§19.23 Integral Representations

In (19.23.1)–(19.23.3) we assume $\mathrm{\Re }y>0$ and $\mathrm{\Re }z>0$.

19.23.1		$$R_F(0,y,z)={\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{-1/2}d\theta ,$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $\sin z$: sine function Referenced by: §19.23, §19.23 Permalink: http://dlmf.nist.gov/19.23.E1 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.2		$$R_G(0,y,z)=\frac{1}{2}{\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{1/2}d\theta ,$$
ⓘ Symbols: $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $\sin z$: sine function Permalink: http://dlmf.nist.gov/19.23.E2 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.3		$$R_D(0,y,z)=3{\int }_0^{\pi /2}{(y{\cos }^2\theta +z{\sin }^2\theta )}^{-3/2}{\sin }^2\theta d\theta .$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral and $\sin z$: sine function Referenced by: §19.23, §19.23 Permalink: http://dlmf.nist.gov/19.23.E3 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.4		$$R_F(0,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(y,z{\cos }^2\theta )d\theta =\frac{2}{\pi }{\int }_0^{\mathrm{\infty }}{R}_C(y{\cosh }^{2}t,z)dt.$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\cosh z$: hyperbolic cosine function and $\int$: integral Referenced by: §19.23 Permalink: http://dlmf.nist.gov/19.23.E4 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.5		$$R_F(x,y,z)=\frac{2}{\pi }{\int }_0^{\pi /2}{R}_C(x,y{\cos }^2\theta +z{\sin }^2\theta )d\theta ,$$
$\mathrm{\Re }y>0$, $\mathrm{\Re }z>0$,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function Referenced by: §19.2(iv), §19.23 Permalink: http://dlmf.nist.gov/19.23.E5 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.6		$$4\pi R_F(x,y,z)={\int }_0^{2\pi }{\int }_0^{\pi }\frac{\sin \theta d\theta d\varphi }{{(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta )}^{1/2}},$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument Referenced by: §19.16(ii), §19.23 Permalink: http://dlmf.nist.gov/19.23.E6 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.6_5		$$R_G(x,y,z)=\frac{1}{4\pi }{\int }_0^{2\pi }{\int }_0^{\pi }{\left(x{\sin }^2\theta {\cos }^2\varphi +y{\sin }^2\theta {\sin }^2\varphi +z{\cos }^2\theta \right)}^{1/2}\sin \theta d\theta d\varphi ,$$
ⓘ Symbols: $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\sin z$: sine function and $\varphi$: real or complex argument Referenced by: (19.16.3), (19.16.3), §19.16(i), §19.16(ii), §19.16(ii), §19.20(ii), §19.23, §19.23, Erratum (V1.1.0) for Rearrangement Permalink: http://dlmf.nist.gov/19.23.E6_5 Encodings: TeX, pMML, png Rearrangement (effective with 1.1.0): This equation, which was originally (19.16.3), was moved here. See also: Annotations for §19.23 and Ch.19

where $x$, $y$, and $z$ have positive real parts—except that at most one of them may be 0.

In (19.23.8)–(19.23.10) one or more of the variables may be 0 if the integral converges. In (19.23.8) $n=2$, and in (19.23.9) $n=3$. Also, in (19.23.8) and (19.23.10) $\mathrm{B}$ denotes the beta function (§5.12).

19.23.7		Moved to (19.16.2_5).
ⓘ Referenced by: (19.16.2_5), §19.16(i), §19.23, Erratum (V1.1.0) for Rearrangement Permalink: http://dlmf.nist.gov/19.23.E7 Rearrangement (effective with 1.1.0): This equation has been moved to (19.16.2_5). See also: Annotations for §19.23 and Ch.19

19.23.8		$$R_{-a}(\mathbf{b};\mathbf{z})=\frac{2}{\mathrm{B}(b_1,b_2)}{\int }_0^{\pi /2}\begin{array}{l}{(z_1{\cos }^2\theta +z_2{\sin }^2\theta )}^{-a}\\ \times {(\cos \theta )}^{2b_1-1}{(\sin \theta )}^{2b_2-1}d\theta ,\end{array}$$
$b_1,b_2>0$; $\mathrm{\Re }{z}_1,\mathrm{\Re }{z}_2>0$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part and $\sin z$: sine function Referenced by: §19.2(iv), §19.23, §19.23, §19.23 Permalink: http://dlmf.nist.gov/19.23.E8 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

With $l_1,l_2,l_3$ denoting any permutation of $\sin \theta \cos \varphi$, $\sin \theta \sin \varphi$, $\cos \theta$,

19.23.9		$$R_{-a}(\mathbf{b};\mathbf{z})=\frac{4\mathrm{\Gamma }\left(b_1+b_2+b_3\right)}{\mathrm{\Gamma }\left(b_1\right)\mathrm{\Gamma }\left(b_2\right)\mathrm{\Gamma }\left(b_3\right)}{\int }_0^{\pi /2}{\int }_0^{\pi /2}{\left(\sum _{j=1}^3{z}_j{l}_j^2\right)}^{-a}\prod _{j=1}^3{l}_j^{2b_j-1}\sin \theta d\theta d\varphi ,$$
$b_j>0$, $\mathrm{\Re }{z}_j>0$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$, $\int$: integral, $\mathrm{\Re }$: real part, $\sin z$: sine function, $l$: nonnegative integer and $\varphi$: real or complex argument Referenced by: §19.23, §19.23 Permalink: http://dlmf.nist.gov/19.23.E9 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.10		$$R_{-a}(\mathbf{b};\mathbf{z})=\frac{1}{\mathrm{B}(a,a^{\prime })}{\int }_0^1{u}^{a-1}{(1-u)}^{a^{\prime }-1}\prod _{j=1}^n{(1-u+u{z}_j)}^{-b_j}du,$$
$a,a^{\prime }>0$; $a+a^{\prime }={\sum }_{j=1}^n{b}_j$; $z_j\in ℂ\setminus (-\mathrm{\infty },0]$.
ⓘ Symbols: $R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(\mathbf{b};\mathbf{z})$: multivariate hypergeometric function, $\mathrm{B}(a,b)$: beta function, $ℂ$: complex plane, $dx$: differential of $x$, $\in$: element of, $\int$: integral, $(a,b]$: half-closed interval, $\setminus$: set subtraction and $n$: nonnegative integer Referenced by: §19.19, §19.23, §19.23 Permalink: http://dlmf.nist.gov/19.23.E10 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

For generalizations of (19.23.6_5) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).