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1: 9.1 Special Notation

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$k$	nonnegative integer, except in §9.9(iii).
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►The main functions treated in this chapter are the Airy functions $\mathrm{Ai}\left(z\right)$ and $\mathrm{Bi}\left(z\right)$, and the Scorer functions $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ (also known as inhomogeneous Airy functions). …

2: 9.12 Scorer Functions

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9.12.4 $$\mathrm{Gi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt+\mathrm{Ai}\left(z\right){\int }_0^z\mathrm{Bi}\left(t\right)dt,$$

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Defines:

$\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §9.12(i), §9.12 and Ch.9

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9.12.5 $$\mathrm{Hi}\left(z\right)=\mathrm{Bi}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt-\mathrm{Ai}\left(z\right){\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt.$$

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Defines:

$\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $dx$: 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:

pMML, png, TeX

See also:

Annotations for §9.12(i), §9.12 and Ch.9

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9.12.6 $$\mathrm{Gi}\left(0\right)=\frac{1}{2}\mathrm{Hi}\left(0\right)=\frac{1}{3}\mathrm{Bi}\left(0\right)=1/\left(3^{7/6}\mathrm{\Gamma }\left(\frac{2}{3}\right)\right)=\mathrm{0.20497\hspace{0.33em}55424}\mathrm{\dots },$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Source:

Olver (1997b, p. 431)

Permalink:

http://dlmf.nist.gov/9.12.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iii), §9.12 and Ch.9

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9.12.7 $${\mathrm{Gi}}^{\prime }\left(0\right)=\frac{1}{2}{\mathrm{Hi}}^{\prime }\left(0\right)=\frac{1}{3}{\mathrm{Bi}}^{\prime }\left(0\right)=1/\left(3^{5/6}\mathrm{\Gamma }\left(\frac{1}{3}\right)\right)=\mathrm{0.14942\hspace{0.33em}94524}\mathrm{\dots }.$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function) and $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function)

Source:

Olver (1997b, p. 431)

Permalink:

http://dlmf.nist.gov/9.12.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §9.12(iii), §9.12 and Ch.9

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

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: 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:

pMML, png, TeX

See also:

Annotations for §9.12(v), §9.12 and Ch.9

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3: 14.29 Generalizations

… ►For inhomogeneous versions of the associated Legendre equation, and properties of their solutions, see Babister (1967, pp. 252–264).

4: 11.13 Methods of Computation

… ►A comprehensive approach is to integrate the defining inhomogeneous differential equations (11.2.7) and (11.2.9) numerically, using methods described in §3.7. … ►Sequences of values of ${𝐇}_{\nu }\left(z\right)$ and ${𝐋}_{\nu }\left(z\right)$, with $z$ fixed, can be computed by application of the inhomogeneous difference equations (11.4.23) and (11.4.25). …

5: 3.6 Linear Difference Equations

… ►If $d_n=0$, $\forall n$, then the difference equation is homogeneous; otherwise it is inhomogeneous. … ►

§3.6(iv) Inhomogeneous Equations

… ►It is applicable equally to the computation of the recessive solution of the homogeneous equation (3.6.3) or the computation of any solution $w_n$ of the inhomogeneous equation (3.6.1) for which the conditions of §3.6(iv) are satisfied. … ►Thus in the inhomogeneous case it may sometimes be necessary to recur backwards to achieve stability. … ►or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. …

6: 11.9 Lommel Functions

… ►The inhomogeneous Bessel differential equation … ►For uniform asymptotic expansions, for large $\nu$ and fixed $\mu =-1,0,1,2,\mathrm{\dots }$, of solutions of the inhomogeneous modified Bessel differential equation that corresponds to (11.9.1) see Olver (1997b, pp. 388–390). For an error bound for (11.9.9) and an exponentially-improved extension see Nemes (2015b). …

7: 1.13 Differential Equations

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§1.13(iii) Inhomogeneous Equations

►The inhomogeneous (or nonhomogeneous) equation … ►

Variation of Parameters

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8: 3.7 Ordinary Differential Equations

… ►If $h=0$ the differential equation is homogeneous, otherwise it is inhomogeneous. … … ►(This can happen only for inhomogeneous equations.) … ►The latter is especially useful if the endpoint $b$ of $𝒫$ is at $\mathrm{\infty }$, or if the differential equation is inhomogeneous. …

9: 11.2 Definitions

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§11.2(ii) Differential Equations

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Modified Struve’s Equation

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10: 9.10 Integrals

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9.10.1 $${\int }_z^{\mathrm{\infty }}\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.4) and its differentiated form with the first of (9.10.11)

Referenced by:

(9.10.2)

Permalink:

http://dlmf.nist.gov/9.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.2 $${\int }_{-\mathrm{\infty }}^z\mathrm{Ai}\left(t\right)dt=\pi \left(\mathrm{Ai}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Ai}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right),$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.11) and its differentiated form with (9.10.1), then apply (9.2.7) and (9.10.11)

Permalink:

http://dlmf.nist.gov/9.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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9.10.3 $${\int }_{-\mathrm{\infty }}^z\mathrm{Bi}\left(t\right)dt={\int }_0^z\mathrm{Bi}\left(t\right)dt=\pi \left({\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Gi}\left(z\right)-\mathrm{Bi}\left(z\right){\mathrm{Gi}}^{\prime }\left(z\right)\right)=\pi \left(\mathrm{Bi}\left(z\right){\mathrm{Hi}}^{\prime }\left(z\right)-{\mathrm{Bi}}^{\prime }\left(z\right)\mathrm{Hi}\left(z\right)\right).$$

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Symbols:

$\mathrm{Bi}\left(z\right)$: Airy function, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral and $z$: complex variable

Proof sketch:

Combine (9.12.5) and its differentiated form with (9.2.7).

Permalink:

http://dlmf.nist.gov/9.10.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §9.10(i), §9.10 and Ch.9

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