convolution integrals

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1: 2.5 Mellin Transform Methods

… ►with . ►One of the two convolution integrals associated with the Mellin transform is of the form ►

2.5.3 $$I(x)={\int }_0^{\mathrm{\infty }}f(t)h(xt)dt,$$ $x>0$,

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Symbols:

$dx$: differential of $x$ , $\int$: integral, $f(x)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function

Referenced by:

§2.5(ii), §2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E3

Encodings:

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See also:

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

… ►

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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See also:

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

… ►

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

Encodings:

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See also:

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

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2: 2.6 Distributional Methods

… ►We now derive an asymptotic expansion of ${𝐼}^{\mu }f(x)$ for large positive values of $x$. … ►The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►The method of distributions can be further extended to derive asymptotic expansions for convolution integrals: ►

2.6.54 $$I(x)={\int }_0^{\mathrm{\infty }}f(t)h(xt)dt.$$

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Symbols:

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

Referenced by:

§2.6(iv)

Permalink:

http://dlmf.nist.gov/2.6.E54

Encodings:

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See also:

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

… ►

2.6.62 $$I(x)=\sum _{j=0}^{n-1}{a}_jℳh\left(j+\alpha \right)x^{-j-\alpha }+\sum _{k=0}^{n-1}{b}_kℳf\left(1-k-\beta \right)x^{-k-\beta }+{\delta }_n(x)$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(t)$: locally integrable function, $I(x)$: convolution integral, $n$: positive integer, $a_n$: coefficients, $b_n$: coefficients, $h(x)$: function and ${\delta }_n(x)$: integral

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.6.E62

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

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

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3: 30.10 Series and Integrals

… ►

4: 35.2 Laplace Transform

… ►

35.2.3

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Defines:

$\ast$: convolution operator (locally)

Symbols:

$\int$: integral, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, : less-than for matrices, $f(𝐗)$: complex-valued function of matrix argument and $𝟎$: $m\times m$ matrix of zeros

Referenced by:

§35.6(ii)

Permalink:

http://dlmf.nist.gov/35.2.E3

Encodings:

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See also:

Annotations for §35.2, §35.2 and Ch.35

5: 10.22 Integrals

… ►

Convolutions

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6: 1.14 Integral Transforms

… ►

1.14.5 $$(f\ast g)(t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t-s)g(s)ds.$$

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Defines:

$\ast$: convolution (Fourier) (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.14.E5

Encodings:

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See also:

Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

… ►

1.14.6 $$(f\ast g)(t)=\frac{1}{\sqrt{2\pi }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}F(x)G(x){\mathrm{e}}^{-\mathrm{i}tx}dx,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $F(x)$: Fourier transform of $f(t)$ , $\ast$: convolution (Fourier) and $G(x)$: Fourier transform of $g(t)$

Permalink:

http://dlmf.nist.gov/1.14.E6

Encodings:

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Annotations for §1.14(i), §1.14(i), §1.14 and Ch.1

… ►

1.14.30 $$(f\ast g)(t)={\int }_0^{t}f(u)g(t-u)du.$$

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Defines:

$\ast$: convolution (Laplace) (locally)

Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.14.E30

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(iii), §1.14(iii), §1.14 and Ch.1

… ►

1.14.39 $$(f\ast g)(x)={\int }_0^{\mathrm{\infty }}f(y)g\left(\frac{x}{y}\right)\frac{dy}{y}.$$

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Defines:

$\ast$: convolution (Mellin) (locally)

Symbols:

$dx$: differential of $x$ and $\int$: integral

Permalink:

http://dlmf.nist.gov/1.14.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

… ►

1.14.40 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(f\ast g)(x)dx=ℳf\left(s\right)ℳg\left(s\right).$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $dx$: differential of $x$ , $\int$: integral and $\ast$: convolution (Mellin)

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/1.14.E40

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(s\right)$ from $ℳ(f;s)$.

See also:

Annotations for §1.14(iv), §1.14(iv), §1.14 and Ch.1

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7: 7.21 Physical Applications

§7.21 Physical Applications

►The error functions, Fresnel integrals, and related functions occur in a variety of physical applications. … ►Carslaw and Jaeger (1959) gives many applications and points out the importance of the repeated integrals of the complementary error function ${\mathrm{i}}^n\mathrm{erfc}\left(z\right)$. … ►Voigt functions $𝖴(x,t)$, $𝖵(x,t)$, can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects. …Dawson’s integral appears in de-convolving even more complex motional effects; see Pratt (2007). …

8: 18.17 Integrals

§18.17 Integrals

►

§18.17(i) Indefinite Integrals

… ►

§18.17(iv) Fractional Integrals

… ►

Ultraspherical

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§18.17(viii) Other Integrals

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9: Bibliography W

… ►

Some useful integrals of $\mathrm{Si}(x),$ $\mathrm{Ci}(x)$ and related integrals. Optica Acta 14 (3), pp. 317–322.

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External Links:

Document, MathReview (S. D. Bajpai)

Referenced by:

§6.14(iii), §6.7(iv).

Encodings:

BibTeX

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Tables of Summable Series and Integrals Involving Bessel Functions. Holden-Day, San Francisco, CA.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.22(vi), §10.23(iv), §10.43(vi).

Encodings:

BibTeX

… ►

A class of integral transforms. Proc. Edinburgh Math. Soc. (2) 14, pp. 33–40.

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External Links:

Document, MathReview (B. R. Bhonsle), ZentralBlatt

Referenced by:

§13.23(iv).

Encodings:

BibTeX

… ►

On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§8.11(v).

Encodings:

BibTeX

… ►

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

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External Links:

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

Referenced by:

§2.6(ii), §2.6(iv).

Encodings:

BibTeX

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10: Bibliography G

… ►

Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

… ►

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:

BibTeX

… ►

Definite integrals of the complete elliptic integral $K$ . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.13(i).

Encodings:

BibTeX

… ►

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:

BibTeX

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