Mellin transforms

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21: 10.22 Integrals

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10.22.43 $${\int }_0^{\mathrm{\infty }}{t}^{\mu }{J}_{\nu }\left(t\right)dt=2^{\mu }\frac{\mathrm{\Gamma }\left(\frac{1}{2}\nu +\frac{1}{2}\mu +\frac{1}{2}\right)}{\mathrm{\Gamma }\left(\frac{1}{2}\nu -\frac{1}{2}\mu +\frac{1}{2}\right)},$$ $\mathrm{\Re }\left(\mu +\nu \right)>-1$, ,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellintransform

A&S Ref:

11.4.16

Referenced by:

§10.22(iii), §10.59

Permalink:

http://dlmf.nist.gov/10.22.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.45 $${\int }_0^{\mathrm{\infty }}\frac{1-J_0\left(t\right)}{t^{\mu }}dt=-\frac{\pi \sec \left(\frac{1}{2}\mu \pi \right)}{2^{\mu }{\mathrm{\Gamma }}^2\left(\frac{1}{2}\mu +\frac{1}{2}\right)},$$ .

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sec z$: secant function

Keywords:

Mellintransform

A&S Ref:

11.4.18 (modified)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E45

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.47 $${\int }_0^{\mathrm{\infty }}\frac{t^{\nu }{Y}_{\nu }\left(at\right)}{t^2+b^2}dt=-b^{\nu -1}{K}_{\nu }\left(ab\right),$$ .

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Symbols:

$Y_{\nu }\left(z\right)$: Bessel function of the second kind, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $\nu$: complex parameter

Keywords:

Mellintransform

A&S Ref:

11.4.46 (is the special case $\nu =0$.)

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E47

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.49 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -1}{\mathrm{e}}^{-at}{J}_{\nu }\left(bt\right)dt=\frac{{(\frac{1}{2}b)}^{\nu }}{a^{\mu +\nu }}\mathrm{\Gamma }\left(\mu +\nu \right)𝐅(\frac{\mu +\nu }{2},\frac{\mu +\nu +1}{2};\nu +1;-\frac{b^2}{a^2}),$$ $\mathrm{\Re }\left(\mu +\nu \right)>0,\mathrm{\Re }\left(a\pm \mathrm{i}b\right)>0$,

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: 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 and $\nu$: complex parameter

Keywords:

Laplace transform, Mellintransform

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E49

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22 and Ch.10

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10.22.62

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Symbols:

$J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral and $\nu$: complex parameter

Keywords:

Mellintransform

A&S Ref:

11.4.41

Referenced by:

§10.22(iv)

Permalink:

http://dlmf.nist.gov/10.22.E62

Encodings:

pMML, png, TeX

See also:

Annotations for §10.22(iv), §10.22(iv), §10.22 and Ch.10

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22: 11.5 Integral Representations

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11.5.8 $${(\frac{1}{2}x)}^{-\nu -1}{𝐇}_{\nu }\left(x\right)=-\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(\frac{1}{4}{x}^2)}^{s}ds,$$ $x>0$, $\mathrm{\Re }\nu >-1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${𝐇}_{\nu }\left(z\right)$: Struve function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $x$: real variable and $\nu$: real or complex order

Keywords:

Mellintransform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

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11.5.9 $${(\frac{1}{2}z)}^{-\nu -1}{𝐋}_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }}^{(0+)}\frac{\pi \csc \left(\pi s\right)}{\mathrm{\Gamma }\left(\frac{3}{2}+s\right)\mathrm{\Gamma }\left(\frac{3}{2}+\nu +s\right)}{(-\frac{1}{4}{z}^2)}^{s}ds.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\csc z$: cosecant function, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, ${𝐋}_{\nu }\left(z\right)$: modified Struve function, $z$: complex variable and $\nu$: real or complex order

Keywords:

Mellintransform

Referenced by:

§11.5(ii), §11.5(ii)

Permalink:

http://dlmf.nist.gov/11.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §11.5(ii), §11.5(ii), §11.5 and Ch.11

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23: 12.5 Integral Representations

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12.5.8 $$U(a,z)=\frac{{\mathrm{e}}^{-\frac{1}{4}{z}^2}{z}^{-a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}+a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}+a-2t\right)2^t{z}^{2t}dt,$$ $a\ne -\frac{1}{2},-\frac{3}{2},-\frac{5}{2},\mathrm{\dots }$, ,

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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, $\int$: integral, $U(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

Keywords:

Mellintransform

A&S Ref:

19.5.13 (modification and correction of)

Referenced by:

§12.5(iii)

Permalink:

http://dlmf.nist.gov/12.5.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(iii), §12.5 and Ch.12

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12.5.9 $$V(a,z)=\sqrt{\frac{2}{\pi }}\frac{{\mathrm{e}}^{\frac{1}{4}{z}^2}{z}^{a-\frac{1}{2}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}-a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(\frac{1}{2}-a-2t\right)2^t{z}^{2t}\cos \left(\pi t\right)dt,$$ $a\ne \frac{1}{2},\frac{3}{2},\frac{5}{2},\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $V(a,z)$: parabolic cylinder function, $\mathrm{ph}$: phase, $z$: complex variable and $a$: real or complex parameter

Keywords:

Mellintransform

A&S Ref:

19.5.13 (modification and correction of)

Referenced by:

§12.5(iii), §12.8(ii)

Permalink:

http://dlmf.nist.gov/12.5.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §12.5(iii), §12.5 and Ch.12

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24: 10.32 Integral Representations

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

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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 $x$ , $\mathrm{i}$: 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:

Mellintransform

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

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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:

Mellintransform

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

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

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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 $x$ , $\mathrm{i}$: 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:

Mellintransform

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

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25: 13.4 Integral Representations

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13.4.16 $$𝐌(a,b,-z)=\frac{1}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(b+t\right)}{z}^{t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $𝐌(a,b,z)$: Olver’s confluent hypergeometric 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 and $z$: complex variable

Keywords:

Mellintransform

A&S Ref:

13.2.9 (in different form)

Permalink:

http://dlmf.nist.gov/13.4.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.17 $$U(a,b,z)=\frac{z^{-a}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(a+t\right)\mathrm{\Gamma }\left(1+a-b+t\right)\mathrm{\Gamma }\left(-t\right)}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{z}^{-t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric 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 and $z$: complex variable

Keywords:

Mellintransform

A&S Ref:

13.2.10

Referenced by:

§13.6(vi)

Permalink:

http://dlmf.nist.gov/13.4.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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13.4.18 $$U(a,b,z)=\frac{z^{1-b}{\mathrm{e}}^z}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(b-1+t\right)\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(a+t\right)}{z}^{-t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric 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, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable

Keywords:

Mellintransform

Permalink:

http://dlmf.nist.gov/13.4.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §13.4(iii), §13.4 and Ch.13

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26: 24.7 Integral Representations

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24.7.11 $$B_n\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-c-\mathrm{i}\mathrm{\infty }}^{-c+\mathrm{i}\mathrm{\infty }}{(x+t)}^n{\left(\frac{\pi }{\sin \left(\pi t\right)}\right)}^{2}dt,$$ .

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Symbols:

$B_n\left(x\right)$: Bernoulli polynomials, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $\sin z$: sine function, $n$: integer, $t$: real or complex and $x$: real or complex

Keywords:

Mellintransform

Referenced by:

§24.7(ii)

Permalink:

http://dlmf.nist.gov/24.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §24.7(ii), §24.7(ii), §24.7 and Ch.24

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27: Bibliography O

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

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External Links:

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).

Encodings:

BibTeX

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28: 10.9 Integral Representations

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10.9.22 $$J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right){(\frac{1}{2}x)}^{\nu +2t}}{\mathrm{\Gamma }\left(\nu +t+1\right)}dt,$$ $\mathrm{\Re }\nu >0$, $x>0$,

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Symbols:

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

Keywords:

Mellintransform

A&S Ref:

9.1.26

Referenced by:

§10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.23 $$J_{\nu }\left(z\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }-\mathrm{i}c}^{-\mathrm{\infty }+\mathrm{i}c}\frac{\mathrm{\Gamma }\left(t\right)}{\mathrm{\Gamma }\left(\nu -t+1\right)}{(\frac{1}{2}z)}^{\nu -2t}dt,$$

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Symbols:

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

Keywords:

Mellintransform

Permalink:

http://dlmf.nist.gov/10.9.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.24 $$H_{\nu }^{(1)}\left(z\right)=-\frac{{\mathrm{e}}^{-\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(-\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(1)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellintransform

Referenced by:

§10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E24

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.25 $$H_{\nu }^{(2)}\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}\nu \pi \mathrm{i}}}{2{\pi }^2}{\int }_{c-\mathrm{i}\mathrm{\infty }}^{c+\mathrm{i}\mathrm{\infty }}\mathrm{\Gamma }\left(t\right)\mathrm{\Gamma }\left(t-\nu \right){(\frac{1}{2}\mathrm{i}z)}^{\nu -2t}dt,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $H_{\nu }^{(2)}\left(z\right)$: Bessel function of the third kind (or Hankel 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, $\int$: integral, $\mathrm{ph}$: phase, $z$: complex variable, $\nu$: complex parameter and $c$: constant

Keywords:

Mellintransform

Referenced by:

§10.9(ii), §10.9(ii)

Permalink:

http://dlmf.nist.gov/10.9.E25

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(ii), §10.9(ii), §10.9 and Ch.10

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10.9.29 $$J_{\mu }\left(x\right)J_{\nu }\left(x\right)=\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(-t\right)\mathrm{\Gamma }\left(2t+\mu +\nu +1\right){(\frac{1}{2}x)}^{\mu +\nu +2t}}{\mathrm{\Gamma }\left(t+\mu +1\right)\mathrm{\Gamma }\left(t+\nu +1\right)\mathrm{\Gamma }\left(t+\mu +\nu +1\right)}dt,$$ $x>0$,

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Symbols:

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

Keywords:

Mellintransform

Permalink:

http://dlmf.nist.gov/10.9.E29

Encodings:

pMML, png, TeX

See also:

Annotations for §10.9(iii), §10.9(iii), §10.9 and Ch.10

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29: Bibliography F

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

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External Links:

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

Referenced by:

§2.3(vi).

Encodings:

BibTeX

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30: Errata

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There have been extensive changes in the notation used for the integral transforms defined in §1.14. These changes are applied throughout the DLMF. The following table summarizes the changes.

Transform	New	Abbreviated	Old
	Notation	Notation	Notation
Fourier	$ℱ\left(f\right)\left(x\right)$	$ℱf\left(x\right)$
Fourier Cosine	${ℱ}_c\left(f\right)\left(x\right)$	${ℱ}_{c}f\left(x\right)$
Fourier Sine	${ℱ}_s\left(f\right)\left(x\right)$	${ℱ}_{s}f\left(x\right)$
Laplace	$ℒ\left(f\right)\left(s\right)$	$ℒf\left(s\right)$	$ℒ(f(t);s)$
Mellin	$ℳ\left(f\right)\left(s\right)$	$ℳf\left(s\right)$	$ℳ(f;s)$
Hilbert	$\mathscr{H}\left(f\right)\left(s\right)$	$\mathscr{H}f\left(s\right)$	$\mathscr{H}(f;s)$
Stieltjes	$𝒮\left(f\right)\left(s\right)$	$𝒮f\left(s\right)$	$𝒮(f;s)$

Previously, for the Fourier, Fourier cosine and Fourier sine transforms, either temporary local notations were used or the Fourier integrals were written out explicitly.

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