Mellin transform methods

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1: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

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§2.5(ii) Extensions

… ►To apply the Mellin transform method outlined in §2.5(i), we require the transforms $ℳf\left(1-z\right)$ and $ℳh\left(z\right)$ to have a common strip of analyticity. … ►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). See also Brüning (1984) for a different approach. …

2: 2.3 Integrals of a Real Variable

… ►where $ℳf\left(\alpha \right)$ is the Mellin transform of $f(t)$ (§2.5(i)). … ►

§2.3(iii) Laplace’s Method

… ►The integral (2.3.24) transforms into … ►

§2.3(vi) Asymptotics of Mellin Transforms

►For the asymptotics of the Mellin transform $ℳf\left(z\right)={\int }_0^{\mathrm{\infty }}{t}^{z-1}f(t)dt$ as $z\to \mathrm{\infty }$ see Frenzen (1987b), Sidi (1985, 2011).

3: 3.5 Quadrature

… ►Stroud and Secrest (1966) includes computational methods and extensive tables. … ►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature. … ►Further methods are given in Clendenin (1966) and Lyness (1985). … ► ►Other contour integrals occur in standard integral transforms or their inverses, for example, Hankel transforms (§10.22(v)), Kontorovich–Lebedev transforms (§10.43(v)), and Mellin transforms (§1.14(iv)). …

4: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

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

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ISBN 0-387-06942-9, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.5(iii), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(iii), §13.23(i), §14.17(vi), §15.14, §18.17(ix), §20.10(i), §5.13, §6.14(iii), §6.7(iv), §7.14(iii), §7.7(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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ISBN 0-444-82351-4, Pdf, MathReview (Konrad Engel), ZentralBlatt

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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.

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§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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Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

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5: 2.6 Distributional Methods

§2.6 Distributional Methods

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§2.6(ii) Stieltjes Transform

… ► $ℳf\left(z\right)$ being the Mellin transform of $f(t)$ or its analytic continuation (§2.5(ii)). … ►For a more detailed discussion of the derivation of asymptotic expansions of Stieltjes transforms by the distribution method, see McClure and Wong (1978) and Wong (1989, Chapter 6). … ►where $ℳf\left(z\right)$ is the Mellin transform of $f$ or its analytic continuation. …

6: Bibliography S

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

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Document, MathReview (R. K. Nesbet)

Referenced by:

§8.24(i).

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Asymptotic expansions of Mellin transforms and analogues of Watson’s lemma. SIAM J. Math. Anal. 16 (4), pp. 896–906.

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ISSN 0036-1410, Document, Link, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.3(vi).

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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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Asymptotic expansion of Mellin transforms in the complex plane. Int. J. Pure Appl. Math. 71 (3), pp. 465–480.

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ISSN 1311-8080, MathReview Entry, ZentralBlatt

Referenced by:

§2.3(vi).

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7: Bibliography W

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The cubic transformation of the hypergeometric function. Quart. J. Pure and Applied Math. 41, pp. 70–79.

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ZentralBlatt

Referenced by:

§15.8(v), §15.8(v).

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Wave functions for large arguments by the amplitude-phase method. Phys. Rev. 52, pp. 1123–1127.

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External Links:

Document, ZentralBlatt

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§33.5(ii).

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The method of Darboux. J. Approximation Theory 10 (2), pp. 159–171.

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Document, MathReview (M. E. Noble), ZentralBlatt

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§2.10(iv).

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On a uniform treatment of Darboux’s method. Constr. Approx. 21 (2), pp. 225–255.

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ISSN 0176-4276, Document, MathReview Entry, ZentralBlatt

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§2.10(iv).

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Explicit error terms for asymptotic expansions of Mellin convolutions. J. Math. Anal. Appl. 72 (2), pp. 740–756.

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ISSN 0022-247X, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.6(ii), §2.6(iv).

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8: Bibliography F

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Un théorème de Paley-Wiener pour la transformation de Fourier sur un espace riemannien symétrique de rang un. J. Funct. Anal. 49 (2), pp. 230–268.

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ISSN 0022-1236, Document, Link, MathReview (Mogens Flensted-Jensen), ZentralBlatt

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§18.39(i).

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Calculation of elliptic integrals of the third kind by means of Gauss’ transformation. Math. Comp. 19 (89), pp. 97–104.

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FullText, MathReview (J. E. L. Peck), ZentralBlatt

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§19.36(ii).

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Algebraic methods to compute Mathieu functions. J. Phys. A 34 (17), pp. 3541–3551.

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ISSN 0305-4470, Document, Link, MathReview (Aleksander Denisiuk), ZentralBlatt

Referenced by:

§28.6(i), §28.6(i), §28.6(ii), §28.6(ii), §28.8(i), §28.8(i), §28.8(ii), §28.8(ii), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii).

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On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt

Referenced by:

§2.3(vi).

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Evaluation, design and extrapolation methods for optical signals, based on use of the prolate functions. In Progress in Optics, E. Wolf (Ed.), Vol. 9, pp. 311–407.

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Referenced by:

§30.15(i), §30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).

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9: Bibliography P

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Asymptotics and Mellin-Barnes Integrals. Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-79001-8, MathReview, ZentralBlatt

Referenced by:

§1.14(iv), §10.32(iii), §10.32(ii), §10.46, §10.9(iii), §10.9(ii), §16.11(i), §16.11(ii), §2.10(iii), §2.11(iii), §2.11(v), §2.5(ii), §2.5(ii), Ch.2, §24.7(ii), §25.9, §5.11(ii), §5.11(iii), §5.13, §5.19(ii), §5.6(ii), Ch.5, §8.6(ii), Profile Richard B. Paris.

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Smoothing of the Stokes phenomenon using Mellin-Barnes integrals. J. Comput. Appl. Math. 41 (1-2), pp. 117–133.

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ISSN 0377-0427, Document, MathReview (Anatoly Kilbas), ZentralBlatt

Referenced by:

§2.11(v).

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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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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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