19 Elliptic Integrals Symmetric Integrals 19.22 Quadratic Transformations 19.24 Inequalities

§19.23 Integral Representations

ⓘ

Keywords:: integral representations, symmetric elliptic integrals
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 $\cos\theta=\operatorname{sech}t$ . For (19.23.10) see Carlson (1977b, (6.8-2)).
Referenced by:: Erratum (V1.1.0) for Rearrangement
Permalink:: http://dlmf.nist.gov/19.23
Rearrangement (effective with 1.1.0):: The integral representation (19.23.6_5), which was previously (19.16.3), is a better fit here. Likewise, (19.23.7) has been moved to (19.16.2_5).
See also:: Annotations for Ch.19

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

19.23.1		$R_{F}\left(0,y,z\right)=\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)^% {-1/2}\,\mathrm{d}\theta,$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : 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		$R_{G}\left(0,y,z\right)=\frac{1}{2}\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^% {2}\theta)^{1/2}\,\mathrm{d}\theta,$
ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : 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		$R_{D}\left(0,y,z\right)=3\int_{0}^{\pi/2}(y{\cos}^{2}\theta+z{\sin}^{2}\theta)% ^{-3/2}{\sin}^{2}\theta\,\mathrm{d}\theta.$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\sin\NVar{z}$ : 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		$R_{F}\left(0,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(y,z{\cos}^{2}% \theta\right)\,\mathrm{d}\theta=\frac{2}{\pi}\int_{0}^{\infty}R_{C}\left(y{% \cosh}^{2}t,z\right)\,\mathrm{d}t.$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\cosh\NVar{z}$ : 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		$R_{F}\left(x,y,z\right)=\frac{2}{\pi}\int_{0}^{\pi/2}R_{C}\left(x,y{\cos}^{2}% \theta+z{\sin}^{2}\theta\right)\,\mathrm{d}\theta,$
$\Re y>0$ , $\Re z>0$ ,
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\sin\NVar{z}$ : 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		$4\pi R_{F}\left(x,y,z\right)=\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}\frac{\sin% \theta\,\mathrm{d}\theta\,\mathrm{d}\phi}{(x{\sin}^{2}\theta{\cos}^{2}\phi+y{% \sin}^{2}\theta{\sin}^{2}\phi+z{\cos}^{2}\theta)^{1/2}},$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $\phi$ : 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		$R_{G}\left(x,y,z\right)=\frac{1}{4\pi}\int_{0}^{2\pi}\!\!\!\!\int_{0}^{\pi}% \left(x{\sin}^{2}\theta{\cos}^{2}\phi+y{\sin}^{2}\theta{\sin}^{2}\phi+z{\cos}^% {2}\theta\right)^{1/2}\sin\theta\,\mathrm{d}\theta\,\mathrm{d}\phi,$
ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function and $\phi$ : 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, §19.25(vi), 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		$R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{2}{\mathrm{B}\left(b_{1},b_{2}% \right)}\int_{0}^{\pi/2}{(z_{1}{\cos}^{2}\theta+z_{2}{\sin}^{2}\theta)}^{-a}\*% (\cos\theta)^{2b_{1}-1}(\sin\theta)^{2b_{2}-1}\,\mathrm{d}\theta,$
$b_{1},b_{2}>0$ ; $\Re z_{1},\Re z_{2}>0$ .
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part and $\sin\NVar{z}$ : 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 $\sin\theta\cos\phi$ , $\sin\theta\sin\phi$ , $\cos\theta$ ,

19.23.9		$R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{4\Gamma\left(b_{1}+b_{2}+b_{3}% \right)}{\Gamma\left(b_{1}\right)\Gamma\left(b_{2}\right)\Gamma\left(b_{3}% \right)}\int_{0}^{\pi/2}\!\!\!\!\int_{0}^{\pi/2}\left(\sum_{j=1}^{3}z_{j}l_{j}% ^{2}\right)^{-a}\*\prod_{j=1}^{3}l_{j}^{2b_{j}-1}\sin\theta\,\mathrm{d}\theta% \,\mathrm{d}\phi,$
$b_{j}>0$ , $\Re z_{j}>0$ .
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\Re$ : real part, $\sin\NVar{z}$ : 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 Encodings: TeX, pMML, png See also: Annotations for §19.23 and Ch.19

19.23.10		$R_{-a}\left(\mathbf{b};\mathbf{z}\right)=\frac{1}{\mathrm{B}\left(a,a^{\prime}% \right)}\int_{0}^{1}u^{a-1}(1-u)^{a^{\prime}-1}\*\prod_{j=1}^{n}(1-u+uz_{j})^{% -b_{j}}\,\mathrm{d}u,$
$a,a^{\prime}>0$ ; $a+a^{\prime}=\sum_{j=1}^{n}b_{j}$ ; $z_{j}\in\mathbb{C}\setminus(-\infty,0]$ .
ⓘ Symbols: $R_{\NVar{-a}}\left(\NVar{b_{1}},\dots,\NVar{b_{n}};\NVar{z_{1}},\dots,\NVar{z_% {n}}\right)$ or $R_{\NVar{-a}}\left(\NVar{\mathbf{b}};\NVar{\mathbf{z}}\right)$ : multivariate hypergeometric function, $\mathrm{B}\left(\NVar{a},\NVar{b}\right)$ : beta function, $\mathbb{C}$ : complex plane, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\int$ : integral, $(\NVar{a},\NVar{b}]$ : 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)).

19.22 Quadratic Transformations 19.24 Inequalities

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§19.23 Integral Representations

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