§19.23 Integral Representations

Notes:: For (19.23.8) and (19.23.9) see Carlson (1977b, Exercises 5.9-19, 5.9-20, and p. 306). By §19.16(iii), (19.23.8) implies (19.23.1)–(19.23.3), and (19.23.9) implies (19.23.6). Use(19.23.8) to integrate over $\theta$ in (19.23.6) and then permute variables to prove (19.23.5). To prove (19.23.4) put $z=0$ in (19.23.5), relabel variables, and substitute $\mathop{\cos\/}\nolimits\theta=\mathop{\mathrm{sech}\/}\nolimits t$ . For (19.23.7) and (19.23.10) see Carlson (1977b, (9.1-9) and (6.8-2), respectively).
Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.23

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

19.23.1 $\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)=\int _{0}^{{\pi/2}}(y{\mathop{\cos\/}\nolimits^{{2}}}\theta+z{\mathop{\sin\/}\nolimits^{{2}}}\theta)^{{-1/2}}d\theta,$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E1
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19.23.2 $\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)=\frac{1}{2}\int _{0}^{{\pi/2}}(y{\mathop{\cos\/}\nolimits^{{2}}}\theta+z{\mathop{\sin\/}\nolimits^{{2}}}\theta)^{{1/2}}d\theta,$

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Permalink:: http://dlmf.nist.gov/19.23.E2
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19.23.3 $\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)=3\int _{0}^{{\pi/2}}(y{\mathop{\cos\/}\nolimits^{{2}}}\theta+z{\mathop{\sin\/}\nolimits^{{2}}}\theta)^{{-3/2}}{\mathop{\sin\/}\nolimits^{{2}}}\theta d\theta.$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E3
Encodings:: TeX, pMML, png

19.23.4 $\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)=\frac{2}{\pi}\int _{0}^{{\pi/2}}\mathop{R_{C}\/}\nolimits\!\left(y,z{\mathop{\cos\/}\nolimits^{{2}}}\theta\right)d\theta=\frac{2}{\pi}\int _{0}^{{\infty}}\mathop{R_{C}\/}\nolimits\!\left(y{\mathop{\cosh\/}\nolimits^{{2}}}t,z\right)dt.$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\mathop{\cosh\/}\nolimits z$ : hyperbolic cosine function and $\int$ : integral
Referenced by:: §19.23
Permalink:: http://dlmf.nist.gov/19.23.E4
Encodings:: TeX, pMML, png

19.23.5 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\frac{2}{\pi}\int _{0}^{{\pi/2}}\mathop{R_{C}\/}\nolimits\!\left(x,y{\mathop{\cos\/}\nolimits^{{2}}}\theta+z{\mathop{\sin\/}\nolimits^{{2}}}\theta\right)d\theta,$ $\realpart{y}>0$ , $\realpart{z}>0$ ,

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.2(iv), §19.23
Permalink:: http://dlmf.nist.gov/19.23.E5
Encodings:: TeX, pMML, png

19.23.6 $4\pi\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\int _{0}^{{2\pi}}\!\!\!\!\int _{0}^{{\pi}}\frac{\mathop{\sin\/}\nolimits\theta d\theta d\phi}{(x{\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{\cos\/}\nolimits^{{2}}}\phi+y{\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{\sin\/}\nolimits^{{2}}}\phi+z{\mathop{\cos\/}\nolimits^{{2}}}\theta)^{{1/2}}},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $\phi$ : real or complex argument
Referenced by:: §19.23
Permalink:: http://dlmf.nist.gov/19.23.E6
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where $x$ , $y$ , and $z$ have positive real parts—except that at most one of them may be 0.

In (19.23.7)–(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) $\mathop{\mathrm{B}\/}\nolimits$ denotes the beta function (§5.12).

19.23.7 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{4}\int _{0}^{{\infty}}\frac{1}{\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}}\*\left(\frac{x}{t+x}+\frac{y}{t+y}+\frac{z}{t+z}\right)tdt,$ $x,y,z\in\Complex\setminus(-\infty,0]$ .

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\Complex$ : complex plane (excluding infinity), $dx$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(a,b]$ : half-closed interval and $\setminus$ : set subtraction
Referenced by:: §19.21(ii), §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E7
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19.23.8 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{2}{\mathop{\mathrm{B}\/}\nolimits\!\left(b_{1},b_{2}\right)}\int _{0}^{{\pi/2}}{(z_{1}{\mathop{\cos\/}\nolimits^{{2}}}\theta+z_{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta)}^{{-a}}\*(\mathop{\cos\/}\nolimits\theta)^{{2b_{1}-1}}(\mathop{\sin\/}\nolimits\theta)^{{2b_{2}-1}}d\theta,$ $b_{1},b_{2}>0$ ; $\realpart{z_{1}},\realpart{z_{2}}>0$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part and $\mathop{\sin\/}\nolimits z$ : sine function
Referenced by:: §19.2(iv), §19.23, §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E8
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With $l_{1},l_{2},l_{3}$ denoting any permutation of $\mathop{\sin\/}\nolimits\theta\mathop{\cos\/}\nolimits\phi$ , $\mathop{\sin\/}\nolimits\theta\mathop{\sin\/}\nolimits\phi$ , $\mathop{\cos\/}\nolimits\theta$ ,

19.23.9 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{4\mathop{\Gamma\/}\nolimits\!\left(b_{1}+b_{2}+b_{3}\right)}{\mathop{\Gamma\/}\nolimits\!\left(b_{1}\right)\mathop{\Gamma\/}\nolimits\!\left(b_{2}\right)\mathop{\Gamma\/}\nolimits\!\left(b_{3}\right)}\int _{0}^{{\pi/2}}\!\!\!\!\int _{0}^{{\pi/2}}\left(\sum _{{j=1}}^{3}z_{j}l_{j}^{2}\right)^{{-a}}\*\prod _{{j=1}}^{3}l_{j}^{{2b_{j}-1}}\mathop{\sin\/}\nolimits\theta d\theta d\phi,$ $b_{j}>0$ , $\realpart{z_{j}}>0$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $dx$ : differential of $x$ , $\int$ : integral, $\realpart{}$ : real part, $\mathop{\sin\/}\nolimits z$ : sine function, $l$ : nonnegative integer and $\phi$ : real or complex argument
Referenced by:: §19.23, §19.23
Permalink:: http://dlmf.nist.gov/19.23.E9
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19.23.10 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}\right)}\int _{0}^{1}u^{{a-1}}(1-u)^{{a^{{\prime}}-1}}\*\prod _{{j=1}}^{n}(1-u+uz_{j})^{{-b_{j}}}du,$ $a,a^{{\prime}}>0$ ; $a+a^{{\prime}}=\sum _{{j=1}}^{n}b_{j}$ ; $z_{j}\in\Complex\setminus(-\infty,0]$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\Complex$ : complex plane (excluding infinity), $dx$ : 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

For generalizations of (19.16.3) and (19.23.8) see Carlson (1964, (6.2), (6.12), and (6.1)).