inhomogeneous Airy functions

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

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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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9.12.12 $$\mathrm{Gi}\left(z\right)=\frac{1}{2}{\mathrm{e}}^{\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{-2\pi \mathrm{i}/3}\right)+\frac{1}{2}{\mathrm{e}}^{-\pi \mathrm{i}/3}\mathrm{Hi}\left(z{\mathrm{e}}^{2\pi \mathrm{i}/3}\right),$$

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Symbols:

$\mathrm{Gi}\left(z\right)$: 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, $\mathrm{e}$: 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:

(9.12.30)

Permalink:

http://dlmf.nist.gov/9.12.E12

Encodings:

pMML, png, TeX

See also:

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

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

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Symbols:

$\mathrm{Ai}\left(z\right)$: 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, $\mathrm{e}$: 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).

Permalink:

http://dlmf.nist.gov/9.12.E13

Encodings:

pMML, png, TeX

See also:

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

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3: 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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4: Bibliography E

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The asymptotic behaviour of the inhomogeneous Airy function $\mathrm{Hi}(z)$ . Math. Chronicle 12, pp. 99–104.

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External Links:

ISSN 0581-1155, MathReview (Ram Kishore Saxena), ZentralBlatt

Referenced by:

(9.12.24), §9.12(vii).

Encodings:

BibTeX

5: 11.11 Asymptotic Expansions of Anger–Weber Functions

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11.11.17 $${𝐀}_{-\nu }\left(\nu +a{\nu }^{1/3}\right)=2^{1/3}{\nu }^{-1/3}\mathrm{Hi}\left(-2^{1/3}a\right)+O\left({\nu }^{-1}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), ${𝐀}_{\nu }\left(z\right)$: Anger–Weber function, $\nu$: real or complex order and $a_k(\lambda )$: expansion function

Sources:

Olver (1997b, pp. 103 and 352); Nemes (2020)

Referenced by:

§11.11(iii), §11.11(iii), §11.11(iii)

Permalink:

http://dlmf.nist.gov/11.11.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §11.11(iii), §11.11 and Ch.11

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6: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

… ►Swanson and Headley (1967) define independent solutions $A_n\left(z\right)$ and $B_n\left(z\right)$ of (9.13.1) by … ► … ►

7: Bibliography L

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The inhomogeneous Airy functions, $\mathrm{Gi}(z)$ and $\mathrm{Hi}(z)$ . J. Chem. Phys. 72 (1), pp. 332–336.

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External Links:

ISSN 0021-9606, Document, MathReview (V. L. Deshpande)

Referenced by:

(9.12.21), §9.12(vii), §9.17(iii).

Encodings:

BibTeX

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8: Bibliography G

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On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.

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External Links:

Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§3.5(ix), §9.17(iii).

Encodings:

BibTeX

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9: Bibliography M

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Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.

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External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

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10: Bibliography B

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Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge.

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Notes:

Reprint of the 1897 original, with a foreword by Igor Krichever.

External Links:

ISBN 0-521-49877-5, MathReview (John B. Little), ZentralBlatt

Referenced by:

§20.9(ii).

Encodings:

BibTeX

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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External Links:

ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

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Coefficient functions for an inhomogeneous turning-point problem. Mathematika 38 (2), pp. 217–238.

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External Links:

ISSN 0025-5793, MathReview (Archibald D. MacGillivray), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Automatic computation of zeros of Bessel functions and other special functions. SIAM J. Sci. Comput. 21 (4), pp. 1458–1464.

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External Links:

ISSN 1064-8275, Document, MathReview (Andrea Laforgia), ZentralBlatt

Referenced by:

§10.74(vi), §3.8(iii).

Encodings:

BibTeX

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Mathieu functions of general order: Connection formulae, base functions and asymptotic formulae. I–V. Philos. Trans. Roy. Soc. London Ser. A 301, pp. 75–162.

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External Links:

ISSN 0080-4614, FullText, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

BibTeX

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