Laplace equation
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21: 2.3 Integrals of a Real Variable
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►Assume that the Laplace transform
…Then
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§2.3(iii) Laplace’s Method
… ►For error bounds for Watson’s lemma and Laplace’s method see Boyd (1993) and Olver (1997b, Chapter 3). These references and Wong (1989, Chapter 2) also contain examples. …22: Errata
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Additions
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Equations (15.2.3_5), (19.11.6_5)
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Equation (14.15.23)
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Equation (10.13.4)
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Equations (4.45.8), (4.45.9)
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Equation (16.16.5_5).
These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.
Four of the terms were rewritten for improved clarity.
has been generalized to cover an additional case.
These equations have been rewritten to improve the numerical computation of .
23: Bibliography H
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The Laplace transform for expressions that contain a probability function.
Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
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Asymptotic expansion of Laplace transforms near the origin.
SIAM J. Math. Anal. 1 (1), pp. 118–130.
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Poncelet Polygons and the Painlevé Equations.
In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.),
pp. 151–185.
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Estimates of the stability intervals for Hill’s equation.
Proc. Amer. Math. Soc. 14 (6), pp. 930–932.
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Differential Equations: A Modern Approach.
Holt, Rinehart and Winston, New York.
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24: 9.10 Integrals
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►Let be any solution of Airy’s equation (9.2.1).
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§9.10(v) Laplace Transforms
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9.10.15
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9.10.16
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►For Laplace transforms of products of Airy functions see Shawagfeh (1992).
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25: 16.15 Integral Representations and Integrals
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16.15.3
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, ,
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►For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
26: Bibliography O
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Tables of Laplace Transforms.
Springer-Verlag, Berlin-New York.
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Studies on the Painlevé equations. I. Sixth Painlevé equation
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Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
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Studies on the Painlevé equations. II. Fifth Painlevé equation
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Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
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Studies on the Painlevé equations. IV. Third Painlevé equation
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Funkcial. Ekvac. 30 (2-3), pp. 305–332.
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Hyperasymptotics for nonlinear ODEs. II. The first Painlevé equation and a second-order Riccati equation.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461 (2062), pp. 3005–3021.
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27: 10.43 Integrals
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10.43.22
►For the second equation there is a cut in the -plane along the interval , and all quantities assume their principal values (§4.2(i)).
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28: Bibliography
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Asymptotics of solutions of the generalized sine-Gordon equation, the third Painlevé equation and the d’Alembert equation.
Dokl. Akad. Nauk SSSR 280 (2), pp. 265–268 (Russian).
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Nonlinear chains and Painlevé equations.
Phys. D 73 (4), pp. 335–351.
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The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates.
Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.
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Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations.
Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
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Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum.
SIAM J. Math. Anal. 5, pp. 119–124.
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29: 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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Problem 72-21, Laplace transforms of Airy functions.
SIAM Rev. 15 (4), pp. 796–798.
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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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Special classes of solutions of Painlevé equations.
Differ. Uravn. 18 (3), pp. 419–429 (Russian).
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Theory of Painlevé’s equations.
Differ. Uravn. 11 (11), pp. 373–376 (Russian).
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