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1: 2.4 Contour Integrals
§2.4(iii) Laplace’s Method
2: 2.3 Integrals of a Real Variable
§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. … When x + Laplace’s method2.3(iii)) applies, but the form of the resulting approximation is discontinuous at α = 0 . … The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …
3: Bibliography N
  • G. Nemes (2013a) An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.
  • 4: 11.6 Asymptotic Expansions
    5: 35.7 Gaussian Hypergeometric Function of Matrix Argument
    Butler and Wood (2002) applies Laplace’s method2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …
    6: 2.10 Sums and Sequences
    By application of Laplace’s method2.3(iii)) and use again of (5.11.7), we obtain …
    7: 2.11 Remainder Terms; Stokes Phenomenon
    Then by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have …
    8: Bibliography K
  • V. I. Krylov and N. S. Skoblya (1985) A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.
  • 9: 2.5 Mellin Transform Methods
    §2.5 Mellin Transform Methods
    §2.5(ii) Extensions
    See also Brüning (1984) for a different approach.
    §2.5(iii) Laplace Transforms with Small Parameters
    Let h ( t ) satisfy (2.5.18) and (2.5.20) with c > - 1 , and consider the Laplace transform …
    10: 3.5 Quadrature
    Stroud and Secrest (1966) includes computational methods and extensive tables. … Further methods are given in Clendenin (1966) and Lyness (1985). …
    Example. Laplace Transform Inversion
    In fact from (7.14.4) and the inversion formula for the Laplace transform (§1.14(iii)) we have … For integrals in higher dimensions, Monte Carlo methods are another—often the only—alternative. …