Euler transformation

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1: 24.13 Integrals

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§24.13(iii) Compendia

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2: 3.9 Acceleration of Convergence

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§3.9(ii) Euler’s Transformation of Series

… ►Euler’s transformation is usually applied to alternating series. …

3: 2.11 Remainder Terms; Stokes Phenomenon

… ►The transformations in §3.9 for summing slowly convergent series can also be very effective when applied to divergent asymptotic series. ►A simple example is provided by Euler’s transformation (§3.9(ii)) applied to the asymptotic expansion for the exponential integral (§6.12(i)): … ► … ►Further improvements in accuracy can be realized by making a second application of the Euler transformation; see Olver (1997b, pp. 540–543). …

4: 2.5 Mellin Transform Methods

… ►Since $ℳ{\mathrm{e}}^{-t}\left(z\right)=\mathrm{\Gamma }\left(z\right)$, by the Parseval formula (2.5.5), there are real numbers $p_1$ and $p_2$ such that , , and ►

2.5.40 $$I_j(x)=\frac{1}{2\pi \mathrm{i}}{\int }_{p_j-\mathrm{i}\mathrm{\infty }}^{p_j+\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_j\left(z\right)dz,$$ $j=1,2$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $h(x)$: locally integrable function, $I_j(x)$: integral and $p_j$: real numbers

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E40

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

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

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2.5.41 $$I_1(x)=ℳh_1\left(1\right)x^{-1}+\frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho +\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_1\left(z\right)dz,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral, $h(x)$: locally integrable function, $I_j(x)$: integral and $\rho$: parameter

Keywords:

Mellin transform

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E41

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

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

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2.5.42 $$I_2(x)=\sum _{\mathrm{\Re }{\beta }_0\le \mathrm{\Re }z\le 1}res\left[-x^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)\right]+\frac{1}{2\pi \mathrm{i}}{\int }_{\rho -\mathrm{i}\mathrm{\infty }}^{\rho +\mathrm{i}\mathrm{\infty }}{x}^{-z}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)dz.$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\pi$: 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, $res$: residue, $h(x)$: locally integrable function, $I_j(x)$: integral and $\rho$: parameter

Keywords:

Mellin transform

Referenced by:

§2.5(iii)

Permalink:

http://dlmf.nist.gov/2.5.E42

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

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

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2.5.47 $${res}_{z=1}\left[-{\zeta }^{z-1}\mathrm{\Gamma }\left(1-z\right)ℳh_2\left(z\right)\right]=\left(-\ln \zeta -\gamma \right)-ℳh_1\left(1\right),$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\gamma$: Euler’s constant, $ℳ\left(f\right)\left(s\right)$: Mellin transform, $\ln z$: principal branch of logarithm function, $res$: residue and $h(x)$: locally integrable function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E47

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

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

See also:

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

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5: 15.14 Integrals

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

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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

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6: 13.23 Integrals

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13.23.1 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)}{{\left(z+\frac{1}{2}\right)}^{\mu +\nu +\frac{1}{2}}}{}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{\frac{1}{2}+\mu -\kappa ,\frac{1}{2}+\mu +\nu }{1+2\mu };\frac{1}{z+\frac{1}{2}}),$$ $\mathrm{\Re }\mu +\nu +\frac{1}{2}>0$, $\mathrm{\Re }z>\frac{1}{2}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${}_2{}^{}F_1^{}(a,b;c;z)$: $=F(a,b;c;z)$ notation for Gauss’ hypergeometric function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.2 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{\mu -\frac{1}{2}}{M}_{\kappa ,\mu }\left(t\right)dt=\mathrm{\Gamma }\left(2\mu +1\right){\left(z+\frac{1}{2}\right)}^{-\kappa -\mu -\frac{1}{2}}{\left(z-\frac{1}{2}\right)}^{\kappa -\mu -\frac{1}{2}},$$ $\mathrm{\Re }\mu >-\frac{1}{2}$, $\mathrm{\Re }z>\frac{1}{2}$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Laplace transform, Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.3 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-\frac{1}{2}t}{t}^{\nu -1}{M}_{\kappa ,\mu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\kappa -\nu \right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\nu \right)},$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker confluent hypergeometric function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Laplace transform, Mellin transform

Referenced by:

§13.23(i)

Permalink:

http://dlmf.nist.gov/13.23.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.6 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{M}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{I}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker 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, $I_{\nu }\left(z\right)$: modified Bessel function of the first kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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13.23.7 $$\frac{1}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt+\frac{1}{2}{t}^{-1}}{t}^{\kappa }{W}_{\kappa ,\mu }\left(t^{-1}\right)dt=\frac{2z^{-\kappa -\frac{1}{2}}}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{K}_{2\mu }\left(2\sqrt{z}\right),$$ $\mathrm{\Re }z>0$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker 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, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: real part and $z$: complex variable

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.23.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §13.23(i), §13.23 and Ch.13

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7: 11.7 Integrals and Sums

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11.7.10 $${\int }_0^{\mathrm{\infty }}{t}^{-\nu -1}{𝐇}_{\nu }\left(t\right)dt=\frac{\pi }{2^{\nu +1}\mathrm{\Gamma }\left(\nu +1\right)},$$ $\mathrm{\Re }\nu >-\frac{3}{2}$,

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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, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/11.7.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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11.7.11 $${\int }_0^{\mathrm{\infty }}{t}^{\mu -\nu -1}{𝐇}_{\nu }\left(t\right)dt=\frac{\mathrm{\Gamma }\left(\frac{1}{2}\mu \right)2^{\mu -\nu -1}\tan \left(\frac{1}{2}\pi \mu \right)}{\mathrm{\Gamma }\left(\nu -\frac{1}{2}\mu +1\right)},$$ , $\mathrm{\Re }\nu >\mathrm{\Re }\mu -\frac{3}{2}$,

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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, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $\tan z$: tangent function and $\nu$: real or complex order

Keywords:

Mellin transform

A&S Ref:

12.1.27

Permalink:

http://dlmf.nist.gov/11.7.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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11.7.12 $${\int }_0^{\mathrm{\infty }}{t}^{-\mu -\nu }{𝐇}_{\mu }\left(t\right){𝐇}_{\nu }\left(t\right)dt=\frac{\sqrt{\pi }\mathrm{\Gamma }\left(\mu +\nu \right)}{2^{\mu +\nu }\mathrm{\Gamma }\left(\mu +\nu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\mu +\frac{1}{2}\right)\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right)},$$ $\mathrm{\Re }\left(\mu +\nu \right)>0$.

ⓘ

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, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\nu$: real or complex order

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/11.7.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §11.7(ii), §11.7 and Ch.11

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8: 13.16 Integral Representations

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13.16.10 $$\frac{1}{\mathrm{\Gamma }\left(1+2\mu \right)}{M}_{\kappa ,\mu }\left({\mathrm{e}}^{\pm \pi \mathrm{i}}z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}z\pm (\frac{1}{2}+\mu )\pi \mathrm{i}}}{2\pi \mathrm{i}\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(t-\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}+\mu -t\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)}{z}^{t}dt,$$ ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $M_{\kappa ,\mu }\left(z\right)$: Whittaker 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:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

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13.16.11 $$W_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{-\frac{1}{2}z}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)\mathrm{\Gamma }\left(-\kappa -t\right)}{\mathrm{\Gamma }\left(\frac{1}{2}+\mu -\kappa \right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu -\kappa \right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker 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:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

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13.16.12 $$W_{\kappa ,\mu }\left(z\right)=\frac{{\mathrm{e}}^{\frac{1}{2}z}}{2\pi \mathrm{i}}{\int }_{-\mathrm{i}\mathrm{\infty }}^{\mathrm{i}\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\frac{1}{2}+\mu +t\right)\mathrm{\Gamma }\left(\frac{1}{2}-\mu +t\right)}{\mathrm{\Gamma }\left(1-\kappa +t\right)}{z}^{-t}dt,$$ ,

ⓘ

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $W_{\kappa ,\mu }\left(z\right)$: Whittaker 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:

Mellin transform

Permalink:

http://dlmf.nist.gov/13.16.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.16(iii), §13.16 and Ch.13

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9: Bibliography M

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Euler-type transformations for the generalized hypergeometric function ${}_{r+2}{}^{}F_{r+1}^{}(x)$ . Z. Angew. Math. Phys. 62 (1), pp. 31–45.

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External Links:

ISSN 0044-2275, Document, FullText, MathReview (Mridula Garg), ZentralBlatt

Referenced by:

§16.6, §16.6.

Encodings:

BibTeX

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10: 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:

Mellin transform

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:

Mellin transform

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

ⓘ

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:

Mellin transform

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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