§13.4 Integral Representations

13.4.1		$$\mathbf{M}(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(b-a\right)}{\int }_0^1{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,$$
$\mathrm{\Re }b>\mathrm{\Re }a>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$$\mathbf{M}(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(b-c\right)}{\int }_0^1\mathbf{M}(a,c,zt)t^{c-1}{(1-t)}^{b-c-1}dt,$$
$\mathrm{\Re }b>\mathrm{\Re }c>0$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: 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		$$\mathbf{M}(a,b,-z)=\frac{z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{J}_{b-1}\left(2\sqrt{zt}\right)dt,$$
$\mathrm{\Re }a>0$.
ⓘ Symbols: $J_{\nu }\left(z\right)$: Bessel function of the first kind, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{\Re }$: 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		$$U(a,b,z)=\frac{1}{\mathrm{\Gamma }\left(a\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$
$\mathrm{\Re }a>0$, ,
ⓘ 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, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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		$$U(a,b,z)=\frac{z^{1-a}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{U(b-a,b,t){\mathrm{e}}^{-t}{t}^{a-1}}{t+z}dt,$$
, $\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,
ⓘ 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, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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		$$U(a,b,z)=\frac{{(-1)}^n{z}^{1-b-n}}{\mathrm{\Gamma }\left(1+a-b\right)}{\int }_0^{\mathrm{\infty }}\frac{\mathbf{M}(b-a,b,t){\mathrm{e}}^{-t}{t}^{b+n-1}}{t+z}dt,$$
, $n=0,1,2,\mathrm{\dots }$, ,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\mathrm{ph}$: phase, $\mathrm{\Re }$: 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		$$U(a,b,z)=\frac{2z^{\frac{1}{2}-\frac{1}{2}b}}{\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-t}{t}^{a-\frac{1}{2}b-\frac{1}{2}}{K}_{b-1}\left(2\sqrt{zt}\right)dt,$$
$\mathrm{\Re }a>\max (\mathrm{\Re }b-1,0)$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $U(a,b,z)$: Kummer confluent hypergeometric function, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{\Re }$: 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		$$U(a,b,z)=z^{c-a}{\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-zt}{t}^{c-1}{}_2\mathbf{F}_1(a,a-b+1;c;-t)dt,$$
,
ⓘ Symbols: $U(a,b,z)$: 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, ${}_p\mathbf{F}_q(\mathbf{a};\mathbf{b};z)$ or ${}_p\mathbf{F}_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$: scaled (or Olver’s) generalized hypergeometric function, $\int$: integral, $\mathrm{ph}$: 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, $\mathrm{\Re }c>0$. For the functions $K_{b-1}$ and ${}_2\mathbf{F}_1$ see §10.25(ii) and §§15.1, 15.2(i).

13.4.9		$$\mathbf{M}(a,b,z)=\frac{\mathrm{\Gamma }\left(1+a-b\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(a\right)}{\int }_0^{(1+)}{\mathrm{e}}^{zt}{t}^{a-1}{(t-1)}^{b-a-1}dt,$$
$b-a\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }a>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: 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		$$\mathbf{M}(a,b,z)={\mathrm{e}}^{-a\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-a\right)}{2\pi \mathrm{i}\mathrm{\Gamma }\left(b-a\right)}{\int }_1^{(0+)}{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,$$
$a\ne 1,2,3,\mathrm{\dots }$, $\mathrm{\Re }\left(b-a\right)>0$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: 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

13.4.11		$$\mathbf{M}(a,b,z)=\begin{array}{l}{\mathrm{e}}^{-b\pi \mathrm{i}}\mathrm{\Gamma }\left(1-a\right)\mathrm{\Gamma }\left(1+a-b\right)\frac{1}{4{\pi }^2}\\ \times {\int }_{\alpha }^{(0+,1+,0-,1-)}{\mathrm{e}}^{zt}{t}^{a-1}{(1-t)}^{b-a-1}dt,\end{array}$$
$a,b-a\ne 1,2,3,\mathrm{\dots }$.
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: 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		$$\mathbf{M}(a,c,z)=\frac{\mathrm{\Gamma }\left(b\right)}{2\pi \mathrm{i}}{z}^{1-b}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{}_2\mathbf{F}_1(a,b;c;1/t)dt,$$
$b\ne 0,-1,-2,\mathrm{\dots }$, .
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, ${}_p\mathbf{F}_q(\mathbf{a};\mathbf{b};z)$ or ${}_p\mathbf{F}_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$: scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: 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 $(1,\mathrm{\infty })$, $t^{-b}$ 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		$$\mathbf{M}(a,b,z)=\frac{z^{1-b}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+,1+)}{\mathrm{e}}^{zt}{t}^{-b}{\left(1-\frac{1}{t}\right)}^{-a}dt,$$
.
ⓘ Symbols: $\mathbf{M}(a,b,z)$: Olver’s confluent hypergeometric function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $(a,b)$: open interval, $\mathrm{ph}$: 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		$$U(a,b,z)={\mathrm{e}}^{-a\pi \mathrm{i}}\frac{\mathrm{\Gamma }\left(1-a\right)}{2\pi \mathrm{i}}{\int }_{\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{-zt}{t}^{a-1}{(1+t)}^{b-a-1}dt,$$
$a\ne 1,2,3,\mathrm{\dots }$, .
ⓘ 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 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 $\mathrm{ph}t=\pi$ and $\mathrm{ph}\left(1+t\right)=0$.

13.4.15		$$\frac{U(a,b,z)}{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(c-b+1\right)}=\frac{z^{1-c}}{2\pi \mathrm{i}}{\int }_{-\mathrm{\infty }}^{(0+)}{\mathrm{e}}^{zt}{t}^{-c}{}_2\mathbf{F}_1(a,c;a+c-b+1;1-\frac{1}{t})dt,$$
.
ⓘ 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, ${}_p\mathbf{F}_q(\mathbf{a};\mathbf{b};z)$ or ${}_p\mathbf{F}_q(\genfrac{}{}{0pt}{}{\mathbf{a}}{\mathbf{b}};z)$: scaled (or Olver’s) generalized hypergeometric function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{ph}$: phase and $z$: complex variable 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 intersects the positive real axis.