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
§13.4(ii) Contour Integrals
§13.4(iii) Mellin–Barnes Integrals

§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 ${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)\Gamma\left(b-a% \right)}\int_{0}^{1}e^{zt}t^{a-1}(1-t)^{b-a-1}\,\mathrm{d}t,$ $\Re b>\Re a>0$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : 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 ${\mathbf{M}}\left(a,b,z\right)=\frac{1}{\Gamma\left(b-c\right)}\int_{0}^{1}{% \mathbf{M}}\left(a,c,zt\right)t^{c-1}(1-t)^{b-c-1}\,\mathrm{d}t,$ $\Re b>\Re c>0$ ,

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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 , $\int$ : integral, $\Re$ : real part, : complex variable and : 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 ${\mathbf{M}}\left(a,b,-z\right)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma% \left(a\right)}\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2}}J_{b-1}% \left(2\sqrt{zt}\right)\,\mathrm{d}t,$ $\Re a>0$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part and : 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 $U\left(a,b,z\right)=\frac{1}{\Gamma\left(a\right)}\int_{0}^{\infty}e^{-zt}t^{a% -1}(1+t)^{b-a-1}\,\mathrm{d}t,$ $\Re a>0$ , $|\operatorname{ph}{z}|<\frac{1}{2}\pi$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part and : 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 $U\left(a,b,z\right)=\frac{z^{1-a}}{\Gamma\left(a\right)\Gamma\left(1+a-b\right% )}\int_{0}^{\infty}\frac{U\left(b-a,b,t\right)e^{-t}t^{a-1}}{t+z}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\pi$ , $\Re a>\max\left(\Re b-1,0\right)$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part and : 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 $U\left(a,b,z\right)=\frac{(-1)^{n}z^{1-b-n}}{\Gamma\left(1+a-b\right)}\int_{0}% ^{\infty}\frac{{\mathbf{M}}\left(b-a,b,t\right)e^{-t}t^{b+n-1}}{t+z}\,\mathrm{% d}t,$ $\left|\operatorname{ph}z\right|<\pi$ , $n=0,1,2,\dots$ , $-\Re b<n<1+\Re\left(a-b\right)$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{ph}$ : phase, $\Re$ : real part, : nonnegative integer and : 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 $U\left(a,b,z\right)=\frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\Gamma\left(a\right)% \Gamma\left(a-b+1\right)}\*\int_{0}^{\infty}e^{-t}t^{a-\frac{1}{2}b-\frac{1}{2% }}K_{b-1}\left(2\sqrt{zt}\right)\,\mathrm{d}t,$ $\Re a>\max\left(\Re b-1,0\right)$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\Re$ : real part and : 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 $U\left(a,b,z\right)=z^{c-a}\*\int_{0}^{\infty}e^{-zt}t^{c-1}{{}_{2}{\mathbf{F}% }_{1}}\left(a,a-b+1;c;-t\right)\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\frac{1}{2}\pi$ ,

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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 , $\mathrm{e}$ : 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, : complex variable and : 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 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 ${\mathbf{M}}\left(a,b,z\right)=\frac{\Gamma\left(1+a-b\right)}{2\pi\mathrm{i}% \Gamma\left(a\right)}\int_{0}^{(1+)}e^{zt}t^{a-1}{(t-1)^{b-a-1}}\,\mathrm{d}t,$ $b-a\neq 1,2,3,\dots$ , $\Re a>0$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and : 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 ${\mathbf{M}}\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(1-a\right)% }{2\pi\mathrm{i}\Gamma\left(b-a\right)}\int_{1}^{(0+)}e^{zt}t^{a-1}{(1-t)^{b-a% -1}}\,\mathrm{d}t,$ $a\neq 1,2,3,\dots$ , $\Re\left(b-a\right)>0$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and : 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

Figure 13.4.1: Contour of integration in (13.4.11). (Compare Figure 5.12.3.) Magnify

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Keywords:: Pochhammer double-loop contour
Referenced by:: §13.4(ii)
Permalink:: http://dlmf.nist.gov/13.4.F1
Encodings:: pdf, png
See also:: Annotations for §13.4(ii), §13.4 and Ch.13

13.4.11 ${\mathbf{M}}\left(a,b,z\right)=e^{-b\pi\mathrm{i}}\Gamma\left(1-a\right)\Gamma% \left(1+a-b\right)\*\frac{1}{4\pi^{2}}\int_{\alpha}^{(0+,1+,0-,1-)}e^{zt}t^{a-% 1}{(1-t)^{b-a-1}}\,\mathrm{d}t,$ $a,b-a\neq 1,2,3,\dots$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral and : 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 and 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 ${\mathbf{M}}\left(a,c,z\right)=\frac{\Gamma\left(b\right)}{2\pi\mathrm{i}}z^{1% -b}\int_{-\infty}^{(0+,1+)}e^{zt}t^{-b}{{}_{2}{\mathbf{F}}_{1}}\left(a,b;c;% \ifrac{1}{t}\right)\,\mathrm{d}t,$ $b\neq 0,-1,-2,\dots$ , $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$ .

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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 , $\mathrm{e}$ : 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 : 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 ${\mathbf{M}}\left(a,b,z\right)=\frac{z^{1-b}}{2\pi\mathrm{i}}\int_{-\infty}^{(% 0+,1+)}e^{zt}t^{-b}\!\left(1-\frac{1}{t}\right)^{-a}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\frac{1}{2}\pi$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $(\NVar{a},\NVar{b})$ : open interval, $\operatorname{ph}$ : phase and : 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 $U\left(a,b,z\right)=e^{-a\pi\mathrm{i}}\frac{\Gamma\left(1-a\right)}{2\pi% \mathrm{i}}\int_{\infty}^{(0+)}e^{-zt}t^{a-1}{(1+t)^{b-a-1}}\,\mathrm{d}t,$ $a\neq 1,2,3,\dots$ , $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$ .

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and : 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 $\frac{U\left(a,b,z\right)}{\Gamma\left(c\right)\Gamma\left(c-b+1\right)}=\frac% {z^{1-c}}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}e^{zt}t^{-c}{{}_{2}{\mathbf{F}}_% {1}}\left(a,c;a+c-b+1;1-\frac{1}{t}\right)\,\mathrm{d}t,$ $\left|\operatorname{ph}z\right|<\frac{1}{2}\pi$ .

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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 , $\mathrm{e}$ : 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 : 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 ${\mathbf{M}}\left(a,b,-z\right)=\frac{1}{2\pi\mathrm{i}\Gamma\left(a\right)}% \int_{-\mathrm{i}\infty}^{\mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma% \left(-t\right)}{\Gamma\left(b+t\right)}z^{t}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$ ,

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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 , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and : 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 and $a-b+1\neq 0,-1,-2,\dots$ , then

13.4.17 $U\left(a,b,z\right)=\frac{z^{-a}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty}^{% \mathrm{i}\infty}\frac{\Gamma\left(a+t\right)\Gamma\left(1+a-b+t\right)\Gamma% \left(-t\right)}{\Gamma\left(a\right)\Gamma\left(1+a-b\right)}z^{-t}\,\mathrm{% d}t,$ $|\operatorname{ph}{z}|<\tfrac{3}{2}\pi$ ,

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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 , $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and : 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 $U\left(a,b,z\right)=\frac{z^{1-b}e^{z}}{2\pi\mathrm{i}}\int_{-\mathrm{i}\infty% }^{\mathrm{i}\infty}\frac{\Gamma\left(b-1+t\right)\Gamma\left(t\right)}{\Gamma% \left(a+t\right)}z^{-t}\,\mathrm{d}t,$ $|\operatorname{ph}{z}|<\tfrac{1}{2}\pi$ ,

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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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase and : 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