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1: 2.5 Mellin Transform Methods

§2.5 Mellin Transform Methods

… ►The Mellin transform of $f(t)$ is defined by …The inversion formula is given by … ► … ►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. …

2: 5.19 Mathematical Applications

… ►

§5.19(ii) Mellin–Barnes Integrals

►Many special functions $f(z)$ can be represented as a Mellin–Barnes integral, that is, an integral of a product of gamma functions, reciprocals of gamma functions, and a power of $z$, the integration contour being doubly-infinite and eventually parallel to the imaginary axis at both ends. …

3: 1.14 Integral Transforms

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§1.14(iv) Mellin Transform

►The Mellin transform of a real- or complex-valued function $f(x)$ is defined by … ►

Inversion

… ►

Parseval-type Formulas

… ►

Convolution

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4: Richard B. Paris

… ► Wood), published by Longman Scientific and Technical in 1986, and Asymptotics and Mellin-Barnes Integrals (with D. …

5: 8.6 Integral Representations

… ►

Mellin–Barnes Integrals

… ►

8.6.10 $$\gamma (a,z)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(s\right)}{a-s}{z}^{a-s}ds,$$ , $a\ne 0,-1,-2,\mathrm{\dots }$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\gamma (a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

►

8.6.11 $$\mathrm{\Gamma }(a,z)=\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+a\right)\frac{z^{-s}}{s}ds,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.19(i), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

►

8.6.12 $$\mathrm{\Gamma }(a,z)=-\frac{z^{a-1}{\mathrm{e}}^{-z}}{\mathrm{\Gamma }\left(1-a\right)}\frac{1}{2\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(s+1-a\right)\frac{\pi z^{-s}}{\sin \left(\pi s\right)}ds,$$ , $a\ne 1,2,3,\mathrm{\dots }$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: 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, $\mathrm{\Gamma }(a,z)$: incomplete gamma function, $\int$: integral, $\mathrm{ph}$: phase, $\sin z$: sine function, $z$: complex variable, $a$: parameter and $c$: real constant

Keywords:

Mellin transform

Referenced by:

§8.19(i), §8.6(ii), §8.6(ii), §8.6(ii), Erratum (V1.0.17) for Paragraph Mellin–Barnes Integrals (in §8.6(ii))

Permalink:

http://dlmf.nist.gov/8.6.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §8.6(ii), §8.6(ii), §8.6 and Ch.8

…

6: Bibliography P

… ►

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.

Encodings:

BibTeX

… ►

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

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External Links:

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

Referenced by:

§2.11(v).

Encodings:

BibTeX

… ►

Automatic computation of Bessel function integrals. Comput. Phys. Comm. 25 (3), pp. 289–295.

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Notes:

Double-precision Fortran, maximum accuracy 20S.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

6th item.

Encodings:

BibTeX

… ►

Voronoi type congruences for Bernoulli numbers. In Voronoi’s Impact on Modern Science. Book I, P. Engel and H. Syta (Eds.),

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External Links:

ZentralBlatt

Referenced by:

§24.10(iii).

Encodings:

BibTeX

… ►

Stacking models of vesicles and compact clusters. J. Statist. Phys. 80 (3–4), pp. 755–779.

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External Links:

Document, ZentralBlatt

Referenced by:

§8.24(ii).

Encodings:

BibTeX

…

7: 10.32 Integral Representations

… ►

Mellin–Barnes Type

►

10.32.13 $$K_{\nu }\left(z\right)=\frac{{(\frac{1}{2}z)}^{\nu }}{4\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}z)}^{-2t}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

Erratum (V1.0.11) for Equation (10.32.13)

Permalink:

http://dlmf.nist.gov/10.32.E13

Encodings:

pMML, png, TeX

Errata (effective with 1.0.11):

Originally the constraint was written incorrectly as .

Reported 2015-05-20 by Richard Paris

See also:

Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10

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10.32.14 $$K_{\nu }\left(z\right)=\frac{1}{2{\pi }^2\mathrm{i}}{\left(\frac{\pi }{2z}\right)}^{\frac{1}{2}}{\mathrm{e}}^{-z}\cos \left(\nu \pi \right){\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-t-\nu \right)\mathrm{\Gamma }\left(\frac{1}{2}-t+\nu \right){(2z)}^{t}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\notin$: not an element of, $\mathrm{ph}$: phase, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(ii), §10.74(iii)

Permalink:

http://dlmf.nist.gov/10.32.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(ii), §10.32(ii), §10.32 and Ch.10

… ►

Mellin–Barnes Type

►

10.32.19 $$K_{\mu }\left(z\right)K_{\nu }\left(z\right)=\frac{1}{8\pi \mathrm{i}}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t+\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t+\frac{1}{2}\mu -\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu +\frac{1}{2}\nu \right)\mathrm{\Gamma }\left(t-\frac{1}{2}\mu -\frac{1}{2}\nu \right)}{\mathrm{\Gamma }\left(2t\right)}{(\frac{1}{2}z)}^{-2t}dt,$$ .

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $z$: complex variable and $\nu$: complex parameter

Keywords:

Mellin transform

Referenced by:

§10.32(iii)

Permalink:

http://dlmf.nist.gov/10.32.E19

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(iii), §10.32(iii), §10.32 and Ch.10

…

8: 20.10 Integrals

… ►

§20.10(i) Mellin Transforms with respect to the Lattice Parameter

►

20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.2 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}({\theta }_3\left(0\right|\mathrm{i}{x}^2)-1)dx={\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E2

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_3\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_3\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.32) and (20.4.4).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

►

20.10.3 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}(1-{\theta }_4\left(0\right|\mathrm{i}{x}^2))dx=(1-2^{1-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E3

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>0$ because $1-{\theta }_4\left(0\right|\mathrm{i}{x}^2)=1-x^{-1}{\theta }_2\left(0\right|\mathrm{i}{x}^{-2})\sim 1$ as $x\to 0+$, which follows from (20.7.33) and (20.4.3).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

…

9: 15.14 Integrals

§15.14 Integrals

►The Mellin transform of the hypergeometric function of negative argument is given by ►

15.14.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}𝐅(\genfrac{}{}{0pt}{}{a,b}{c};-x)dx=\frac{\mathrm{\Gamma }\left(s\right)\mathrm{\Gamma }\left(a-s\right)\mathrm{\Gamma }\left(b-s\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-s\right)},$$ $\min (\mathrm{\Re }a,\mathrm{\Re }b)>\mathrm{\Re }s>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $x$: real variable, $s$: nonnegative integer, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.14

Permalink:

http://dlmf.nist.gov/15.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §15.14 and Ch.15

… ►Mellin transforms of hypergeometric functions are given in Erdélyi et al. (1954a, §6.9), Oberhettinger (1974, §1.15), and Marichev (1983, pp. 288–299). Inverse Mellin transforms are given in Erdélyi et al. (1954a, §7.5). …

10: 15.6 Integral Representations

§15.6 Integral Representations

… ►

15.6.1 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^1\frac{t^{b-1}{(1-t)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.19(iii), (15.4.34), §15.6, §15.6, Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9), Erratum (V1.0.23) for Equations (15.6.1)–(15.6.9)

Permalink:

http://dlmf.nist.gov/15.6.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.2 $$𝐅(a,b;c;z)=\frac{\mathrm{\Gamma }\left(1+b-c\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b\right)}{\int }_0^{(1+)}\frac{t^{b-1}{(t-1)}^{c-b-1}}{{(1-zt)}^a}dt,$$ ; $c-b\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6

Permalink:

http://dlmf.nist.gov/15.6.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.0.23):

The Olver hypergeometric function $𝐅(a,b;c;z)$, previously omitted from the left-hand side to make the formula more concise, has been added. The constraint , originally mentioned in the text, has been directly added to the formula.

See also:

Annotations for §15.6 and Ch.15

►

15.6.2_5 $$𝐅(a,b;c;z)=\frac{1}{\mathrm{\Gamma }\left(b\right)\mathrm{\Gamma }\left(c-b\right)}{\int }_0^{\mathrm{\infty }}\frac{t^{b-1}{\left(t+1\right)}^{a-c}}{{\left(t-zt+1\right)}^a}dt,$$ ; $\mathrm{\Re }c>\mathrm{\Re }b>0$.

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $𝐅(a,b;c;z)$ or $𝐅(\genfrac{}{}{0pt}{}{a,b}{c};z)$: $={}_2{}^{}𝐅_1^{}(a,b;c;z)$ Olver’s hypergeometric function, $z$: complex variable, $a$: real or complex parameter, $b$: real or complex parameter and $c$: real or complex parameter

Keywords:

Mellin transform

Referenced by:

§15.6, Erratum (V1.1.5) for Additions

Permalink:

http://dlmf.nist.gov/15.6.E2_5

Encodings:

pMML, png, TeX

Addition (effective with 1.1.5):

This equation was added.

See also:

Annotations for §15.6 and Ch.15

… ►

►►

Figure 15.6.1: $t$-plane. … Magnify