19 Elliptic Integrals Symmetric Integrals 19.25 Relations to Other Functions 19.27 Asymptotic Approximations and Expansions

§19.26 Addition Theorems

Keywords:: symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.26

§19.26(i) General Formulas
§19.26(ii) Case
§19.26(iii) Duplication Formulas

§19.26(i) General Formulas

Notes:: Addition theorems (and therefore duplication theorems) for the symmetric integrals are proved by Zill and Carlson (1970, §8). For other proofs of (19.26.1) see Carlson (1977b, §9.7) and Carlson (1978, Theorem 3). To prove (19.26.13) use (19.2.19) to show that $2\sqrt{\theta}\mathop{R_{C}\/}\nolimits\!\left(\sigma^{2},\sigma^{2}-\theta% \right)=\mathop{\ln\/}\nolimits\!\left((\sigma+\sqrt{\theta})/(\sigma-\sqrt{% \theta})\right)$ , then apply this to all three terms. To prove (19.26.14) put in (19.21.12) and use (19.20.8).
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.i

In this subsection, and also §§19.26(ii) and 19.26(iii), we assume that $\lambda,x,y,z$ are positive, except that at most one of can be 0.

19.26.1 $\mathop{R_{F}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+\mathop{% R_{F}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=\mathop{R_{F}\/}\nolimits\!% \left(x,y,z\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E1
Encodings:: TeX, pMML, png

where $\mu>0$ and

19.26.2 $x+\mu=\lambda^{{-2}}\left(\sqrt{(x+\lambda)yz}+\sqrt{x(y+\lambda)(z+\lambda)}% \right)^{2},$

Symbols:: $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E2
Encodings:: TeX, pMML, png

with corresponding equations for $y+\mu$ and $z+\mu$ obtained by permuting . Also,

19.26.3 $\sqrt{z}=\frac{\xi\zeta^{{\prime}}+\eta^{{\prime}}\zeta-\xi\eta^{{\prime}}}{% \sqrt{\xi\eta\zeta^{{\prime}}}+\sqrt{\xi^{{\prime}}\eta^{{\prime}}\zeta}},$

Symbols:: : positive, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/19.26.E3
Encodings:: TeX, pMML, png

where

19.26.4

$(\xi,\eta,\zeta)=(x+\lambda,y+\lambda,z+\lambda),$

$(\xi^{{\prime}},\eta^{{\prime}},\zeta^{{\prime}})=(x+\mu,y+\mu,z+\mu),$

Symbols:: $\lambda$ : positive, : positive, : positive, : positive, $\mu>0$ : parameter, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/19.26.E4
Encodings:: TeX, TeX, pMML, pMML, png, png

with $\sqrt{x}$ and $\sqrt{y}$ obtained by permuting , , and . (Note that $\xi\zeta^{{\prime}}+\eta^{{\prime}}\zeta-\xi\eta^{{\prime}}=\xi^{{\prime}}% \zeta+\eta\zeta^{{\prime}}-\xi^{{\prime}}\eta$ .) Equivalent forms of (19.26.2) are given by

19.26.5 $\mu=\lambda^{{-2}}\left(\sqrt{xyz}+\sqrt{(x+\lambda)(y+\lambda)(z+\lambda)}% \right)^{2}-\lambda-x-y-z,$

Symbols:: $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E5
Encodings:: TeX, pMML, png

and

19.26.6 $(\lambda\mu-xy-xz-yz)^{2}=4xyz(\lambda+\mu+x+y+z).$

Symbols:: $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E6
Encodings:: TeX, pMML, png

Also,

19.26.7 $\mathop{R_{D}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+\mathop{% R_{D}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=\mathop{R_{D}\/}\nolimits\!% \left(x,y,z\right)-\frac{3}{\sqrt{z(z+\lambda)(z+\mu)}},$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E7
Encodings:: TeX, pMML, png

19.26.8 $2\!\mathop{R_{G}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda\right)+2\!% \mathop{R_{G}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu\right)=2\!\mathop{R_{G}\/}% \nolimits\!\left(x,y,z\right)+\lambda\mathop{R_{F}\/}\nolimits\!\left(x+% \lambda,y+\lambda,z+\lambda\right)+\mu\mathop{R_{F}\/}\nolimits\!\left(x+\mu,y% +\mu,z+\mu\right)+\sqrt{\lambda+\mu+x+y+z}.$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E8
Encodings:: TeX, pMML, png

19.26.9 $\mathop{R_{J}\/}\nolimits\!\left(x+\lambda,y+\lambda,z+\lambda,p+\lambda\right% )+\mathop{R_{J}\/}\nolimits\!\left(x+\mu,y+\mu,z+\mu,p+\mu\right)=\mathop{R_{J% }\/}\nolimits\!\left(x,y,z,p\right)-3\!\mathop{R_{C}\/}\nolimits\!\left(\gamma% -\delta,\gamma\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\lambda$ : positive, $\delta$ , : positive, : positive, : positive, $\mu>0$ : parameter and $\gamma$
Referenced by:: §19.11(i)
Permalink:: http://dlmf.nist.gov/19.26.E9
Encodings:: TeX, pMML, png

where

19.26.10

$\gamma=p(p+\lambda)(p+\mu),$

$\delta=(p-x)(p-y)(p-z).$

Symbols:: $\lambda$ : positive, $\delta$ , : positive, : positive, : positive, $\mu>0$ : parameter and $\gamma$
Permalink:: http://dlmf.nist.gov/19.26.E10
Encodings:: TeX, TeX, pMML, pMML, png, png

Lastly,

19.26.11 $\mathop{R_{C}\/}\nolimits\!\left(x+\lambda,y+\lambda\right)+\mathop{R_{C}\/}% \nolimits\!\left(x+\mu,y+\mu\right)=\mathop{R_{C}\/}\nolimits\!\left(x,y\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$ : positive, : positive, : positive and $\mu>0$ : parameter
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E11
Encodings:: TeX, pMML, png

where $\lambda>0$ , , $x\geq 0$ , and

19.26.12

$x+\mu=\lambda^{{-2}}(\sqrt{x+\lambda}y+\sqrt{x}(y+\lambda))^{2},$

$y+\mu=(y(y+\lambda)/\lambda^{2})(\sqrt{x}+\sqrt{x+\lambda})^{2}.$

Symbols:: $\lambda$ : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

Equivalent forms of (19.26.11) are given by

19.26.13 $\mathop{R_{C}\/}\nolimits\!\left(\alpha^{2},\alpha^{2}-\theta\right)+\mathop{R% _{C}\/}\nolimits\!\left(\beta^{2},\beta^{2}-\theta\right)=\mathop{R_{C}\/}% \nolimits\!\left(\sigma^{2},\sigma^{2}-\theta\right),$ $\sigma=(\alpha\beta+\theta)/(\alpha+\beta)$ ,

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.26(i), §19.26(ii)
Permalink:: http://dlmf.nist.gov/19.26.E13
Encodings:: TeX, pMML, png

where $0<\gamma^{2}-\theta<\gamma^{2}$ for $\gamma=\alpha,\beta,\sigma$ , except that $\sigma^{2}-\theta$ can be 0, and

19.26.14 $(p-y)\mathop{R_{C}\/}\nolimits\!\left(x,p\right)+(q-y)\mathop{R_{C}\/}% \nolimits\!\left(x,q\right)=(\eta-\xi)\mathop{R_{C}\/}\nolimits\!\left(\xi,% \eta\right),$ $x\geq 0$ , $y\geq 0$ ; $p,q\in\Real\setminus\{0\}$ ,

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$ : element of, $\Real$ : real line (excluding infinity), $\{\ldots\}$ : sequence, asymptotic sequence (or scale), or enumerable set, $\setminus$ : set subtraction, : positive, : positive, $\xi$ : coordinate and $\eta$ : coordinate
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.E14
Encodings:: TeX, pMML, png

where

19.26.15

$(p-x)(q-x)=(y-x)^{2},$

$\xi=y^{2}/x,$

$\eta=pq/x,$

$\eta-\xi=p+q-2y.$

Symbols:: : positive, : positive, $\xi$ : coordinate and $\eta$ : coordinate
Permalink:: http://dlmf.nist.gov/19.26.E15
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

§19.26(ii) Case

Notes:: For (19.26.17) put $\theta=-\alpha\beta$ in (19.26.13) and use homogeneity.
Referenced by:: §19.26(i), §19.36(iv)
Permalink:: http://dlmf.nist.gov/19.26.ii

If , then $\lambda\mu=yz$ . For example,

19.26.16 $\mathop{R_{F}\/}\nolimits\!\left(\lambda,y+\lambda,z+\lambda\right)={\mathop{R% _{F}\/}\nolimits\!\left(0,y,z\right)-\mathop{R_{F}\/}\nolimits\!\left(\mu,y+% \mu,z+\mu\right),}$ $\lambda\mu=yz$ .

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, : positive, : positive and $\mu>0$ : parameter
Permalink:: http://dlmf.nist.gov/19.26.E16
Encodings:: TeX, pMML, png

An equivalent version for $\mathop{R_{C}\/}\nolimits$ is

19.26.17 $\sqrt{\alpha}\mathop{R_{C}\/}\nolimits\!\left(\beta,\alpha+\beta\right)+\sqrt{% \beta}\mathop{R_{C}\/}\nolimits\!\left(\alpha,\alpha+\beta\right)=\pi/2,$ $\alpha,\beta\in\Complex\setminus(-\infty,0)$ , $\alpha+\beta>0$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\Complex$ : complex plane (excluding infinity), $\in$ : element of, : open interval, $\setminus$ : set subtraction and $\alpha^{2}$ : real or complex parameter
Referenced by:: §19.26(ii)
Permalink:: http://dlmf.nist.gov/19.26.E17
Encodings:: TeX, pMML, png

§19.26(iii) Duplication Formulas

Notes:: Put $\mu=\lambda$ in the formulas of §19.26(i). For proofs of (19.26.18) not invoking the addition theorem, see Carlson (1977b, §9.6) and Carlson (1998, §2). Equations (19.26.25) and (19.26.20) are degenerate cases of (19.26.18) and (19.26.22), respectively.
Keywords:: symmetric elliptic integrals
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.iii

19.26.18 $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)=2\!\mathop{R_{F}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda\right)=\mathop{R_{F}\/}\nolimits\!\left(% \frac{x+\lambda}{4},\frac{y+\lambda}{4},\frac{z+\lambda}{4}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\lambda$ : positive, : positive, : positive and : positive
Referenced by:: §19.26(iii), ¶ ‣ §19.36(i), §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E18
Encodings:: TeX, pMML, png

where

19.26.19 $\lambda=\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}.$

Symbols:: $\lambda$ : positive, : positive, : positive and : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E19
Encodings:: TeX, pMML, png

19.26.20 $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)=2\!\mathop{R_{D}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda\right)+\frac{3}{\sqrt{z}(z+\lambda)}.$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\lambda$ : positive, : positive, : positive and : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E20
Encodings:: TeX, pMML, png

19.26.21 $2\!\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)=4\!\mathop{R_{G}\/}\nolimits% \!\left(x+\lambda,y+\lambda,z+\lambda\right)-\lambda\mathop{R_{F}\/}\nolimits% \!\left(x,y,z\right)-\sqrt{x}-\sqrt{y}-\sqrt{z}.$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\lambda$ : positive, : positive, : positive and : positive
Referenced by:: ¶ ‣ §19.36(i)
Permalink:: http://dlmf.nist.gov/19.26.E21
Encodings:: TeX, pMML, png

19.26.22 $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)=2\!\mathop{R_{J}\/}\nolimits\!% \left(x+\lambda,y+\lambda,z+\lambda,p+\lambda\right)+3\!\mathop{R_{C}\/}% \nolimits\!\left(\alpha^{2},\beta^{2}\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\lambda$ : positive, $\alpha$ , $\beta$ , : positive, : positive and : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E22
Encodings:: TeX, pMML, png

where

19.26.23

$\alpha=p(\sqrt{x}+\sqrt{y}+\sqrt{z})+\sqrt{x}\sqrt{y}\sqrt{z},$

$\beta=\sqrt{p}(p+\lambda),$

$\beta\pm\alpha=(\sqrt{p}\pm\sqrt{x})(\sqrt{p}\pm\sqrt{y})(\sqrt{p}\pm\sqrt{z}),$

$\beta^{2}-\alpha^{2}=(p-x)(p-y)(p-z),$

Symbols:: $\lambda$ : positive, $\alpha$ , $\beta$ , : positive, : positive and : positive
Permalink:: http://dlmf.nist.gov/19.26.E23
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

either upper or lower signs being taken throughout.

The equations inverse to $z+\lambda=(\sqrt{z}+\sqrt{x})(\sqrt{z}+\sqrt{y})$ and the two other equations obtained by permuting (see (19.26.19)) are

19.26.24 $z=(\xi\zeta+\eta\zeta-\xi\eta)^{2}/(4\xi\eta\zeta),$ $(\xi,\eta,\zeta)=(x+\lambda,y+\lambda,z+\lambda)$ ,

Symbols:: $\lambda$ : positive, : positive, : positive, : positive, $\xi$ : coordinate, $\eta$ : coordinate and $\zeta$ : coordinate
Permalink:: http://dlmf.nist.gov/19.26.E24
Encodings:: TeX, pMML, png

and two similar equations obtained by exchanging with (and $\zeta$ with $\xi$ ), or with (and $\zeta$ with $\eta$ ).

Next,

19.26.25 $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)=2\!\mathop{R_{C}\/}\nolimits\!% \left(x+\lambda,y+\lambda\right),$ $\lambda=y+2\sqrt{x}\sqrt{y}$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$ : positive, : positive and : positive
Referenced by:: §19.26(iii)
Permalink:: http://dlmf.nist.gov/19.26.E25
Encodings:: TeX, pMML, png

Equivalent forms are given by (19.22.22). Also,

19.26.26 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}\right)=\mathop{R_{C}\/}\nolimits% \!\left(a^{2},ay\right),$

, $\realpart{x}\geq 0$ , $\realpart{y}>0$ ,

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\realpart{}$ : real part, : positive and : positive
Permalink:: http://dlmf.nist.gov/19.26.E26
Encodings:: TeX, pMML, png

and

19.26.27 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},x^{2}-\theta\right)=2\!\mathop{R_{C}\/}% \nolimits\!\left(s^{2},s^{2}-\theta\right),$ $s=x+\sqrt{x^{2}-\theta}$ , $\theta\neq x^{2}$ or $s^{2}$ .

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions and : positive
Permalink:: http://dlmf.nist.gov/19.26.E27
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. 19.25 Relations to Other Functions 19.27 Asymptotic Approximations and Expansions

§19.26 Addition Theorems

Contents

§19.26(i) General Formulas

§19.26(ii) Case

§19.26(iii) Duplication Formulas