13 Confluent Hypergeometric Functions Kummer Functions 13.3 Recurrence Relations and Derivatives 13.5 Continued Fractions

§13.4 Integral Representations

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Permalink:: http://dlmf.nist.gov/13.4
See also:: Annotations for Ch.13

§13.4(i) Integrals Along the Real Line

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Keywords:: Kummer functions, along the real line, hypergeometric function, integral representations, integrals, integrals of Bessel and Hankel functions, integrals of modified Bessel functions, over infinite intervals
Notes:: See Buchholz (1969, §1.4), Erdélyi et al. (1953a, §6.11), and Slater (1960, Chapter 3). For (13.4.5) use (13.2.41); compare the proof of Lemma 3.1 in Olde Daalhuis and Olver (1994).
Referenced by:: §13.16(i)
Permalink:: http://dlmf.nist.gov/13.4.i
See also:: Annotations for §13.4 and Ch.13

13.4.1		${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\,\mathrm{d}t,$
$\Re b>\Re a>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable A&S Ref: 13.2.1 (in different form) Referenced by: §13.10(v), §13.29(iii) Permalink: http://dlmf.nist.gov/13.4.E1 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.2		${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,$
$\Re b>\Re c>0$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $z$ : complex variable and $c$ : arbitrary Referenced by: §13.10(vi) Permalink: http://dlmf.nist.gov/13.4.E2 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.3		${\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,$
$\Re a>0$ .
ⓘ Symbols: $J_{\NVar{\nu}}\left(\NVar{z}\right)$ : Bessel function of the first kind, $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Referenced by: §13.10(v) Permalink: http://dlmf.nist.gov/13.4.E3 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

For the function $J_{b-1}$ see §10.2(ii).

13.4.4		$U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-zt}t^{a% -1}(1+t)^{b-a-1}\,\mathrm{d}t,$
$\Re a>0$ , $\|\operatorname{ph}{z}\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part and $z$ : complex variable A&S Ref: 13.2.5 Referenced by: §13.10(v), §13.29(iii), §7.11, (9.5.8) Permalink: http://dlmf.nist.gov/13.4.E4 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.5		$U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a\right)\Gamma\left(1+a-b\right% )}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^{-t}t^{a-1}}{t+z}\,\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\pi$ , $\Re a>\max\left(\Re b-1,0\right)$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part and $z$ : complex variable Referenced by: §13.10(vi), §13.4(i), (9.10.18) Permalink: http://dlmf.nist.gov/13.4.E5 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.6		$U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{\Gamma\left(1+a-b\right)}\int_{0}% ^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t\right)e^{-t}t^{b+n-1}}{t+z}\,\mathrm{% d}t,$
$\left\|\operatorname{ph}z\right\|<\pi$ , $n=0,1,2,\dots$ , $-\Re b<n<1+\Re\left(a-b\right)$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, $n$ : nonnegative integer and $z$ : complex variable Referenced by: §13.10(vi) Permalink: http://dlmf.nist.gov/13.4.E6 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.7		$U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma\left(a\right)% \Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2% }}K_{b-1}\left(2\sqrt{zt}\right)\,\mathrm{d}t,$
$\Re a>\max\left(\Re b-1,0\right)$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.4.E7 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

13.4.8		$U\left(a,b,z\right)=z^{c-a}\*\int_{0}^{\infty}e^{-zt}t^{c-1}{{}_{2}{\mathbf{F}% }_{1}}\left(a,a-b+1;c;-t\right)\,\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\frac{1}{2}\pi$ ,
ⓘ Symbols: $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\operatorname{ph}$ : phase, $z$ : complex variable and $c$ : arbitrary Permalink: http://dlmf.nist.gov/13.4.E8 Encodings: TeX, pMML, png See also: Annotations for §13.4(i), §13.4 and Ch.13

where $c$ is arbitrary, $\Re c>0$ . For the functions $K_{b-1}$ and ${{}_{2}{\mathbf{F}}_{1}}$ see §10.25(ii) and §§15.1, 15.2(i).

§13.4(ii) Contour Integrals

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Keywords:: Kummer functions, contour integrals, hypergeometric function, integral representations, integrals
Notes:: See Buchholz (1969, §1.4), Erdélyi et al. (1953a, §6.11), and Slater (1960, Chapter 3).
Referenced by:: §13.16(ii)
Permalink:: http://dlmf.nist.gov/13.4.ii
Clarification (effective with 1.2.1):: In regard to the contour of integration, just below (13.4.15), we inserted “(see Figure 5.9.1)”.
See also:: Annotations for §13.4 and Ch.13

13.4.9		${\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1+a-b\right)}{2\pi\mathrm{i}% \Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\,\mathrm{d}t,$
$b-a\neq 1,2,3,\dots$ , $\Re a>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.4.E9 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

13.4.10		${\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(1-a\right)% }{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a% -1}}\,\mathrm{d}t,$
$a\neq 1,2,3,\dots$ , $\Re\left(b-a\right)>0$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $z$ : complex variable Referenced by: §13.8(i) Permalink: http://dlmf.nist.gov/13.4.E10 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

See accompanying text — Figure 13.4.1: Contour of integration in (13.4.11). (Compare Figure 5.12.3.) Magnify

13.4.11		${\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}\Gamma\left(1-a\right)\Gamma% \left(1+a-b\right)\*\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-% 1}{(1-t)^{b-a-1}}\,\mathrm{d}t,$
$a,b-a\neq 1,2,3,\dots$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $z$ : complex variable Referenced by: Figure 13.4.1, Figure 13.4.1 Permalink: http://dlmf.nist.gov/13.4.E11 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

The contour of integration starts and terminates at a point $\alpha$ on the real axis between $0$ and $1$ . It encircles $t=0$ and $t=1$ once in the positive sense, and then once in the negative sense. See Figure 13.4.1. The fractional powers are continuous and assume their principal values at $t=\alpha$ . Similar conventions also apply to the remaining integrals in this subsection.

13.4.12		${\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1% -b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;% \ifrac{1}{t}\right)\,\mathrm{d}t,$
$b\neq 0,-1,-2,\dots$ , $\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.4.E12 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

At the point where the contour crosses the interval $(1,\infty)$ , $t^{-b}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values; compare §§15.1 and 15.2(i). A special case is

13.4.13		${\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi\mathrm{i}}\int_{-\infty}^{(% 0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\,\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\frac{1}{2}\pi$ .
ⓘ Symbols: ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase and $z$ : complex variable Permalink: http://dlmf.nist.gov/13.4.E13 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

13.4.14		$U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(1-a\right)}{2\pi% \mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\,\mathrm{d}t,$
$a\neq 1,2,3,\dots$ , $\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable Referenced by: §13.20(i) Permalink: http://dlmf.nist.gov/13.4.E14 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

The contour cuts the real axis between $-1$ and $0$ . At this point the fractional powers are determined by $\operatorname{ph}{t}=\pi$ and $\operatorname{ph}\left(1+t\right)=0$ .

13.4.15		$\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)\Gamma\left(c-b+1\right)}=\frac% {z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt}t^{-c}{{}_{2}{\mathbf{F}}_% {1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)\,\mathrm{d}t,$
$\left\|\operatorname{ph}z\right\|<\frac{1}{2}\pi$ .
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left(\NVar{\mathbf{a}};\NVar{\mathbf{b}% };\NVar{z}\right)$ or ${{}_{\NVar{p}}{\mathbf{F}}_{\NVar{q}}}\left({\NVar{\mathbf{a}}\atop\NVar{% \mathbf{b}}};\NVar{z}\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable Referenced by: §13.4(ii), Erratum (V1.2.1) for Figure 5.9.1 Permalink: http://dlmf.nist.gov/13.4.E15 Encodings: TeX, pMML, png See also: Annotations for §13.4(ii), §13.4 and Ch.13

Again, $t^{-c}$ and the ${{}_{2}{\mathbf{F}}_{1}}$ function assume their principal values where the contour (see Figure 5.9.1) intersects the positive real axis.

§13.4(iii) Mellin–Barnes Integrals

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Keywords:: Kummer functions, Mellin–Barnes type, contour integrals, integral representations
Notes:: Combine the results of Buchholz (1969, §5.4) with (13.14.2), (13.14.3).
Permalink:: http://dlmf.nist.gov/13.4.iii
See also:: Annotations for §13.4 and Ch.13

If $a\neq 0,-1,-2,\dots$ , then

13.4.16		${\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(-t\right)}{\Gamma\left(b+t\right)}z^{t}\,\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\mathbf{M}}\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Olver’s confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : 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: TeX, pMML, png See also: Annotations for §13.4(iii), §13.4 and Ch.13

where the contour of integration separates the poles of $\Gamma\left(a+t\right)$ from those of $\Gamma\left(-t\right)$ .

If $a$ and $a-b+1\neq 0,-1,-2,\dots$ , then

13.4.17		$U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{% \mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)\Gamma% \left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a-b\right)}z^{-t}\,\mathrm{% d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{3}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : 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: TeX, pMML, png See also: Annotations for §13.4(iii), §13.4 and Ch.13

where the contour of integration separates the poles of $\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)$ from those of $\Gamma\left(-t\right)$ .

13.4.18		$U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty% }^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t\right)\Gamma\left(t\right)}{\Gamma% \left(a+t\right)}z^{-t}\,\mathrm{d}t,$
$\|\operatorname{ph}{z}\|<\tfrac{1}{2}\pi$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, $U\left(\NVar{a},\NVar{b},\NVar{z}\right)$ : Kummer confluent hypergeometric function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and $z$ : complex variable Keywords: Mellin transform Permalink: http://dlmf.nist.gov/13.4.E18 Encodings: TeX, pMML, png See also: Annotations for §13.4(iii), §13.4 and Ch.13

where the contour of integration passes all the poles of $\Gamma\left(b-1+t\right)\Gamma\left(t\right)$ on the right-hand side.

13.3 Recurrence Relations and Derivatives 13.5 Continued Fractions

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§13.4 Integral Representations

Contents

§13.4(i) Integrals Along the Real Line

§13.4(ii) Contour Integrals

§13.4(iii) Mellin–Barnes Integrals

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