inhomogeneous equations

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11: Bibliography O

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On higher-order Stokes phenomena of an inhomogeneous linear ordinary differential equation. J. Comput. Appl. Math. 169 (1), pp. 235–246.

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External Links:

ISSN 0377-0427, Document, MathReview (Jorge Mozo Fernández), ZentralBlatt

Referenced by:

§2.11(iv).

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12: 9.12 Scorer Functions

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9.12.20 $$\mathrm{Hi}\left(z\right)=\frac{1}{\pi }{\int }_0^{\mathrm{\infty }}\exp \left(-\frac{1}{3}{t}^3+zt\right)dt,$$

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Symbols:

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

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

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9.12.30 $${\int }_0^z\mathrm{Gi}\left(t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }-\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{1}{3}\pi -\delta$.

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Gi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: 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.)

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http://dlmf.nist.gov/9.12.E30

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

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9.12.31 $${\int }_0^z\mathrm{Hi}\left(-t\right)dt\sim \frac{1}{\pi }\ln z+\frac{2\gamma +\ln 3}{3\pi }+\frac{1}{\pi }\sum _{k=1}^{\mathrm{\infty }}{(-1)}^{k-1}\frac{(3k-1)!}{k!{(3z^3)}^k},$$ $|\mathrm{ph}z|\le \frac{2}{3}\pi -\delta$,

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Symbols:

$\gamma$: Euler’s constant, $\mathrm{Hi}\left(z\right)$: Scorer function (inhomogeneous Airy function), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $!$: factorial (as in $n!$), $\int$: integral, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: 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 $-x$ ($\le 0$) in (9.12.20) and integrate (§1.5(v)) to obtain $\pi {\int }_0^x\mathrm{Hi}\left(-t\right)dt={\int }_0^{\mathrm{\infty }}(1-{\mathrm{e}}^{-xt}){\mathrm{e}}^{-\frac{1}{3}{t}^3}{t}^{-1}dt$. Next, integrate the right-hand side of this equation by parts—integrating the factor $t^{-1}$ and differentiating the rest. As $x\to \mathrm{\infty }$ the asymptotic expansions of ${\int }_0^{\mathrm{\infty }}x{\mathrm{e}}^{-xt}{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ and ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-xt}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ follow from (2.3.9). Also, ${\int }_0^{\mathrm{\infty }}{t}^2{\mathrm{e}}^{-\frac{1}{3}{t}^3}(\ln t)dt$ 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

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

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13: Bibliography K

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Asymptotic behavior of the solutions of the Painlevé equation of the first kind. Differ. Uravn. 24 (10), pp. 1684–1695 (Russian).

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Notes:

In Russian. English translation: Differential Equations 24(1988), no. 10, pp. 1107–1115 (1989)

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.11(i).

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Asymptotic solution of Maxwell’s equations near caustics. Izv. Vuz. Radiofiz. 7, pp. 1049–1056.

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Referenced by:

§36.14(i).

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Two new asymptotic methods in the theory of wave propagation in inhomogeneous media. Sov. Phys. Acoust. 14, pp. 1–17.

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External Links:

ISSN 0038-562X

Referenced by:

§36.14(iv).

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Quantum-Theoretical Formalism for Inhomogeneous Graded-Index Waveguides. Akademie Verlag, Berlin-New York.

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External Links:

ISBN 3-05-501598-3, MathReview (Apostolos Vourdas), ZentralBlatt

Referenced by:

§10.73(i).

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The Korteweg-de Vries Equation and Related Evolution Equations. In Nonlinear Wave Motion (Proc. AMS-SIAM Summer Sem., Clarkson Coll. Tech., Potsdam, N.Y., 1972), A. C. Newell (Ed.), Lectures in Appl. Math., Vol. 15, pp. 61–83.

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External Links:

MathReview (G. L. Lamb, Jr.), ZentralBlatt

Referenced by:

§22.19(iii).

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14: Bibliography L

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Some differential equations and associated integral equations. Quart. J. Math. (Oxford) 5, pp. 81–97.

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External Links:

Document, ZentralBlatt

Referenced by:

§31.10(i), §31.9(i).

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Solutions to a generalized spheroidal wave equation: Teukolsky’s equations in general relativity, and the two-center problem in molecular quantum mechanics. J. Math. Phys. 27 (5), pp. 1238–1265.

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External Links:

ISSN 0022-2488, Document, MathReview (Basilis C. Xanthopoulos), ZentralBlatt

Referenced by:

§30.12, §31.17(ii).

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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).

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Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation. J. Differential Equations 162 (1), pp. 27–63.

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External Links:

ISSN 0022-0396, Document, MathReview (John Albert), ZentralBlatt

Referenced by:

§22.19(iii).

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The second Painlevé equation. Differ. Uravn. 7 (6), pp. 1124–1125 (Russian).

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Notes:

In Russian. English translation: Differential Equations 7(6), pp. 853–854

External Links:

MathReview (Z. Rubinstein), ZentralBlatt

Referenced by:

§32.7(ii).

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15: 9.13 Generalized Airy Functions

§9.13 Generalized Airy Functions

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§9.13(i) Generalizations from the Differential Equation

►Equations of the form … ► … ►and the difference equation …

16: Bibliography B

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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).

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Integral equations and exact solutions for the fourth Painlevé equation. Proc. Roy. Soc. London Ser. A 437, pp. 1–24.

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ISSN 0962-8444, FullText, MathReview (Mehmet Can), ZentralBlatt

Referenced by:

§32.11(v), §32.2(iii).

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An Introduction to Linear Difference Equations. Dover Publications Inc., New York.

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Notes:

Unaltered republication of the original 1927 Harvard University edition.

External Links:

MathReview, ZentralBlatt

Referenced by:

§8.1, Notations P, Notations Q.

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

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Ordinary differential equations. Fourth edition, John Wiley & Sons, Inc., New York.

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External Links:

ISBN 0-471-86003-4, MathReview (Michal Čverčko)

Referenced by:

§1.13(viii), §1.13(viii), §1.18(iv).

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17: Bibliography E

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Integral equations for Heun functions. Quart. J. Math., Oxford Ser. 13, pp. 107–112.

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External Links:

Document, MathReview (E. Rothe), ZentralBlatt

Referenced by:

§31.10(i).

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The Fuchsian equation of second order with four singularities. Duke Math. J. 9 (1), pp. 48–58.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i).

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Certain expansions of solutions of the Heun equation. Quart. J. Math., Oxford Ser. 15, pp. 62–69.

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External Links:

Document, MathReview (R. E. Langer), ZentralBlatt

Referenced by:

§31.11(i), §31.11(i), §31.9(i).

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A catalogue of Sturm-Liouville differential equations. In Sturm-Liouville theory, pp. 271–331.

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External Links:

Document, Link, MathReview Entry

Referenced by:

§1.18(ix), §1.18(x).

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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).

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18: Bibliography M

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

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

Referenced by:

1st item.

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On reducing the Heun equation to the hypergeometric equation. J. Differential Equations 213 (1), pp. 171–203.

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External Links:

ISSN 0022-0396, Document, MathReview (Wolfgang Bühring), ZentralBlatt

Referenced by:

§31.7(i).

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The 192 solutions of the Heun equation. Math. Comp. 76 (258), pp. 811–843.

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External Links:

ISSN 0025-5718, Document, MathReview (Wadim Zudilin), ZentralBlatt (Masaaki Yoshida)

Referenced by:

§31.3(iii).

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Rational solutions of the Painlevé VI equation. J. Phys. A 34 (11), pp. 2281–2294.

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External Links:

ISSN 0305-4470, Document, MathReview (Shun Shimomura), ZentralBlatt

Referenced by:

§32.8(vi).

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Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321 (1), pp. 157–195.

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External Links:

ISSN 0025-5831, Document, MathReview, ZentralBlatt

Referenced by:

§32.9(iii).

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19: 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

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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

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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

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

… ►Let $w(z)$ be any solution of Airy’s equation (9.2.1). … ►

9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt,$$ .

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Proof sketch:

To prove this equation, combine (13.4.5) with (13.6.11) and use (5.5.3).

Referenced by:

(9.10.19), §9.17(iii), §9.5(ii), Erratum (V1.0.10) for Equation (9.10.18)

Permalink:

http://dlmf.nist.gov/9.10.E18

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Errata (effective with 1.0.10):

The original and incorrect version,

$$\mathrm{Ai}\left(z\right)=\frac{z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{2^{7/2}\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-1/2}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

taken from Schulten et al. (1979), has been corrected.

Reported 2015-03-20 by Walter Gautschi

See also:

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

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20: Bibliography G

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The Computation of Special Functions by Linear Difference Equations. In Advances in Difference Equations (Veszprém, 1995), S. Elaydi, I. Győri, and G. Ladas (Eds.), pp. 213–243.

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External Links:

ISBN 90-5699-521-9, MathReview, ZentralBlatt

Referenced by:

§3.6(iii), §3.6(vii).

Encodings:

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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:

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Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.

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External Links:

ISBN 0-8176-3837-7, MathReview, ZentralBlatt

Referenced by:

§15.17(v).

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Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).

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Notes:

In Russian. English translation: Differential Equations 18(3), pp. 317–326.

External Links:

ISSN 0374-0641, MathReview, ZentralBlatt

Referenced by:

§32.10(iv), §32.10(v), §32.8(iii), §32.8(v), §32.9(ii).

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Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

Encodings:

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