Laplace method

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11: 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 …

12: 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. …

13: Bibliography B

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Asymptotic methods in enumeration. SIAM Rev. 16 (4), pp. 485–515.

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External Links:

Document, MathReview (Bernard Harris), ZentralBlatt

Referenced by:

Ch.2.

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Numerical Methods for Least Squares Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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ISBN 0-89871-360-9, MathReview (Hongyuan Zha), ZentralBlatt

Referenced by:

§3.11(v).

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From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. (3) 90 (1), pp. 167–208.

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ISSN 0024-6115, Document, MathReview (Andrei A. Kapaev), ZentralBlatt

Referenced by:

§32.9(iii).

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Laplace approximations for hypergeometric functions with matrix argument. Ann. Statist. 30 (4), pp. 1155–1177.

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ISSN 0090-5364, Document, MathReview, ZentralBlatt

Referenced by:

§35.6(iv), §35.7(iv), §35.7(iv), §35.9.

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Laplace approximation for Bessel functions of matrix argument. J. Comput. Appl. Math. 155 (2), pp. 359–382.

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ISSN 0377-0427, Document, MathReview (S. Bhargava), ZentralBlatt

Referenced by:

§35.5(ii), §35.5(iii).

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14: Bibliography S

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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms. Math. Tables Aids Comput. 9 (52), pp. 164–177.

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External Links:

FullText, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§3.5(viii).

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The Laplace Transform: Theory and Applications. Undergraduate Texts in Mathematics, Springer-Verlag, New York.

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ISBN 0-387-98698-7, MathReview (Philip Heywood), ZentralBlatt

Referenced by:

§1.14(iii), §1.14(vii).

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A global Newton method for the zeros of cylinder functions. Numer. Algorithms 18 (3-4), pp. 259–276.

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ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

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Practical Extrapolation Methods: Theory and Applications. Cambridge Monographs on Applied and Computational Mathematics, Vol. 10, Cambridge University Press, Cambridge.

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ISBN 0-521-66159-5, MathReview (Thomas A. Atchison), ZentralBlatt

Referenced by:

§3.9(vi).

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15: Frank W. J. Olver

… ►In 1945–1961 he was a founding member of the Mathematics Division and Head of the Numerical Methods Section at the National Physical Laboratory, Teddington, U. … ►Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►In a review of that volume, Jet Wimp of Drexel University said that the papers “exemplify a redoubtable mathematical talent, the work of a man who has done more than almost anyone else in the 20th century to bestow on the discipline of applied mathematics the elegance and rigor that its earliest practitioners, such as Gauss and Laplace, would have wished for it. … ►

1 Algebraic and Analytic Methods

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16: Bibliography P

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Exactification of the method of steepest descents: The Bessel functions of large order and argument. Proc. Roy. Soc. London Ser. A 460, pp. 2737–2759.

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ISSN 1364-5021, Document, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§10.20(ii).

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Modified Clenshaw-Curtis method for the computation of Bessel function integrals. BIT 23 (3), pp. 370–381.

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ISSN 0006-3835, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§10.74(vii).

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Integrals and Series: Direct Laplace Transforms, Vol. 4. Gordon and Breach Science Publishers, New York.

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Notes:

Table erratum: Math. Comp. v. 66 (1997), no. 220, p. 1766.

External Links:

ISBN 2-88124-837-3, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §14.17(v), §15.14, §16.20, §16.5, §18.17(ix), §19.13(iii), §20.10(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.14, §9.10(ix).

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Integrals and Series: Inverse Laplace Transforms, Vol. 5. Gordon and Breach Science Publishers, New York.

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External Links:

ISBN 2-88124-838-1, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §10.22(vi), §10.43(vi), §10.71(iii), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(ii), §13.10(vi), §13.23(i), §13.23(iv), §14.17(v), §15.14, §15.14, §16.15, §16.20, §16.5, §18.17(ix), §19.13(iii), §24.13(iii), §25.11(ix), §25.12(ii), §25.7, §5.13, §6.14(iii), §7.14(iii), §8.6(iii), §9.10(ix).

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A method for computing Bessel function integrals. J. Comput. Phys. 75 (2), pp. 334–344.

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ISSN 0021-9991, Document, MathReview (J. G. Herriot), ZentralBlatt

Referenced by:

§10.74(vii), §10.74(vii).

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17: Bibliography H

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The Laplace transform for expressions that contain a probability function. Bul. Akad. Štiince RSS Moldoven. 1973 (2), pp. 78–80, 93 (Russian).

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External Links:

MathReview (L. Berg)

Referenced by:

§7.14(iii).

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The combination of $R$-matrix and complex coordinate methods: Application to the diamagnetic Rydberg spectra of Ba and Sr. J. Phys. B 26 (12), pp. 1775–1790.

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ISSN 0953-4075, Document, MathReview

Referenced by:

4th item.

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Asymptotic expansion of Laplace transforms near the origin. SIAM J. Math. Anal. 1 (1), pp. 118–130.

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External Links:

Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§2.5(iii), §2.5(ii).

Encodings:

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18: Bibliography O

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Tables of Laplace Transforms. Springer-Verlag, Berlin-New York.

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Notes:

Table errata: Math. Comp. v. 50 (1988), no. 182, pp. 653–654.

External Links:

ISBN 0-387-06350-1, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(ii), §13.23(i), §14.17(v), §15.14, §18.17(ix), §20.10(ii), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(iii), §8.6(iii).

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Asymptotic Enumeration Methods. In Handbook of Combinatorics, Vol. 2, L. Lovász, R. L. Graham, and M. Grötschel (Eds.), pp. 1063–1229.

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External Links:

ISBN 0-444-82351-4, Pdf, MathReview (Konrad Engel), ZentralBlatt

Referenced by:

Ch.2.

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Computing $\pi (x)$: The combinatorial method. Revista do DETUA 4 (6), pp. 759–768.

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Notes:

Online fulltext contains postpublication annotations.

External Links:

FullText

Referenced by:

§27.18.

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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A new method for the evaluation of zeros of Bessel functions and of other solutions of second-order differential equations. Proc. Cambridge Philos. Soc. 46 (4), pp. 570–580.

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External Links:

Document, MathReview (J. Todd), ZentralBlatt

Referenced by:

§10.21(ii).

Encodings:

BibTeX

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19: 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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ISSN 0033-569X, FullText, MathReview (I. Fenyő), ZentralBlatt

Referenced by:

§2.4(i).

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Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.

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Notes:

Asymptotic methods in analysis and combinatorics

External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

§8.18(ii).

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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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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.

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Iterative Methods for the Solution of Equations. Prentice-Hall Series in Automatic Computation, Prentice-Hall Inc., Englewood Cliffs, N.J..

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External Links:

MathReview (W. C. Rheinboldt), ZentralBlatt

Referenced by:

§3.8(iii).

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Some methods for flows past blunt slender bodies. J. Fluid Mech. 18, pp. 619–635.

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ISSN 0022-1120, Document, MathReview (L. E. Payne)

Referenced by:

§18.17(viii).

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BibTeX

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20: Bibliography G

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On high precision methods for computing integrals involving Bessel functions. Math. Comp. 33 (147), pp. 1049–1057.

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ISSN 0025-5718, FullText, MathReview (K. Jetter), ZentralBlatt

Referenced by:

§10.74(vii).

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On the generalization of a method for computing Bessel function integrals. J. Comput. Appl. Math. 6 (2), pp. 167–168.

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ISSN 0771-050X, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

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Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

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A method for evaluating certain Bessel integrals. Z. Angew. Math. Phys. 30 (4), pp. 722–723.

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ISSN 0044-2275, Document, MathReview

Referenced by:

§10.71(ii).

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Spectral Methods and Their Applications. World Scientific Publishing Co. Inc., River Edge, NJ-Singapore.

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External Links:

ISBN 981-02-3333-7, MathReview (K. Moszyński), ZentralBlatt

Referenced by:

§18.38(i).

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