§19.26 Addition Theorems

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 $x,y,z$ can be 0.

19.26.1		$$R_F(x+\lambda ,y+\lambda ,z+\lambda )+R_F(x+\mu ,y+\mu ,z+\mu )=R_F(x,y,z),$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\lambda$: positive, $x$: positive, $y$: positive, $z$: 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, $x$: positive, $y$: positive, $z$: 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 $x,y,z$. 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: $z$: 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, $x$: positive, $y$: positive, $z$: 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 $x$, $y$, and $z$. (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, $x$: positive, $y$: positive, $z$: 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, $x$: positive, $y$: positive, $z$: 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		$$R_D(x+\lambda ,y+\lambda ,z+\lambda )+R_D(x+\mu ,y+\mu ,z+\mu )=R_D(x,y,z)-\frac{3}{\sqrt{z(z+\lambda )(z+\mu )}},$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\lambda$: positive, $x$: positive, $y$: positive, $z$: 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		$$\begin{array}{l}2R_G(x+\lambda ,y+\lambda ,z+\lambda )+2R_G(x+\mu ,y+\mu ,z+\mu )\\ =2R_G(x,y,z)+\lambda R_F(x+\lambda ,y+\lambda ,z+\lambda )+\mu R_F(x+\mu ,y+\mu ,z+\mu )+\sqrt{\lambda +\mu +x+y+z}.\end{array}$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\lambda$: positive, $x$: positive, $y$: positive, $z$: 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		$$R_J(x+\lambda ,y+\lambda ,z+\lambda ,p+\lambda )+R_J(x+\mu ,y+\mu ,z+\mu ,p+\mu )=R_J(x,y,z,p)-3R_C(\gamma -\delta ,\gamma ),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\lambda$: positive, $\delta$, $x$: positive, $y$: positive, $z$: 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, $\delta$, $x$: positive, $y$: positive, $z$: 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		$$R_C(x+\lambda ,y+\lambda )+R_C(x+\mu ,y+\mu )=R_C(x,y),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive, $y$: 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 $\lambda >0$, $y>0$, $x\ge 0$, 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, $x$: positive, $y$: 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		$$R_C({\alpha }^2,{\alpha }^2-\theta )+R_C({\beta }^2,{\beta }^2-\theta )=R_C({\sigma }^2,{\sigma }^2-\theta ),$$
$\sigma =(\alpha \beta +\theta )/(\alpha +\beta )$,
ⓘ Symbols: $R_C(x,y)$: 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 for $\gamma =\alpha ,\beta ,\sigma$, except that ${\sigma }^2-\theta$ can be 0, and

19.26.14		$$(p-y)R_C(x,p)+(q-y)R_C(x,q)=(\eta -\xi )R_C(\xi ,\eta ),$$
$x\ge 0$, $y\ge 0$; $p,q\in ℝ\setminus \{0\}$,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\in$: element of, $ℝ$: real line, $\setminus$: set subtraction, $x$: positive, $y$: 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: $x$: positive, $y$: 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

If $x=0$, then $\lambda \mu =yz$. For example,

19.26.16		$$R_F(\lambda ,y+\lambda ,z+\lambda )=R_F(0,y,z)-R_F(\mu ,y+\mu ,z+\mu ),$$
$\lambda \mu =yz$.
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\lambda$: positive, $y$: positive, $z$: 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		$$\sqrt{\alpha }{R}_C(\beta ,\alpha +\beta )+\sqrt{\beta }{R}_C(\alpha ,\alpha +\beta )=\pi /2,$$
$\alpha ,\beta \in ℂ\setminus (-\mathrm{\infty },0)$, $\alpha +\beta >0$.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\pi$: the ratio of the circumference of a circle to its diameter, $ℂ$: complex plane, $\in$: element of, $(a,b)$: 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

19.26.18		$$R_F(x,y,z)=2R_F(x+\lambda ,y+\lambda ,z+\lambda )=R_F(\frac{x+\lambda }{4},\frac{y+\lambda }{4},\frac{z+\lambda }{4}),$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.26(iii), §19.36(i), §19.36(i) Permalink: http://dlmf.nist.gov/19.26.E18 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

where

19.26.19		$$\lambda =\sqrt{x}\sqrt{y}+\sqrt{y}\sqrt{z}+\sqrt{z}\sqrt{x}.$$
ⓘ Symbols: $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.26(iii) Permalink: http://dlmf.nist.gov/19.26.E19 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

19.26.20		$$R_D(x,y,z)=2R_D(x+\lambda ,y+\lambda ,z+\lambda )+\frac{3}{\sqrt{z}(z+\lambda )}.$$
ⓘ Symbols: $R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.26(iii) Permalink: http://dlmf.nist.gov/19.26.E20 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

19.26.21		$$2R_G(x,y,z)=4R_G(x+\lambda ,y+\lambda ,z+\lambda )-\lambda R_F(x,y,z)-\sqrt{x}-\sqrt{y}-\sqrt{z}.$$
ⓘ Symbols: $R_F(x,y,z)$: symmetric elliptic integral of first kind, $R_G(x,y,z)$: symmetric elliptic integral of second kind, $\lambda$: positive, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.36(i) Permalink: http://dlmf.nist.gov/19.26.E21 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

19.26.22		$$R_J(x,y,z,p)=2R_J(x+\lambda ,y+\lambda ,z+\lambda ,p+\lambda )+3R_C({\alpha }^2,{\beta }^2),$$
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $R_J(x,y,z,p)$: symmetric elliptic integral of third kind, $\lambda$: positive, $\alpha$, $\beta$, $x$: positive, $y$: positive and $z$: positive Referenced by: §19.26(iii) Permalink: http://dlmf.nist.gov/19.26.E22 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

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, $\alpha$, $\beta$, $x$: positive, $y$: positive and $z$: positive Permalink: http://dlmf.nist.gov/19.26.E23 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

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 $x,y,z$ (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, $x$: positive, $y$: positive, $z$: positive, $\xi$: coordinate, $\eta$: coordinate and $\zeta$: coordinate Permalink: http://dlmf.nist.gov/19.26.E24 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

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

Next,

19.26.25		$$R_C(x,y)=2R_C(x+\lambda ,y+\lambda ),$$
$\lambda =y+2\sqrt{x}\sqrt{y}$.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\lambda$: positive, $x$: positive and $y$: positive Referenced by: §19.26(iii) Permalink: http://dlmf.nist.gov/19.26.E25 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

Equivalent forms are given by (19.22.22). Also,

19.26.26		$$R_C(x^2,y^2)=R_C(a^2,ay),$$
$a=(x+y)/2$, $\mathrm{\Re }x\ge 0$, $\mathrm{\Re }y>0$,
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathrm{\Re }$: real part, $x$: positive and $y$: positive Permalink: http://dlmf.nist.gov/19.26.E26 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19

and

19.26.27		$$R_C(x^2,x^2-\theta )=2R_C(s^2,s^2-\theta ),$$
$s=x+\sqrt{x^2-\theta }$, $\theta \ne x^2$ or $s^2$.
ⓘ Symbols: $R_C(x,y)$: Carlson’s combination of inverse circular and inverse hyperbolic functions and $x$: positive Permalink: http://dlmf.nist.gov/19.26.E27 Encodings: TeX, pMML, png See also: Annotations for §19.26(iii), §19.26 and Ch.19