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21: 2.3 Integrals of a Real Variable
Assume that the Laplace transform …Then …
§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
  • Equations (15.2.3_5), (19.11.6_5)

    These equations, originally added in Other Changes and Other Changes, respectively, have been assigned interpolated numbers.

  • Equation (4.8.14)

    The constraint a 0 was added.

  • Equation (14.15.23)

    Four of the terms were rewritten for improved clarity.

  • Equation (10.13.4)

    has been generalized to cover an additional case.

  • Equations (4.45.8), (4.45.9)

    These equations have been rewritten to improve the numerical computation of arctan x .

  • 23: Bibliography H
  • P. I. Hadži (1973) The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).
  • R. A. Handelsman and J. S. Lew (1970) Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.
  • N. J. Hitchin (1995) Poncelet Polygons and the Painlevé Equations. In Geometry and Analysis (Bombay, 1992), Ramanan (Ed.), pp. 151–185.
  • H. Hochstadt (1963) Estimates of the stability intervals for Hill’s equation. Proc. Amer. Math. Soc. 14 (6), pp. 930–932.
  • H. Hochstadt (1964) Differential Equations: A Modern Approach. Holt, Rinehart and Winston, New York.
  • 24: 16.15 Integral Representations and Integrals
    16.15.3 F 3 ( α , α ; β , β ; γ ; x , y ) = Γ ( γ ) Γ ( β ) Γ ( β ) Γ ( γ - β - β ) Δ u β - 1 v β - 1 ( 1 - u - v ) γ - β - β - 1 ( 1 - u x ) α ( 1 - v y ) α d u d v , ( γ - β - β ) > 0 , β > 0 , β > 0 ,
    For inverse Laplace transforms of Appell functions see Prudnikov et al. (1992b, §3.40).
    25: 9.10 Integrals
    Let w ( z ) be any solution of Airy’s equation (9.2.1). …
    §9.10(v) Laplace Transforms
    9.10.15 0 e - p t Ai ( - t ) d t = 1 3 e p 3 / 3 ( Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) + Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) ) , p > 0 ,
    9.10.16 0 e - p t Bi ( - t ) d t = 1 3 e p 3 / 3 ( Γ ( 2 3 , 1 3 p 3 ) Γ ( 2 3 ) - Γ ( 1 3 , 1 3 p 3 ) Γ ( 1 3 ) ) , p > 0 .
    For Laplace transforms of products of Airy functions see Shawagfeh (1992). …
    26: Bibliography O
  • F. Oberhettinger and L. Badii (1973) Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.
  • K. Okamoto (1987a) Studies on the Painlevé equations. I. Sixth Painlevé equation P VI . Ann. Mat. Pura Appl. (4) 146, pp. 337–381.
  • K. Okamoto (1987b) Studies on the Painlevé equations. II. Fifth Painlevé equation P V . Japan. J. Math. (N.S.) 13 (1), pp. 47–76.
  • K. Okamoto (1987c) Studies on the Painlevé equations. IV. Third Painlevé equation P III . Funkcial. Ekvac. 30 (2-3), pp. 305–332.
  • A. B. Olde Daalhuis (2005b) 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.
  • 27: Bibliography
  • A. S. Abdullaev (1985) 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).
  • V. È. Adler (1994) Nonlinear chains and Painlevé equations. Phys. D 73 (4), pp. 335–351.
  • F. M. Arscott (1967) The Whittaker-Hill equation and the wave equation in paraboloidal co-ordinates. Proc. Roy. Soc. Edinburgh Sect. A 67, pp. 265–276.
  • U. M. Ascher and L. R. Petzold (1998) Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.
  • R. Askey (1974) 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.
  • 28: 10.43 Integrals
    10.43.22 0 t μ - 1 e - a t K ν ( t ) d t = { ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( 1 - a 2 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , - 1 < a < 1 , ( 1 2 π ) 1 2 Γ ( μ - ν ) Γ ( μ + ν ) ( a 2 - 1 ) - 1 2 μ + 1 4 P ν - 1 2 - μ + 1 2 ( a ) , a 0 , a 1 .
    For the second equation there is a cut in the a -plane along the interval [ 0 , 1 ] , and all quantities assume their principal values (§4.2(i)). …
    29: Bibliography G
  • W. Gautschi (1997b) 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.
  • A. G. Gibbs (1973) Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.
  • J. J. Gray (2000) Linear Differential Equations and Group Theory from Riemann to Poincaré. 2nd edition, Birkhäuser Boston Inc., Boston, MA.
  • V. I. Gromak and N. A. Lukaševič (1982) Special classes of solutions of Painlevé equations. Differ. Uravn. 18 (3), pp. 419–429 (Russian).
  • V. I. Gromak (1975) Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).
  • 30: 7.7 Integral Representations
    7.7.4 0 e - a t t + z 2 d t = π a e a z 2 erfc ( a z ) , a > 0 , z > 0 .
    7.7.15 0 e - a t cos ( t 2 ) d t = π 2 f ( a 2 π ) , a > 0 ,
    7.7.16 0 e - a t sin ( t 2 ) d t = π 2 g ( a 2 π ) , a > 0 .