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Laplace method

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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.
  • G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.
  • 4: 11.11 Asymptotic Expansions of Anger–Weber Functions
    11.11.10 𝐀 ν ( λ ν ) 1 π k = 0 ( 2 k ) ! a k ( λ ) ν 2 k + 1 , ν , | ph ν | π δ .
    11.11.11 𝐀 ν ( λ ν ) ( 2 π ν ) 1 / 2 e ν μ k = 0 ( 1 2 ) k b k ( λ ) ν k , ν , | ph ν | π 2 δ ,
    5: 11.6 Asymptotic Expansions
    6: 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 …
    7: 2.10 Sums and Sequences
    By application of Laplace’s method2.3(iii)) and use again of (5.11.7), we obtain …
    8: 2.11 Remainder Terms; Stokes Phenomenon
    Then by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have …
    9: 15.4 Special Cases
    15.4.34 F ( 3 a , a ; 2 a ; e i π / 3 ) = π e i π a / 2 2 2 a Γ ( 1 2 + a ) 3 ( 3 a + 1 ) / 2 ( 1 Γ ( 1 3 + a ) Γ ( 2 3 ) + 1 Γ ( 2 3 + a ) Γ ( 1 3 ) ) ,
    10: 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.