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

§19.26 Addition Theorems

ⓘ

Keywords:: addition theorems, symmetric elliptic integrals
Permalink:: http://dlmf.nist.gov/19.26
See also:: Annotations for Ch.19

§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 $2\sqrt{\theta}R_{C}\left(\sigma^{2},\sigma^{2}-\theta\right)=\ln\left((\sigma+% \sqrt{\theta})/(\sigma-\sqrt{\theta})\right)$ , 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
See also:: Annotations for §19.26 and Ch.19

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 $R_{F}\left(x+\lambda,y+\lambda,z+\lambda\right)+R_{F}\left(x+\mu,y+\mu,z+\mu% \right)=R_{F}\left(x,y,z\right),$

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

Also,

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

ⓘ

Symbols:: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

Lastly,

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

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

Equivalent forms of (19.26.11) are given by

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

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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

19.26.14 $(p-y)R_{C}\left(x,p\right)+(q-y)R_{C}\left(x,q\right)=(\eta-\xi)R_{C}\left(\xi% ,\eta\right),$ $x\geq 0$ , $y\geq 0$ ; $p,q\in\mathbb{R}\setminus\{0\}$ ,

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$ : element of, $\mathbb{R}$ : real line, $\setminus$ : 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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

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
See also:: Annotations for §19.26(i), §19.26 and Ch.19

§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
See also:: Annotations for §19.26 and Ch.19

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

19.26.16 $R_{F}\left(\lambda,y+\lambda,z+\lambda\right)={R_{F}\left(0,y,z\right)-R_{F}% \left(\mu,y+\mu,z+\mu\right),}$ $\lambda\mu=yz$ .

ⓘ

Symbols:: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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
See also:: Annotations for §19.26(ii), §19.26 and Ch.19

An equivalent version for $R_{C}$ is

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

ⓘ

Symbols:: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathbb{C}$ : complex plane, $\in$ : element of, $(\NVar{a},\NVar{b})$ : 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
See also:: Annotations for §19.26(ii), §19.26 and Ch.19

§19.26(iii) Duplication Formulas

ⓘ

Keywords:: addition theorems, duplication formulas, symmetric elliptic integrals
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.
Referenced by:: §19.26(i)
Permalink:: http://dlmf.nist.gov/19.26.iii
See also:: Annotations for §19.26 and Ch.19

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

ⓘ