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

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2: 2.3 Integrals of a Real Variable

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§2.3(iii) Laplace’s Method

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3: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

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§2.5(ii) Extensions

… â–ºThe 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). … â–º

§2.5(iii) Laplace Transforms with Small Parameters

… â–ºFor examples in which the integral defining the Mellin transform

\mathscr{M}\mskip-3.0muh\mskip 3.0mu\left(z\right)

does not exist for any value of

z

, 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

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11.11.10

\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim-\frac{1}{\pi}\sum_{k=0}^{\infty}% \frac{(2k)!\,a_{k}(-\lambda)}{\nu^{2k+1}},

\nu\to\infty

|\operatorname{ph}\nu|\leq\pi-\delta

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Symbols:: $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger–Weber function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : phase, $\nu$ : real or complex order, $k$ : nonnegative integer, $\delta$ : arbitrary small positive constant, $\lambda$ : parameter and $a_{k}(\lambda)$ : expansion function
Sources:: Meijer (1932); Watson (1944, §10.15); Olver (1997b, pp. 103 and 352); Nemes (2014b); Nemes (2014c)
Proof sketch:: Apply Laplace’s method to the integral (11.10.4).
Referenced by:: (11.11.18), (11.11.19), §11.11(iii), §11.11(iii)
Permalink:: http://dlmf.nist.gov/11.11.E10
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.2):: The constraint which was originally $\nu\to+\infty$ , has been extended to be $\nu\to\infty$ , $|\operatorname{ph}\nu|\leq\pi-\delta$ .
Suggested 2021-04-05 by GergÅ‘ Nemes
See also:: Annotations for §11.11(iii), §11.11 and Ch.11

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11.11.11

\mathbf{A}_{-\nu}\left(\lambda\nu\right)\sim\left(\frac{2}{\pi\nu}\right)^{1/2% }e^{-\nu\mu}\sum_{k=0}^{\infty}\frac{{\left(\tfrac{1}{2}\right)_{k}}b_{k}(% \lambda)}{\nu^{k}},

\nu\to\infty

|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta

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Symbols:: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathbf{A}_{\NVar{\nu}}\left(\NVar{z}\right)$ : Anger–Weber function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $\nu$ : real or complex order, $k$ : nonnegative integer, $\delta$ : arbitrary small positive constant, $\lambda$ : parameter and $b_{k}(\lambda)$ : expansion function
Sources:: Dingle (1973, p. 388); Olver (1997b, pp. 103 and 352)
Proof sketch:: Derivable by applying Laplace’s method to the integral (11.10.4).
Permalink:: http://dlmf.nist.gov/11.11.E11
Encodings:: pMML, png, TeX
Clarification (effective with 1.1.2):: The constraint which was originally $\nu\to+\infty$ , has been extended to be $\nu\to\infty$ , $|\operatorname{ph}\nu|\leq\frac{\pi}{2}-\delta$ .
Suggested 2021-04-05 by GergÅ‘ Nemes
See also:: Annotations for §11.11(iii), §11.11 and Ch.11

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5: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$ , Fixed $\nu$

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§11.6(ii) Large $|\nu|$ , Fixed $z$

… â–ºMore 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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D. Naylor (1990) On an asymptotic expansion of the Kontorovich-Lebedev transform. Applicable Anal. 39 (4), pp. 249–263.

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External Links:: ISSN 0003-6811, Document, MathReview (Aloknath Chakrabarti), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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D. Naylor (1996) On an asymptotic expansion of the Kontorovich-Lebedev transform. Methods Appl. Anal. 3 (1), pp. 98–108.

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External Links:: ISSN 1073-2772, FullText, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

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G. Nemes (2013a) 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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G. Nemes (2020) An extension of Laplace’s method. Constr. Approx. 51 (2), pp. 247–272.

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External Links:: ISSN 0176-4276, Document, Link, MathReview (José Luis López), ZentralBlatt
Referenced by:: (11.11.16), (11.11.17), §11.11(iii).
Encodings:: BibTeX

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7: 2.11 Remainder Terms; Stokes Phenomenon

… â–º … â–ºThe rest of this section is devoted to general methods for increasing this accuracy. … â–ºThen by application of Laplace’s method (§§2.4(iii) and 2.4(iv)), we have … â–ºA simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … â–º …

8: Bibliography S

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A. Sharples (1971) Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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External Links:: MathReview (J. Meixner), ZentralBlatt
Referenced by:: §28.8(iv).
Encodings:: BibTeX

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I. Shavitt and M. Karplus (1965) Gaussian-transform method for molecular integrals. I. Formulation for energy integrals. J. Chem. Phys. 43 (2), pp. 398–414.

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External Links:: Document, MathReview (R. K. Nesbet)
Referenced by:: §8.24(i).
Encodings:: BibTeX

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A. Sidi (2010) A simple approach to asymptotic expansions for Fourier integrals of singular functions. Appl. Math. Comput. 216 (11), pp. 3378–3385.

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External Links:: ISSN 0096-3003, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §2.3(iv).
Encodings:: BibTeX

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K. Soni (1980) 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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External Links:: ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

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W. F. Sun (1996) Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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External Links:: Abstract
Referenced by:: §18.16(v).
Encodings:: BibTeX

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9: Bibliography T

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N. M. Temme (1985) Laplace type integrals: Transformation to standard form and uniform asymptotic expansions. Quart. Appl. Math. 43 (1), pp. 103–123.

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External Links:: ISSN 0033-569X, FullText, MathReview (I. FenyÅ‘), ZentralBlatt
Referenced by:: §2.4(i).
Encodings:: BibTeX

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N. M. Temme (1990b) 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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External Links:: ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt
Referenced by:: §13.8(iii), §13.8(iii).
Encodings:: BibTeX

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N. M. Temme (1995c) Uniform asymptotic expansions of integrals: A selection of problems. J. Comput. Appl. Math. 65 (1-3), pp. 395–417.

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External Links:: ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt
Referenced by:: Ch.2.
Encodings:: BibTeX

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N. M. Temme (2015) Asymptotic Methods for Integrals. Series in Analysis, Vol. 6, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ.

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External Links:: ISBN 978-981-4612-15-9, MathReview Entry, ZentralBlatt
Referenced by:: §12.9(i), §12.9(i), (13.8.13), (13.8.14), (13.8.15), (13.8.16), §13.8(ii), §13.8(ii), §13.8(iii), §13.8(iii), §14.15(i), §14.15(i), §14.15(iii), §14.15(iii), §15.12(iii), §15.12(iii), §18.15(vi), §18.15(vi), §2.4(v), §2.4(v), §2.4(vi), §2.4(vi), §26.8(vii), §26.8(vii), §33.12(i), §33.12(i), §33.12(ii), §33.12(ii), Profile Nico M. Temme.
Encodings:: BibTeX

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N. M. Temme (2022) 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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External Links:: ISSN 1065-2469, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §13.8(iv).
Encodings:: BibTeX

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10: Bibliography B

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B. C. Berndt and R. J. Evans (1984) Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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External Links:: ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.10(iii).
Encodings:: BibTeX

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N. Bleistein and R. A. Handelsman (1975) Asymptotic Expansions of Integrals. Holt, Rinehart, and Winston, New York.

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Notes:: Reprinted with corrections by Dover Publications Inc., New York, 1986
External Links:: ISBN 0-486-65082-0, MathReview, ZentralBlatt
Referenced by:: §2.3(ii), §2.3(v), §2.4(iv), §2.4(v), §2.5(iii), §2.5(ii), Ch.2, Ch.9.
Encodings:: BibTeX

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R. Bo and R. Wong (1994) Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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W. G. C. Boyd (1995) 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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External Links:: ISSN 1073-2772, Pdf, MathReview (Dan PolisÌ†evski), ZentralBlatt
Referenced by:: §2.4(iv).
Encodings:: BibTeX

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J. Brüning (1984) On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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External Links:: ISSN 0003-889X, Document, MathReview (E. RiekstiÅ†Å¡), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

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Laplace method for asymptotic expansions of integrals

1—10 of 18 matching pages

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

3: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

§2.5(ii) Extensions

§2.5(iii) Laplace Transforms with Small Parameters

4: 11.11 Asymptotic Expansions of Anger–Weber Functions

5: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

§11.6(i) Large | z | , Fixed Î½

§11.6(ii) Large | Î½ | , Fixed z

6: Bibliography N

7: 2.11 Remainder Terms; Stokes Phenomenon

8: Bibliography S

9: Bibliography T

10: Bibliography B

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$