Ch.9

See Figures 9.12.1 and 9.12.2.

Figure 9.12.1: $\operatorname{Gi}\left(x\right)$ , $\operatorname{Gi}'\left(x\right)$ . Magnify

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : 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
See also:: Annotations for §9.12(ii), §9.12 and Ch.9

Figure 9.12.2: $\operatorname{Hi}\left(x\right)$ , $\operatorname{Hi}'\left(x\right)$ . Magnify

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Symbols:: $\operatorname{Hi}\left(\NVar{z}\right)$ : 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
See also:: Annotations for §9.12(ii), §9.12 and Ch.9

§9.12(iii) Initial Values

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Keywords:: Scorer functions, differential equation, initial values
Notes:: See Olver (1997b, p. 431).
Referenced by:: (9.12.11), (9.12.12), (9.12.13), (9.12.14), (9.12.20), §9.12(v)
Permalink:: http://dlmf.nist.gov/9.12.iii
See also:: Annotations for §9.12 and Ch.9

9.12.6 $\operatorname{Gi}\left(0\right)=\tfrac{1}{2}\operatorname{Hi}\left(0\right)=% \tfrac{1}{3}\operatorname{Bi}\left(0\right)={1\Big/\!\left(3^{7/6}\Gamma\left(% \tfrac{2}{3}\right)\right)=0.20497\;55424\ldots,}$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function) and $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function)
Source:: Olver (1997b, p. 431)
Permalink:: http://dlmf.nist.gov/9.12.E6
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(iii), §9.12 and Ch.9

9.12.7 $\operatorname{Gi}'\left(0\right)=\tfrac{1}{2}\operatorname{Hi}'\left(0\right)=% \tfrac{1}{3}\operatorname{Bi}'\left(0\right)=1\Big/\left(3^{5/6}\Gamma\left(% \tfrac{1}{3}\right)\right)=0.14942\;94524\ldots.$

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Symbols:: $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function) and $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function)
Source:: Olver (1997b, p. 431)
Permalink:: http://dlmf.nist.gov/9.12.E7
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(iii), §9.12 and Ch.9

§9.12(iv) Numerically Satisfactory Solutions

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Keywords:: Scorer functions, differential equation, numerically satisfactory solutions
Notes:: Refer to the asymptotic expansions given in §§9.7(ii) and 9.12(viii).
Permalink:: http://dlmf.nist.gov/9.12.iv
See also:: Annotations for §9.12 and Ch.9

$-\operatorname{Gi}\left(x\right)$ 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 $-\infty<x\leq 0$ .

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

9.12.8 $-\operatorname{Gi}\left(z\right),\operatorname{Ai}\left(z\right),\operatorname% {Bi}\left(z\right),$ $|\operatorname{ph}z|\leq\tfrac{1}{3}\pi$ ,

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Bi}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{ph}$ : phase and : complex variable
Source:: Olver (1997b, p. 432)
Permalink:: http://dlmf.nist.gov/9.12.E8
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(iv), §9.12 and Ch.9

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

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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, $\operatorname{ph}$ : phase and : complex variable
Source:: Olver (1997b, p. 432)
Permalink:: http://dlmf.nist.gov/9.12.E9
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(iv), §9.12 and Ch.9

and

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

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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, $\operatorname{ph}$ : phase and : complex variable
Source:: Olver (1997b, p. 432)
Permalink:: http://dlmf.nist.gov/9.12.E10
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(iv), §9.12 and Ch.9

§9.12(v) Connection Formulas

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Keywords:: Scorer functions, connection formulas
Notes:: These results can be verified with the aid of §§9.2(ii) and 9.12(iii).
Permalink:: http://dlmf.nist.gov/9.12.v
See also:: Annotations for §9.12 and Ch.9

9.12.11 $\operatorname{Gi}\left(z\right)+\operatorname{Hi}\left(z\right)=\operatorname{% Bi}\left(z\right),$

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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 : 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 $\operatorname{Gi}\left(z\right)=\tfrac{1}{2}e^{\pi i/3}\operatorname{Hi}\left(% ze^{-2\pi i/3}\right)+\tfrac{1}{2}e^{-\pi i/3}\operatorname{Hi}\left(ze^{2\pi i% /3}\right),$

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous 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 : complex variable
Proof sketch:: Verify with the aid of §§9.2(ii) and 9.12(iii).
Referenced by:: (9.12.30)
Permalink:: http://dlmf.nist.gov/9.12.E12
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(v), §9.12 and Ch.9

9.12.13 $\operatorname{Gi}\left(z\right)=e^{\mp\pi i/3}\operatorname{Hi}\left(ze^{\pm 2% \pi i/3}\right)\pm i\operatorname{Ai}\left(z\right),$

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit and : complex variable
Proof sketch:: Verify with the aid of §§9.2(ii) and 9.12(iii).
Permalink:: http://dlmf.nist.gov/9.12.E13
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(v), §9.12 and Ch.9

9.12.14 $\operatorname{Hi}\left(z\right)=e^{\pm 2\pi i/3}\operatorname{Hi}\left(ze^{\pm 2% \pi i/3}\right)+2e^{\mp\pi i/6}\operatorname{Ai}\left(ze^{\mp 2\pi i/3}\right).$

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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 : 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

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Keywords:: Maclaurin series, Scorer functions
Referenced by:: §9.17(i)
Permalink:: http://dlmf.nist.gov/9.12.vi
See also:: Annotations for §9.12 and Ch.9

9.12.15 $\operatorname{Gi}\left(z\right)=\frac{3^{-2/3}}{\pi}\*\sum_{k=0}^{\infty}\cos% \left(\frac{2k-1}{3}\pi\right)\Gamma\left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)% ^{k}}{k!},$

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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 ), : nonnegative integer and : 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 $\operatorname{Gi}'\left(z\right)=\frac{3^{-1/3}}{\pi}\*\sum_{k=0}^{\infty}\cos% \left(\frac{2k+1}{3}\pi\right)\Gamma\left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)% ^{k}}{k!}.$

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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 ), : nonnegative integer and : complex variable
Proof sketch:: Differentiate (9.12.15).
Permalink:: http://dlmf.nist.gov/9.12.E16
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vi), §9.12 and Ch.9

9.12.17 $\operatorname{Hi}\left(z\right)=\frac{3^{-2/3}}{\pi}\sum_{k=0}^{\infty}\Gamma% \left(\frac{k+1}{3}\right)\frac{(3^{1/3}z)^{k}}{k!},$

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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 ), : nonnegative integer and : complex variable
Proof sketch:: Expand the integral in (9.12.20) in powers of 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 $\operatorname{Hi}'\left(z\right)=\frac{3^{-1/3}}{\pi}\sum_{k=0}^{\infty}\Gamma% \left(\frac{k+2}{3}\right)\frac{(3^{1/3}z)^{k}}{k!}.$

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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 ), : nonnegative integer and : complex variable
Proof sketch:: Differentiate (9.12.17).
Permalink:: http://dlmf.nist.gov/9.12.E18
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vi), §9.12 and Ch.9

§9.12(vii) Integral Representations

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Keywords:: Scorer functions, integral representations, integrals of modified Bessel functions, over infinite intervals
Notes:: See Lee (1980), Gordon (1970, Appendix A), Exton (1983).
Permalink:: http://dlmf.nist.gov/9.12.vii
See also:: Annotations for §9.12 and Ch.9

9.12.19 $\operatorname{Gi}\left(x\right)=\frac{1}{\pi}\int_{0}^{\infty}\sin\left(\tfrac% {1}{3}t^{3}+xt\right)\,\mathrm{d}t,$ $x\in\mathbb{R}$ .

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\in$ : element of, $\int$ : integral, $\mathbb{R}$ : real line, $\sin\NVar{z}$ : sine function and : real variable
Proof sketch:: Combine (9.5.3) and (9.12.11).
Permalink:: http://dlmf.nist.gov/9.12.E19
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

9.12.20 $\operatorname{Hi}\left(z\right)=\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-% \tfrac{1}{3}t^{3}+zt\right)\,\mathrm{d}t,$

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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 , $\exp\NVar{z}$ : exponential function, $\int$ : integral and : 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 $\operatorname{Gi}\left(z\right)=-\frac{1}{\pi}\int_{0}^{\infty}\exp\left(-% \tfrac{1}{3}t^{3}-\tfrac{1}{2}zt\right)\cos\left(\tfrac{1}{2}\sqrt{3}zt+\tfrac% {2}{3}\pi\right)\,\mathrm{d}t.$

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of , $\exp\NVar{z}$ : exponential function, $\int$ : integral and : complex variable
Source:: Lee (1980, (2.20))
Permalink:: http://dlmf.nist.gov/9.12.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

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

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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 , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\operatorname{ph}$ : phase, : complex variable and $\zeta$ : variable
Source:: Gordon (1970, Appendix A)
Permalink:: http://dlmf.nist.gov/9.12.E22
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

9.12.23 $\operatorname{Gi}\left(x\right)=\frac{4x^{2}}{3^{3/2}\pi^{2}}\pvint_{0}^{% \infty}\frac{K_{1/3}\left(t\right)}{\zeta^{2}-t^{2}}\,\mathrm{d}t,$

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\pvint_{\NVar{a}}^{\NVar{b}}$ : Cauchy principal value, : real variable and $\zeta$ : variable
Source:: Gordon (1970, Appendix A)
Permalink:: http://dlmf.nist.gov/9.12.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

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

Mellin–Barnes Type Integral

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Keywords:: Scorer functions, integral representations
See also:: Annotations for §9.12(vii), §9.12 and Ch.9

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

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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, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : complex variable
Keywords:: Mellin transform
Source:: Exton (1983, (2.1), p. 100)
Permalink:: http://dlmf.nist.gov/9.12.E24
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(vii), §9.12(vii), §9.12 and Ch.9

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

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Keywords:: Scorer functions, asymptotic expansions
Notes:: see Olver (1997b, pp. 431–432), Rothman (1954a).
Referenced by:: §9.12(iv), §9.17(i)
Permalink:: http://dlmf.nist.gov/9.12.viii
See also:: Annotations for §9.12 and Ch.9

Functions and Derivatives

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See also:: Annotations for §9.12(viii), §9.12 and Ch.9

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

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

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Proof sketch:: See Olver (1997b, pp. 431–432).
Referenced by:: (9.12.30)
Permalink:: http://dlmf.nist.gov/9.12.E25
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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

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Symbols:: $\operatorname{Gi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Proof sketch:: Refer to §2.1(ii).
Permalink:: http://dlmf.nist.gov/9.12.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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

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Symbols:: $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Proof sketch:: See Olver (1997b, pp. 431–432).
Referenced by:: (9.12.31)
Permalink:: http://dlmf.nist.gov/9.12.E27
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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

ⓘ

Symbols:: $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, : factorial (as in ), $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable and $\delta$ : small positive constant
Proof sketch:: Refer to §2.1(ii).
Permalink:: http://dlmf.nist.gov/9.12.E28
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

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

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Symbols:: $\operatorname{Hi}\left(\NVar{z}\right)$ : Scorer function (inhomogeneous Airy function), $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\operatorname{ph}$ : phase, : nonnegative integer, : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient
Permalink:: http://dlmf.nist.gov/9.12.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §9.12(viii), §9.12(viii), §9.12 and Ch.9

Integrals

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Keywords:: Scorer functions, asymptotic expansions, integrals
See also:: Annotations for §9.12(viii), §9.12 and Ch.9

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

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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 , : factorial (as in ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, : nonnegative integer, : 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 , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\operatorname{ph}$ : phase, : nonnegative integer, : real variable, : 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 by ( $\leq 0$ ) 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 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

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Keywords:: Scorer functions, zeros
Permalink:: http://dlmf.nist.gov/9.12.ix
See also:: Annotations for §9.12 and Ch.9

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.

9.11 Products 9.13 Generalized Airy Functions

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