## §19.22(i) Complete Integrals

Let $\Re{x}>0$, $\Re{y}>0$, $a=(x+y)/2$, and $p\neq 0$. Then

 19.22.1 $\displaystyle\mathop{R_{F}\/}\nolimits\!\left(0,x^{2},y^{2}\right)$ $\displaystyle=\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right),$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.22(i), §19.28 Permalink: http://dlmf.nist.gov/19.22.E1 Encodings: TeX, pMML, png See also: Annotations for 19.22(i) 19.22.2 $\displaystyle 2\!\mathop{R_{G}\/}\nolimits\!\left(0,x^{2},y^{2}\right)$ $\displaystyle=4\!\mathop{R_{G}\/}\nolimits\!\left(0,xy,a^{2}\right)-xy\mathop{% R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right),$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Referenced by: §19.22(i) Permalink: http://dlmf.nist.gov/19.22.E2 Encodings: TeX, pMML, png See also: Annotations for 19.22(i) 19.22.3 $\displaystyle 2y^{2}\mathop{R_{D}\/}\nolimits\!\left(0,x^{2},y^{2}\right)$ $\displaystyle=\tfrac{1}{4}(y^{2}-x^{2})\mathop{R_{D}\/}\nolimits\!\left(0,xy,a% ^{2}\right)+3\!\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right).$ Symbols: $\mathop{R_{D}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.22(i), §19.22(i), §19.22(iii) Permalink: http://dlmf.nist.gov/19.22.E3 Encodings: TeX, pMML, png See also: Annotations for 19.22(i)
 19.22.4 $(p_{\pm}^{2}-p_{\mp}^{2})\mathop{R_{J}\/}\nolimits\!\left(0,x^{2},y^{2},p^{2}% \right)=2(p_{\pm}^{2}-a^{2})\mathop{R_{J}\/}\nolimits\!\left(0,xy,a^{2},p_{\pm% }^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right)+3\pi/(2p),$

where

 19.22.5 $2p_{\pm}=\sqrt{(p+x)(p+y)}\pm\sqrt{(p-x)(p-y)},$ Referenced by: §19.22(iii), §19.22(iii) Permalink: http://dlmf.nist.gov/19.22.E5 Encodings: TeX, pMML, png See also: Annotations for 19.22(i)

and hence

 19.22.6 $\displaystyle p_{+}p_{-}$ $\displaystyle=pa,$ $\displaystyle p_{+}^{2}+p_{-}^{2}$ $\displaystyle=p^{2}+xy,$ $\displaystyle p_{+}^{2}-p_{-}^{2}$ $\displaystyle=\sqrt{(p^{2}-x^{2})(p^{2}-y^{2})},$ $\displaystyle 4(p_{\pm}^{2}-a^{2})$ $\displaystyle=(\sqrt{p^{2}-x^{2}}\pm\sqrt{p^{2}-y^{2}})^{2}.$ Referenced by: §19.22(iii) Permalink: http://dlmf.nist.gov/19.22.E6 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 19.22(i)

### Bartky’s Transformation

 19.22.7 $2p^{2}\mathop{R_{J}\/}\nolimits\!\left(0,x^{2},y^{2},p^{2}\right)=v_{+}v_{-}% \mathop{R_{J}\/}\nolimits\!\left(0,xy,a^{2},v^{2}_{+}\right)+3\!\mathop{R_{F}% \/}\nolimits\!\left(0,xy,a^{2}\right),$ $v_{\pm}=(p^{2}\pm xy)/(2p)$. Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$: symmetric elliptic integral of third kind Referenced by: §19.22(i), §19.22(i) Permalink: http://dlmf.nist.gov/19.22.E7 Encodings: TeX, pMML, png See also: Annotations for 19.22(i)

If $p=y$, then (19.22.7) reduces to (19.22.3), but if $p=x$ or $p=y$, then both sides of (19.22.4) are 0 by (19.20.9). If $x or $y, then $p_{+}$ and $p_{-}$ are complex conjugates.

## §19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)

The AGM, $\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)$, of two positive numbers $a_{0}$ and $g_{0}$ is defined in §19.8(i). Again, we assume that $a_{0}\geq g_{0}$ (except in (19.22.10)), and define $c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}$. Then

 19.22.8 $\frac{2}{\pi}\mathop{R_{F}\/}\nolimits\!\left(0,a_{0}^{2},g_{0}^{2}\right)=% \frac{1}{\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)},$
 19.22.9 $\frac{4}{\pi}\mathop{R_{G}\/}\nolimits\!\left(0,a_{0}^{2},g_{0}^{2}\right)=% \frac{1}{\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)}\left(a_{0}^{2}-\sum_% {n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{\mathop{M\/}\nolimits\!\left(a_% {0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}\right),$

and

 19.22.10 $\mathop{R_{D}\/}\nolimits\!\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4\!% \mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)a_{0}^{2}}\sum_{n=0}^{\infty}Q_% {n},$

where

 19.22.11 $\displaystyle Q_{0}$ $\displaystyle=1,$ $\displaystyle Q_{n+1}$ $\displaystyle=\tfrac{1}{2}Q_{n}\frac{a_{n}-g_{n}}{a_{n}+g_{n}}.$ Symbols: $n$: nonnegative integer, $Q_{n}$, $a_{n}$: iterate and $g_{n}$: iterate Referenced by: §19.22(ii) Permalink: http://dlmf.nist.gov/19.22.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 19.22(ii)

$Q_{n}$ has the same sign as $a_{0}-g_{0}$ for $n\geq 1$.

 19.22.12 $\mathop{R_{J}\/}\nolimits\!\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)=\frac{% 3\pi}{4\!\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)p_{0}^{2}}\sum_{n=0}^{% \infty}Q_{n},$

where $p_{0}>0$ and

 19.22.13 $\displaystyle p_{n+1}$ $\displaystyle=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}},$ $\displaystyle\varepsilon_{n}$ $\displaystyle=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}},$ $\displaystyle Q_{0}$ $\displaystyle=1,$ $\displaystyle Q_{n+1}$ $\displaystyle=\tfrac{1}{2}Q_{n}\varepsilon_{n}.$ Symbols: $n$: nonnegative integer, $Q_{n}$, $\varepsilon_{n}$, $a_{n}$: iterate and $g_{n}$: iterate Referenced by: §19.22(ii) Permalink: http://dlmf.nist.gov/19.22.E13 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 19.22(ii)

(If $p_{0}=a_{0}$, then $p_{n}=a_{n}$ and (19.22.13) reduces to (19.22.11).) As $n\to\infty$, $p_{n}$ and $\varepsilon_{n}$ converge quadratically to $\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)$ and 0, respectively, and $Q_{n}$ converges to 0 faster than quadratically. If the last variable of $\mathop{R_{J}\/}\nolimits$ is negative, then the Cauchy principal value is

 19.22.14 $\mathop{R_{J}\/}\nolimits\!\left(0,g_{0}^{2},a_{0}^{2},-q_{0}^{2}\right)=\frac% {-3\pi}{4\!\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)(q_{0}^{2}+a_{0}^{2}% )}\*\left(2+\frac{a_{0}^{2}-g_{0}^{2}}{q_{0}^{2}+g_{0}^{2}}\sum_{n=0}^{\infty}% Q_{n}\right),$

and (19.22.13) still applies, provided that

 19.22.15 $p_{0}^{2}=a_{0}^{2}(q_{0}^{2}+g_{0}^{2})/(q_{0}^{2}+a_{0}^{2}).$ Symbols: $a_{n}$: iterate and $g_{n}$: iterate Permalink: http://dlmf.nist.gov/19.22.E15 Encodings: TeX, pMML, png See also: Annotations for 19.22(ii)

## §19.22(iii) Incomplete Integrals

Let $x$, $y$, and $z$ have positive real parts, assume $p\neq 0$, and retain (19.22.5) and (19.22.6). Define

 19.22.16 $\displaystyle a$ $\displaystyle=(x+y)/2,$ $\displaystyle 2z_{\pm}$ $\displaystyle=\sqrt{(z+x)(z+y)}\pm\sqrt{(z-x)(z-y)},$ Referenced by: §19.22(iii), §19.36(ii) Permalink: http://dlmf.nist.gov/19.22.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 19.22(iii)

so that

 19.22.17 $\displaystyle z_{+}z_{-}$ $\displaystyle=za,$ $\displaystyle z_{+}^{2}+z_{-}^{2}$ $\displaystyle=z^{2}+xy,$ $\displaystyle z_{+}^{2}-z_{-}^{2}$ $\displaystyle=\sqrt{(z^{2}-x^{2})(z^{2}-y^{2})},$ $\displaystyle 4(z_{\pm}^{2}-a^{2})$ $\displaystyle=(\sqrt{z^{2}-x^{2}}\pm\sqrt{z^{2}-y^{2}})^{2}.$ Referenced by: §19.36(ii) Permalink: http://dlmf.nist.gov/19.22.E17 Encodings: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png See also: Annotations for 19.22(iii)

Then

 19.22.18 $\mathop{R_{F}\/}\nolimits\!\left(x^{2},y^{2},z^{2}\right)=\mathop{R_{F}\/}% \nolimits\!\left(a^{2},z_{-}^{2},z_{+}^{2}\right),$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind Referenced by: §19.22(i), §19.22(iii), §19.22(iii), §19.29(iii), §19.36(ii), §19.36(ii) Permalink: http://dlmf.nist.gov/19.22.E18 Encodings: TeX, pMML, png See also: Annotations for 19.22(iii)
 19.22.19 $(z_{\pm}^{2}-z_{\mp}^{2})\mathop{R_{D}\/}\nolimits\!\left(x^{2},y^{2},z^{2}% \right)={2(z_{\pm}^{2}-a^{2})}\mathop{R_{D}\/}\nolimits\!\left(a^{2},z_{\mp}^{% 2},z_{\pm}^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(x^{2},y^{2},z^{2}% \right)+(3/z),$
 19.22.20 $(p_{\pm}^{2}-p_{\mp}^{2})\mathop{R_{J}\/}\nolimits\!\left(x^{2},y^{2},z^{2},p^% {2}\right)=2(p_{\pm}^{2}-a^{2})\mathop{R_{J}\/}\nolimits\!\left(a^{2},z_{+}^{2% },z_{-}^{2},p_{\pm}^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(x^{2},y^{2}% ,z^{2}\right)+3\!\mathop{R_{C}\/}\nolimits\!\left(z^{2},p^{2}\right),$
 19.22.21 $2\!\mathop{R_{G}\/}\nolimits\!\left(x^{2},y^{2},z^{2}\right)=4\!\mathop{R_{G}% \/}\nolimits\!\left(a^{2},z_{+}^{2},z_{-}^{2}\right)-xy\mathop{R_{F}\/}% \nolimits\!\left(x^{2},y^{2},z^{2}\right)-z,$ Symbols: $\mathop{R_{F}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(\NVar{x},\NVar{y},\NVar{z}\right)$: symmetric elliptic integral of second kind Referenced by: §19.22(i), §19.22(iii), §19.22(iii) Permalink: http://dlmf.nist.gov/19.22.E21 Encodings: TeX, pMML, png See also: Annotations for 19.22(iii)
 19.22.22 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}\right)=\mathop{R_{C}\/}\nolimits% \!\left(a^{2},ay\right).$ Symbols: $\mathop{R_{C}\/}\nolimits\!\left(\NVar{x},\NVar{y}\right)$: Carlson’s combination of inverse circular and inverse hyperbolic functions Referenced by: §19.15, §19.22(iii), §19.22(iii), §19.26(iii) Permalink: http://dlmf.nist.gov/19.22.E22 Encodings: TeX, pMML, png See also: Annotations for 19.22(iii)

If $x,y,z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when $x,y (implying $a), and descending Gauss transformations when $z (implying $z_{+}). Ascent and descent correspond respectively to increase and decrease of $k$ in Legendre’s notation. Descending Gauss transformations include, as special cases, transformations of complete integrals into complete integrals; ascending Landen transformations do not.

If $p=x$ or $p=y$, then (19.22.20) reduces to $0=0$ by (19.20.13), and if $z=x$ or $z=y$ then (19.22.19) reduces to $0=0$ by (19.20.20) and (19.22.22). If $x or $y, then $z_{+}$ and $z_{-}$ are complex conjugates. However, if $x$ and $y$ are complex conjugates and $z$ and $p$ are real, then the right-hand sides of all transformations in §§19.22(i) and 19.22(iii)—except (19.22.3) and (19.22.22)—are free of complex numbers and $p_{\pm}^{2}-p_{\mp}^{2}=\pm|p^{2}-x^{2}|\neq 0$.

The transformations inverse to the ones just described are the descending Landen transformations and the ascending Gauss transformations. The equations inverse to (19.22.5) and (19.22.16) are given by

 19.22.23 $\displaystyle x+y$ $\displaystyle=2a,$ $\displaystyle x-y$ $\displaystyle=(\ifrac{2}{a})\sqrt{(a^{2}-z_{+}^{2})(a^{2}-z_{-}^{2})},$ $\displaystyle z$ $\displaystyle=\ifrac{z_{+}z_{-}}{a},$ Permalink: http://dlmf.nist.gov/19.22.E23 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for 19.22(iii)

and the corresponding equations with $z$, $z_{+}$, and $z_{-}$ replaced by $p$, $p_{+}$, and $p_{-}$, respectively. These relations need to be used with caution because $y$ is negative when $0.