19 Elliptic Integrals Symmetric Integrals 19.15 Advantages of Symmetry 19.17 Graphics

§19.16 Definitions

Permalink:: http://dlmf.nist.gov/19.16

§19.16(i) Symmetric Integrals
§19.16(ii) $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$
§19.16(iii) Various Cases of $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

§19.16(i) Symmetric Integrals

Keywords:: hyperelliptic integrals, symmetric elliptic integrals
Referenced by:: §19.20(i), §19.29(i), §20.9(i)
Permalink:: http://dlmf.nist.gov/19.16.i

19.16.1 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{2}\int_{0}^{{\infty}}% \frac{dt}{s(t)},$

Defines:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Symbols:: : differential of , $\int$ : integral and : function
Referenced by:: §19.16(i), §19.16(i), §19.18(i), §19.25(vi)
Permalink:: http://dlmf.nist.gov/19.16.E1
Encodings:: TeX, pMML, png

19.16.2 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=\frac{3}{2}\int_{0}^{{\infty}}% \frac{dt}{s(t)(t+p)},$

Defines:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Symbols:: : differential of , $\int$ : integral and : function
Referenced by:: §19.16(i), §19.19, §19.20(iii), §19.28
Permalink:: http://dlmf.nist.gov/19.16.E2
Encodings:: TeX, pMML, png

19.16.3 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\frac{1}{4\pi}\int_{0}^{{2\pi}}% \!\!\!\!\int_{0}^{{\pi}}\left(x{\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{% \cos\/}\nolimits^{{2}}}\phi+y{\mathop{\sin\/}\nolimits^{{2}}}\theta{\mathop{% \sin\/}\nolimits^{{2}}}\phi+z{\mathop{\cos\/}\nolimits^{{2}}}\theta\right)^{{% \frac{1}{2}}}\mathop{\sin\/}\nolimits\theta d\theta d\phi,$

Defines:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $\phi$ : real or complex argument
Referenced by:: §19.16(i), §19.16(ii), §19.20(ii), §19.23
Permalink:: http://dlmf.nist.gov/19.16.E3
Encodings:: TeX, pMML, png

where ( $\neq 0$ ) is a real or complex constant, and

19.16.4 $s(t)=\sqrt{t+x}\sqrt{t+y}\sqrt{t+z}.$

Symbols:: : function
Permalink:: http://dlmf.nist.gov/19.16.E4
Encodings:: TeX, pMML, png

In (19.16.1) and (19.16.2), $x,y,z\in\Complex\setminus(-\infty,0]$ except that one or more of may be 0 when the corresponding integral converges. In (19.16.2) the Cauchy principal value is taken when is real and negative. In (19.16.3) $\realpart{x}$ , $\realpart{y}$ , $\realpart{z}\geq 0$ . It should be noted that the integrals (19.16.1)–(19.16.3) have been normalized so that $\mathop{R_{F}\/}\nolimits\!\left(1,1,1\right)=\mathop{R_{J}\/}\nolimits\!\left% (1,1,1,1\right)=\mathop{R_{G}\/}\nolimits\!\left(1,1,1\right)=1$ .

A fourth integral that is symmetric in only two variables is defined by

19.16.5 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{J}\/}\nolimits\!\left% (x,y,z,z\right)=\frac{3}{2}\int_{0}^{{\infty}}\frac{dt}{s(t)(t+z)},$

Defines:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, : differential of , $\int$ : integral and : function
Referenced by:: §19.18(i), §19.19, §19.20(iii), §19.20(iv), ¶ ‣ §19.24(ii), §19.6(iv)
Permalink:: http://dlmf.nist.gov/19.16.E5
Encodings:: TeX, pMML, png

with the same conditions on , , as for (19.16.1), but now $z\neq 0$ .

Just as the elementary function $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ (§19.2(iv)) is the degenerate case

19.16.6 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\mathop{R_{F}\/}\nolimits\!\left(x% ,y,y\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.19, ¶ ‣ §19.24(ii), §19.6(ii)
Permalink:: http://dlmf.nist.gov/19.16.E6
Encodings:: TeX, pMML, png

and $\mathop{R_{D}\/}\nolimits$ is a degenerate case of $\mathop{R_{J}\/}\nolimits$ , so is $\mathop{R_{J}\/}\nolimits$ a degenerate case of the hyperelliptic integral,

19.16.7 $\frac{3}{2}\int_{0}^{{\infty}}\frac{dt}{\prod_{{j=1}}^{5}\sqrt{t+x_{j}}}.$

Symbols:: : differential of and $\int$ : integral
Permalink:: http://dlmf.nist.gov/19.16.E7
Encodings:: TeX, pMML, png

§19.16(ii) $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

Notes:: See Carlson (1977b, (6.8-6), Ex. 6.8-8, and (5.9-1)). To prove (19.16.12) put $t={\mathop{\csc\/}\nolimits^{{2}}}\theta-{\mathop{\csc\/}\nolimits^{{2}}}\phi$ in the first integral in (19.16.9).
Keywords:: hypergeometric -function, hypergeometric function, multivariate hypergeometric function, symmetric elliptic integrals
Referenced by:: §20.9(i), Other Changes
Permalink:: http://dlmf.nist.gov/19.16.ii
Clarification (effective with 1.0.1):: ”unchanged” was substituted for ”symmetric” in text after 19.16.8.
Reported 2010-11-23

All elliptic integrals of the form (19.2.3) and many multiple integrals, including (19.16.3), are special cases of a multivariate hypergeometric function

19.16.8 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\mathop{R_{{-% a}}\/}\nolimits\!\left(b_{1},\dots,b_{n};z_{1},\dots,z_{n}\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and : nonnegative integer
Referenced by:: §19.16(ii)
Permalink:: http://dlmf.nist.gov/19.16.E8
Encodings:: TeX, pMML, png

which is homogeneous and of degree in the ’s, and unchanged when the same permutation is applied to both sets of subscripts $1,\dots,n$ . Thus $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is symmetric in the variables $z_{j}$ and $z_{{\ell}}$ if the parameters $b_{j}$ and $b_{{\ell}}$ are equal. The -function is often used to make a unified statement of a property of several elliptic integrals. Before 1969 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ was denoted by $R(a;\mathbf{b};\mathbf{z})$ .

19.16.9 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{% \mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}\right)}\int_{0}^{{\infty}% }t^{{a^{{\prime}}-1}}\prod^{n}_{{j=1}}(t+z_{j})^{{-b_{j}}}dt=\frac{1}{\mathop{% \mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}\right)}\int_{0}^{{\infty}}t^{{a-1% }}\prod^{n}_{{j=1}}(1+tz_{j})^{{-b_{j}}}dt,$ $a,a^{{\prime}}>0$ , $z_{j}\in\Complex\setminus(-\infty,0]$ ,

Defines:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function
Symbols:: $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, $\Complex$ : complex plane (excluding infinity), : differential of , $\in$ : element of, $\int$ : integral, : half-closed interval, $\setminus$ : set subtraction and : nonnegative integer
Referenced by:: §19.16(ii), §19.16(ii), §19.19, §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.16.E9
Encodings:: TeX, pMML, png

where $\mathop{\mathrm{B}\/}\nolimits\!\left(x,y\right)$ is the beta function (§5.12) and

19.16.10 $a^{{\prime}}=-a+\sum_{{j=1}}^{n}b_{j}.$

Symbols:: : nonnegative integer
Permalink:: http://dlmf.nist.gov/19.16.E10
Encodings:: TeX, pMML, png

19.16.11

$\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\lambda\mathbf{z}\right)=% \lambda^{{-a}}\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right),$

$\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};x\boldsymbol{{1}}\right)=x^{{-a}}$ , $\boldsymbol{{1}}=(1,\dots,1)$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function
Referenced by:: §19.19, §19.20(v)
Permalink:: http://dlmf.nist.gov/19.16.E11
Encodings:: TeX, TeX, pMML, pMML, png, png

When a useful version of (19.16.9) is given by

19.16.12 $\mathop{R_{{-a}}\/}\nolimits\!\left(b_{1},\dots,b_{4};c-1,c-k^{2},c,c-\alpha^{% 2}\right)=\frac{2({\mathop{\sin\/}\nolimits^{{2}}}\phi)^{{1-a^{{\prime}}}}}{% \mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}\right)}\int_{0}^{{\phi}}(% \mathop{\sin\/}\nolimits\theta)^{{2a-1}}{({\mathop{\sin\/}\nolimits^{{2}}}\phi% -{\mathop{\sin\/}\nolimits^{{2}}}\theta)}^{{a^{{\prime}}-1}}\*(\mathop{\cos\/}% \nolimits\theta)^{{1-2b_{1}}}{(1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta)}% ^{{-b_{2}}}{(1-\alpha^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta)}^{{-b_{4}}}d\theta,$

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, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.16(ii), §19.25(i)
Permalink:: http://dlmf.nist.gov/19.16.E12
Encodings:: TeX, pMML, png

where

19.16.13

$c={\mathop{\csc\/}\nolimits^{{2}}}\phi;$

$a,a^{{\prime}}>0;$

$b_{3}=a+a^{{\prime}}-b_{1}-b_{2}-b_{4}.$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function and $\phi$ : real or complex argument
Permalink:: http://dlmf.nist.gov/19.16.E13
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

For further information, especially representation of the -function as a Dirichlet average, see Carlson (1977b).

§19.16(iii) Various Cases of $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

Notes:: For (19.16.19) and (19.16.23) see Carlson (1977b, (5.9-19) and (8.3-4)). To derive (19.16.24) exchange subscripts 1 and in Carlson (1963, (7.4)), put $t=s/z_{1}$ , and use (19.16.19).
Keywords:: hypergeometric -function, symmetric elliptic integrals
Referenced by:: §19.2(iv), §19.23, §19.28, §19.31, §20.9(i), Other Changes, Subsection 19.16(iii)
Permalink:: http://dlmf.nist.gov/19.16.iii
Errata (effective with 1.0.1):: Originally it was implied that $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ is an elliptic integral. It was clarified that $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the stated conditions hold; originally these conditions were stated as sufficient but not necessary. In particular, $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ does not satisfy these conditions.
Reported 2010-11-23

$\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ is an elliptic integral iff the ’s are distinct and exactly four of the parameters $a,a^{{\prime}},b_{1},\dots,b_{n}$ are half-odd-integers, the rest are integers, and none of , $a^{{\prime}}$ , $a+a^{{\prime}}$ is zero or a negative integer. The only cases that are integrals of the first kind are the four in which each of and $a^{{\prime}}$ is either $\frac{1}{2}$ or 1 and each $b_{j}$ is $\frac{1}{2}$ . The only cases that are integrals of the third kind are those in which at least one $b_{j}$ is a positive integer. All other elliptic cases are integrals of the second kind.

19.16.14 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{{-\frac{1}{2}}}\/}% \nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.16(iii), §19.18(i), §19.18(ii), §19.19, §19.19, §19.24(ii)
Permalink:: http://dlmf.nist.gov/19.16.E14
Encodings:: TeX, pMML, png

19.16.15 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{{-\frac{3}{2}}}\/}% \nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{3}{2};x,y,z\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.20(iv)
Permalink:: http://dlmf.nist.gov/19.16.E15
Encodings:: TeX, pMML, png

19.16.16 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=\mathop{R_{{-\frac{3}{2}}}\/}% \nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2},1;x,y,z,p\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Permalink:: http://dlmf.nist.gov/19.16.E16
Encodings:: TeX, pMML, png

19.16.17 $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=\mathop{R_{{\frac{1}{2}}}\/}% \nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},\tfrac{1}{2};x,y,z\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.16(iii), §19.24(ii)
Permalink:: http://dlmf.nist.gov/19.16.E17
Encodings:: TeX, pMML, png

19.16.18 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=\mathop{R_{{-\frac{1}{2}}}\/}% \nolimits\!\left(\tfrac{1}{2},1;x,y\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions
Referenced by:: §19.18(i), §19.18(ii), §19.19, §19.19, §19.2(iv)
Permalink:: http://dlmf.nist.gov/19.16.E18
Encodings:: TeX, pMML, png

(Note that $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ is not an elliptic integral.)

When one variable is 0 without destroying convergence, any one of (19.16.14)–(19.16.17) is said to be complete and can be written as an -function with one less variable:

19.16.19 $\mathop{R_{{-a}}\/}\nolimits\!\left(b_{1},\dots,b_{n};0,z_{2},\dots,z_{n}% \right)=\frac{\mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}-b_{1}\right% )}{\mathop{\mathrm{B}\/}\nolimits\!\left(a,a^{{\prime}}\right)}\mathop{R_{{-a}% }\/}\nolimits\!\left(b_{2},\dots,b_{n};z_{2},\dots,z_{n}\right),$ $a+a^{{\prime}}>0$ , $a^{{\prime}}>b_{1}$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function and : nonnegative integer
Referenced by:: §19.16(iii), §19.20(v)
Permalink:: http://dlmf.nist.gov/19.16.E19
Encodings:: TeX, pMML, png

Thus

19.16.20 $\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)=\tfrac{1}{2}\pi\mathop{R_{{-% \frac{1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};y,z\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind
Referenced by:: §19.16(iii), §19.18(i), §19.18(ii), §19.18(ii), §19.19, §19.19, §19.22(ii), ¶ ‣ §19.24(ii), §19.24(i), §19.28
Permalink:: http://dlmf.nist.gov/19.16.E20
Encodings:: TeX, pMML, png

19.16.21 $\mathop{R_{D}\/}\nolimits\!\left(0,y,z\right)=\tfrac{3}{4}\pi\mathop{R_{{-% \frac{3}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{3}{2};y,z\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables
Referenced by:: §19.9(i)
Permalink:: http://dlmf.nist.gov/19.16.E21
Encodings:: TeX, pMML, png

19.16.22 $\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)=\tfrac{3}{4}\pi\mathop{R_{{-% \frac{3}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2},1;y,z,p\right),$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: §19.24(i)
Permalink:: http://dlmf.nist.gov/19.16.E22
Encodings:: TeX, pMML, png

19.16.23 $\mathop{R_{G}\/}\nolimits\!\left(0,y,z\right)=\tfrac{1}{4}\pi\mathop{R_{{\frac% {1}{2}}}\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};y,z\right)=\tfrac{1}{4}% \pi z\mathop{R_{{-\frac{1}{2}}}\/}\nolimits\!\left(-\tfrac{1}{2},\tfrac{3}{2};% y,z\right).$

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.16(iii), §19.16(iii), §19.18(i), §19.18(ii), §19.18(ii), §19.19, §19.19, §19.22(ii), ¶ ‣ §19.24(ii), §19.24(i), §19.33(i)
Permalink:: http://dlmf.nist.gov/19.16.E23
Encodings:: TeX, pMML, png

The last -function has $a=a^{{\prime}}=\frac{1}{2}$ .

Each of the four complete integrals (19.16.20)–(19.16.23) can be integrated to recover the incomplete integral:

19.16.24 $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)=\frac{z_{1}^{% {a^{{\prime}}-b_{1}}}}{\mathop{\mathrm{B}\/}\nolimits\!\left(b_{1},a^{{\prime}% }-b_{1}\right)}\int_{0}^{{\infty}}t^{{b_{1}-1}}(t+z_{1})^{{-a^{{\prime}}}}\*% \mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};0,t+z_{2},\dots,t+z_{n}\right)dt,$ $a^{{\prime}}>b_{1}$ , $a+a^{{\prime}}>b_{1}>0$ .

Symbols:: $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$ : multivariate hypergeometric function, $\mathop{\mathrm{B}\/}\nolimits\!\left(a,b\right)$ : beta function, : differential of , $\int$ : integral and : nonnegative integer
Referenced by:: §19.16(iii), §19.28
Permalink:: http://dlmf.nist.gov/19.16.E24
Encodings:: TeX, pMML, png

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.15 Advantages of Symmetry 19.17 Graphics

§19.16 Definitions

Contents

§19.16(i) Symmetric Integrals

§19.16(ii)

§19.16(iii) Various Cases of

§19.16(ii) $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$

§19.16(iii) Various Cases of $\mathop{R_{{-a}}\/}\nolimits\!\left(\mathbf{b};\mathbf{z}\right)$