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

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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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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

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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

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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

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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:

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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:

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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

… ►

§18.17(viii) Other Integrals

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9: 37.20 Mathematical Applications

… ►For the unit ball and the simplex, these quantities can be written as an one-variable integral involving the Jacobi polynomials. … ►For regular domains, such as square, sphere, ball, simplex, and conic domains, they are used to study convolution structure, maximal functions, and interpolation spaces, as well as localized kernel and localized frames. …

10: 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

… ►

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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