Laplace method

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1: 2.4 Contour Integrals

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§2.4(iii) Laplace’s Method

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2: 2.3 Integrals of a Real Variable

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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. … ►When $x\to +\mathrm{\infty }$ Laplace’s method (§2.3(iii)) applies, but the form of the resulting approximation is discontinuous at $\alpha =0$. … ►The desired uniform expansion is then obtained formally as in Watson’s lemma and Laplace’s method. …

3: Bibliography N

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An explicit formula for the coefficients in Laplace’s method. Constr. Approx. 38 (3), pp. 471–487.

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External Links:

ISSN 0176-4276, Document, Link, MathReview (Pedro J. Pagola), ZentralBlatt

Referenced by:

§5.11(i), §5.11(i).

Encodings:

BibTeX

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4: 11.6 Asymptotic Expansions

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5: 35.7 Gaussian Hypergeometric Function of Matrix Argument

… ►Butler and Wood (2002) applies Laplace’s method (§2.3(iii)) to (35.7.5) to derive uniform asymptotic approximations for the functions …

6: 2.10 Sums and Sequences

… ►By application of Laplace’s method (§2.3(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

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A Handbook of Methods of Approximate Fourier Transformation and Inversion of the Laplace Transformation. Mir, Moscow.

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Notes:

Translated from the Russian by George Yankovsky. Reprint of the 1977 translation

External Links:

MathReview, ZentralBlatt

Referenced by:

§3.5(viii).

Encodings:

BibTeX

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9: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

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§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. …