§13.4 Integral Representations

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

§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

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).
Keywords:: Kummer functions, hypergeometric function, integrals, integrals of Bessel and Hankel functions, integrals of modified Bessel functions
Referenced by:: §13.16(i)
Permalink:: http://dlmf.nist.gov/13.4.i

13.4.1 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}\int _{{0}}^{{1}}e^{{zt}}t^{{a-1}}(1-t)^{{b-a-1}}dt,$ $\realpart{b}>\realpart{a}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : 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
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13.4.2 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(b-c\right)}\int _{{0}}^{{1}}\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,c,zt\right)t^{{c-1}}(1-t)^{{b-c-1}}dt,$ $\realpart{b}>\realpart{c}>0$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : 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

13.4.3 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,-z\right)=\frac{z^{{\frac{1}{2}-\frac{1}{2}b}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\int _{{0}}^{{\infty}}e^{{-t}}t^{{a-\frac{1}{2}b-\frac{1}{2}}}\mathop{J_{{b-1}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{a}>0$ .

Symbols:: $\mathop{J_{{\nu}}\/}\nolimits\!\left(z\right)$ : Bessel function of the first kind, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.10(v)
Permalink:: http://dlmf.nist.gov/13.4.E3
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For the function $\mathop{J_{{b-1}}\/}\nolimits$ see §10.2(ii).

13.4.4 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(a\right)}\int _{{0}}^{{\infty}}e^{{-zt}}t^{{a-1}}(1+t)^{{b-a-1}}dt,$ $\realpart{a}>0$ , $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part and $z$ : complex variable
A&S Ref:: 13.2.5
Referenced by:: §13.10(v), §13.29(iii), §7.11
Permalink:: http://dlmf.nist.gov/13.4.E4
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13.4.5 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{z^{{1-a}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)}\int _{{0}}^{{\infty}}\frac{\mathop{U\/}\nolimits\!\left(b-a,b,t\right)e^{{-t}}t^{{a-1}}}{t+z}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\pi$ , $\realpart{a}>\max\left(\realpart{b-1},0\right)$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.10(vi), §13.4(i)
Permalink:: http://dlmf.nist.gov/13.4.E5
Encodings:: TeX, pMML, png

13.4.6 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{(-1)^{n}z^{{1-b-n}}}{\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)}\int _{0}^{\infty}\frac{\mathop{{\mathbf{M}}\/}\nolimits\!\left(b-a,b,t\right)e^{{-t}}t^{{b+n-1}}}{t+z}dt,$ $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\pi$ , $n=0,1,2,\dots$ , $-\realpart{b}<n<1+\realpart{(a-b)}$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\realpart{}$ : real part, $n$ : nonnegative integer and $z$ : complex variable
Referenced by:: §13.10(vi)
Permalink:: http://dlmf.nist.gov/13.4.E6
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13.4.7 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{2z^{{\frac{1}{2}-\frac{1}{2}b}}}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(a-b+1\right)}\*\int _{{0}}^{{\infty}}e^{{-t}}t^{{a-\frac{1}{2}b-\frac{1}{2}}}\mathop{K_{{b-1}}\/}\nolimits\!\left(2\sqrt{zt}\right)dt,$ $\realpart{a}>\max\left(\realpart{b-1},0\right)$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $(a,b)$ : open interval, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E7
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13.4.8 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=z^{{c-a}}\*\int _{{0}}^{{\infty}}e^{{-zt}}t^{{c-1}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,a-b+1;c;-t\right)dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$ ,

Symbols:: $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $z$ : complex variable and $c$ : arbitrary
Permalink:: http://dlmf.nist.gov/13.4.E8
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where $c$ is arbitrary, $\realpart{c}>0$ . For the functions $\mathop{K_{{b-1}}\/}\nolimits$ and $\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits$ see §10.25(ii) and §§15.1, 15.2(i).

§13.4(ii) Contour Integrals

Notes:: See Buchholz (1969, §1.4), Erdélyi et al. (1953a, §6.11), and Slater (1960, Chapter 3).
Keywords:: Kummer functions, hypergeometric function, integrals
Referenced by:: §13.16(ii)
Permalink:: http://dlmf.nist.gov/13.4.ii

13.4.9 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(a\right)}\int _{0}^{{(1+)}}e^{{zt}}t^{{a-1}}{(t-1)^{{b-a-1}}}dt,$ $b-a\neq 1,2,3,\dots$ , $\realpart{a}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E9
Encodings:: TeX, pMML, png

13.4.10 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=e^{{-a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(b-a\right)}\int _{1}^{{(0+)}}e^{{zt}}t^{{a-1}}{(1-t)^{{b-a-1}}}dt,$ $a\neq 1,2,3,\dots$ , $\realpart{(b-a)}>0$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $z$ : complex variable
Referenced by:: §13.8(i)
Permalink:: http://dlmf.nist.gov/13.4.E10
Encodings:: TeX, pMML, png

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

Keywords:: Pochhammer double-loop contour
Referenced by:: §13.4(ii)
Permalink:: http://dlmf.nist.gov/13.4.F1
Encodings:: pdf, png

13.4.11 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=e^{{-b\pi i}}\mathop{\Gamma\/}\nolimits\!\left(1-a\right)\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)\*\frac{1}{4\pi^{2}}\int _{{\alpha}}^{{(0+,1+,0-,1-)}}e^{{zt}}t^{{a-1}}{(1-t)^{{b-a-1}}}dt,$ $a,b-a\neq 1,2,3,\dots$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral and $z$ : complex variable
Referenced by:: Figure 13.4.1, Figure 13.4.1
Permalink:: http://dlmf.nist.gov/13.4.E11
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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 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,c,z\right)=\frac{\mathop{\Gamma\/}\nolimits\!\left(b\right)}{2\pi i}z^{{1-b}}\int _{{-\infty}}^{{(0+,1+)}}e^{{zt}}t^{{-b}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,b;c;\ifrac{1}{t}\right)dt,$ $b\neq 0,-1,-2,\dots$ , $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E12
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At the point where the contour crosses the interval $(1,\infty)$ , $t^{{-b}}$ and the $\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits$ function assume their principal values; compare §§15.1 and 15.2(i). A special case is

13.4.13 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)=\frac{z^{{1-b}}}{2\pi i}\int _{{-\infty}}^{{(0+,1+)}}e^{{zt}}t^{{-b}}\!\left(1-\frac{1}{t}\right)^{{-a}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $(a,b)$ : open interval, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E13
Encodings:: TeX, pMML, png

13.4.14 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=e^{{-a\pi i}}\frac{\mathop{\Gamma\/}\nolimits\!\left(1-a\right)}{2\pi i}\int _{{\infty}}^{{(0+)}}e^{{-zt}}t^{{a-1}}{(1+t)^{{b-a-1}}}dt,$ $a\neq 1,2,3,\dots$ , $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Referenced by:: §13.20(i)
Permalink:: http://dlmf.nist.gov/13.4.E14
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The contour cuts the real axis between −1 and 0. At this point the fractional powers are determined by $\mathop{\mathrm{ph}\/}\nolimits{t}=\pi$ and $\mathop{\mathrm{ph}\/}\nolimits(1+t)=0$ .

13.4.15 $\frac{\mathop{U\/}\nolimits\!\left(a,b,z\right)}{\mathop{\Gamma\/}\nolimits\!\left(c\right)\mathop{\Gamma\/}\nolimits\!\left(c-b+1\right)}=\frac{z^{{1-c}}}{2\pi i}\int _{{-\infty}}^{{(0+)}}e^{{zt}}t^{{-c}}\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits\!\left(a,c;a+c-b+1;1-\frac{1}{t}\right)dt,$ $\left|\mathop{\mathrm{ph}\/}\nolimits z\right|<\frac{1}{2}\pi$ .

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{{{}_{{p}}{\mathbf{F}}_{{q}}}\/}\nolimits\!\left({\mathbf{a}\atop\mathbf{b}};z\right)$ : scaled (or Olver’s) generalized hypergeometric function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E15
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Again, $t^{{-c}}$ and the $\mathop{{{}_{{2}}{\mathbf{F}}_{{1}}}\/}\nolimits$ function assume their principal values where the contour intersects the positive real axis.

§13.4(iii) Mellin–Barnes Integrals

Notes:: Combine the results of Buchholz (1969, §5.4) with (13.14.2), (13.14.3).
Keywords:: Kummer functions
Permalink:: http://dlmf.nist.gov/13.4.iii

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

13.4.16 $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,-z\right)=\frac{1}{2\pi i\mathop{\Gamma\/}\nolimits\!\left(a\right)}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+t\right)\mathop{\Gamma\/}\nolimits\!\left(-t\right)}{\mathop{\Gamma\/}\nolimits\!\left(b+t\right)}z^{{t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{{\mathbf{M}}\/}\nolimits\!\left(a,b,z\right)$ : Olver’s confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 13.2.9 (in different form)
Permalink:: http://dlmf.nist.gov/13.4.E16
Encodings:: TeX, pMML, png

where the contour of integration separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(a+t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(-t\right)$ .

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

13.4.17 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{z^{{-a}}}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(a+t\right)\mathop{\Gamma\/}\nolimits\!\left(1+a-b+t\right)\mathop{\Gamma\/}\nolimits\!\left(-t\right)}{\mathop{\Gamma\/}\nolimits\!\left(a\right)\mathop{\Gamma\/}\nolimits\!\left(1+a-b\right)}z^{{-t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{3}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
A&S Ref:: 13.2.10
Referenced by:: §13.6(vi)
Permalink:: http://dlmf.nist.gov/13.4.E17
Encodings:: TeX, pMML, png

where the contour of integration separates the poles of $\mathop{\Gamma\/}\nolimits\!\left(a+t\right)\mathop{\Gamma\/}\nolimits\!\left(1+a-b+t\right)$ from those of $\mathop{\Gamma\/}\nolimits\!\left(-t\right)$ .

13.4.18 $\mathop{U\/}\nolimits\!\left(a,b,z\right)=\frac{z^{{1-b}}e^{{z}}}{2\pi i}\int _{{-i\infty}}^{{i\infty}}\frac{\mathop{\Gamma\/}\nolimits\!\left(b-1+t\right)\mathop{\Gamma\/}\nolimits\!\left(t\right)}{\mathop{\Gamma\/}\nolimits\!\left(a+t\right)}z^{{-t}}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits{z}|<\tfrac{1}{2}\pi$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ : Kummer confluent hypergeometric function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/13.4.E18
Encodings:: TeX, pMML, png

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

§13.4 Integral Representations

Contents

§13.4(i) Integrals Along the Real Line

§13.4(ii) Contour Integrals

§13.4(iii) Mellin–Barnes Integrals