Laplace method for asymptotic expansions of integrals
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1: 2.4 Contour Integrals
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§2.4(iii) Laplace’s Method
…2: 2.3 Integrals of a Real Variable
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§2.3(iii) Laplace’s Method
…3: 2.5 Mellin Transform Methods
§2.5 Mellin Transform Methods
… ►§2.5(ii) Extensions
… ►The Mellin transform method can also be extended to derive asymptotic expansions of multidimensional integrals having algebraic or logarithmic singularities, or both; see Wong (1989, Chapter 3), Paris and Kaminski (2001, Chapter 7), and McClure and Wong (1987). … ►§2.5(iii) Laplace Transforms with Small Parameters
… ►For examples in which the integral defining the Mellin transform does not exist for any value of , see Wong (1989, Chapter 3), Bleistein and Handelsman (1975, Chapter 4), and Handelsman and Lew (1970).4: 11.11 Asymptotic Expansions of Anger–Weber Functions
5: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
►§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …6: Bibliography N
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Applicable Anal. 39 (4), pp. 249–263.
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On an asymptotic expansion of the Kontorovich-Lebedev transform.
Methods Appl. Anal. 3 (1), pp. 98–108.
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Uniform asymptotic expansion for the incomplete beta function.
SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.
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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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An extension of Laplace’s method.
Constr. Approx. 51 (2), pp. 247–272.
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7: 2.11 Remainder Terms; Stokes Phenomenon
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►The rest of this section is devoted to general methods for increasing this accuracy.
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►Then by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have
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►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)):
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8: Bibliography S
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Uniform asymptotic expansions of modified Mathieu functions.
J. Reine Angew. Math. 247, pp. 1–17.
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Gaussian-transform method for molecular integrals. I. Formulation for energy integrals.
J. Chem. Phys. 43 (2), pp. 398–414.
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A simple approach to asymptotic expansions for Fourier integrals of singular functions.
Appl. Math. Comput. 216 (11), pp. 3378–3385.
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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels.
SIAM J. Math. Anal. 11 (5), pp. 828–841.
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Uniform asymptotic expansions of Hermite polynomials.
M. Phil. thesis, City University of Hong Kong.
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9: Bibliography T
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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions.
Quart. Appl. Math. 43 (1), pp. 103–123.
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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions.
SIAM J. Math. Anal. 21 (1), pp. 241–261.
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Uniform asymptotic expansions of integrals: A selection of problems.
J. Comput. Appl. Math. 65 (1-3), pp. 395–417.
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Asymptotic Methods for Integrals.
Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.
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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters.
Integral Transforms Spec. Funct. 33 (1), pp. 16–31.
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10: Bibliography B
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Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions.
Expo. Math. 2 (4), pp. 289–347.
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Asymptotic Expansions of Integrals.
Holt, Rinehart, and Winston, New York.
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Uniform asymptotic expansion of Charlier polynomials.
Methods Appl. Anal. 1 (3), pp. 294–313.
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Approximations for the late coefficients in asymptotic expansions arising in the method of steepest descents.
Methods Appl. Anal. 2 (4), pp. 475–489.
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On the asymptotic expansion of some integrals.
Arch. Math. (Basel) 42 (3), pp. 253–259.
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