convolution product

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1: 27.5 Inversion Formulas

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27.5.1 $$h(n)=\sum _{d|n}f(d)g\left(\frac{n}{d}\right),$$

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Symbols:

$d$: positive integer, $n$: positive integer, $f$: function, $g$: function and $h$: function

Permalink:

http://dlmf.nist.gov/27.5.E1

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Annotations for §27.5 and Ch.27

►called the Dirichlet product (or convolution) of $f$ and $g$. …

2: 30.10 Series and Integrals

… ►For product formulas and convolutions see Connett et al. (1993). …

3: 2.6 Distributional Methods

… ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. ►In terms of the convolution product ►

2.6.34 $$(f\ast g)(x)={\int }_0^{x}f(x-t)g(t)dt$$

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Defines:

$\ast$: convolution (locally)

Symbols:

$dx$: differential of $x$ , $\int$: integral, $g(t)$: locally integrable function and $f(t)$: locally integrable function

Referenced by:

§2.6(iii)

Permalink:

http://dlmf.nist.gov/2.6.E34

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Annotations for §2.6(iii), §2.6 and Ch.2

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4: Bibliography C

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Product formulas and convolutions for angular and radial spheroidal wave functions. Trans. Amer. Math. Soc. 338 (2), pp. 695–710.

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External Links:

ISSN 0002-9947, FullText, MathReview (Jean-Claude Lucquiaud), ZentralBlatt

Referenced by:

§30.10, §30.11(vi).

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5: 2.5 Mellin Transform Methods

… ►with . ►One of the two convolution integrals associated with the Mellin transform is of the form … ►In the half-plane $\mathrm{\Re }z>\max (0,-2\nu )$, the product $ℳf\left(1-z\right)ℳh\left(z\right)$ has a pole of order two at each positive integer, and … ►

2.5.29 $$I(x)=\sum _{j,k=1}^2{I}_{jk}(x),$$

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Symbols:

$I(x)$: convolution integral

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E29

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Annotations for §2.5(ii), §2.5 and Ch.2

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2.5.31 $$I_{21}(x)=0,$$ for $x\ge 1$.

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Symbols:

$I(x)$: convolution integral

Permalink:

http://dlmf.nist.gov/2.5.E31

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Annotations for §2.5(ii), §2.5 and Ch.2

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6: Bibliography L

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Numerical evaluation of integrals containing a spherical Bessel function by product integration. J. Math. Phys. 22 (7), pp. 1399–1413.

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External Links:

ISSN 0022-2488, Document, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Error bounds for asymptotic expansions of Laplace convolutions. SIAM J. Math. Anal. 25 (6), pp. 1537–1553.

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External Links:

ISSN 0036-1410, Document, MathReview (Walter Schempp), ZentralBlatt

Referenced by:

§2.6(iii).

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A note on the computation of integrals involving products of trigonometric and Bessel functions. Math. Comp. 27 (124), pp. 871–872.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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Evaluating infinite integrals involving products of Bessel functions of arbitrary order. J. Comput. Appl. Math. 64 (3), pp. 269–282.

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External Links:

ISSN 0377-0427, Document, MathReview Entry, ZentralBlatt

Referenced by:

§10.74(vii).

Encodings:

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7: 10.22 Integrals

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Products

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Products

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Convolutions

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Other Double Products

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Triple Products

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8: 18.17 Integrals

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§18.17(ii) Integral Representations for Products

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Ultraspherical

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Legendre

… ►For addition formulas corresponding to (18.17.5) and (18.17.6) see (18.18.8) and (18.18.9), respectively. …

9: Bibliography G

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Dirichlet convolution of cotangent numbers and relative class number formulas. Monatsh. Math. 110 (3-4), pp. 231–256.

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External Links:

ISSN 0026-9255, MathReview (Peter Stevenhagen), ZentralBlatt

Referenced by:

§24.16(iii).

Encodings:

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Recurrence relations for cross-products of Bessel functions. Quart. J. Mech. Appl. Math. 2 (1), pp. 72–74.

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External Links:

ISSN 0033-5614, Document, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§10.6(iii).

Encodings:

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On the exceptional zeros of cross-products of derivatives of spherical Bessel functions. Z. Angew. Math. Phys. 36 (3), pp. 491–494.

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External Links:

ISSN 0044-2275, Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§10.58.

Encodings:

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Table of Integrals, Series, and Products. 6th edition, Academic Press Inc., San Diego, CA.

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Notes:

Translated from the Russian. Translation edited and with a preface by Alan Jeffrey and Daniel Zwillinger. For a review of the fourth edition (1980), with errata list, see Math. Comp. v. 36 (1981), pp. 310–314. For a review of the fifth edition (1994) see Math. Comp. v. 64 (1995), pp. 439–441.

External Links:

Errata in the 4th, 5th, and 6th editions, ISBN 0-12-294757-6, Link, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §1.4(iv), §1.8(v), §10.22(vi), §10.23(iv), §10.43(vi), §10.60(iv), §11.7(v), §11.9(iv), §12.12, §12.5(iv), §13.10(vi), §13.23(v), §14.18(iv), §15.14, §18.17(ix), §18.18(ix), §19.25(v), §19.29(i), §19.29(ii), §20.10(iii), §22.14(ii), §22.14(iii), §23.14, §24.7(iii), §28.10(iii), §28.28(v), §4.10(iii), §4.11, §4.26(v), §4.27, §4.40(v), §4.41, §5.13, §6.14(iii), §7.14(iii), §8.14.

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Riccati equations and convolution formulae for functions of Rayleigh type. J. Phys. A 33 (7), pp. 1363–1368.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

§10.21(xiii).

Encodings:

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