19 Elliptic Integrals Symmetric Integrals 19.21 Connection Formulas 19.23 Integral Representations

§19.22 Quadratic Transformations

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Permalink:: http://dlmf.nist.gov/19.22
See also:: Annotations for Ch.19

§19.22(i) Complete Integrals
§19.22(ii) Gauss’s Arithmetic-Geometric Mean (AGM)
§19.22(iii) Incomplete Integrals

§19.22(i) Complete Integrals

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Notes:: In (19.22.18), (19.22.21), and (19.22.20), put $z=0$ to obtain (19.22.1), (19.22.2), and (19.22.4), respectively. (19.22.3) is derivable from (19.22.2) and (19.21.3), or more directly by putting $p=y$ in (19.22.7). For (19.22.7) see Carlson (1976, (4.14), (4.13)), where $(\pi/4)R_{L}(y,z,p)=R_{F}\left(0,y,z\right)-(p/3)R_{J}\left(0,y,z,p\right)$ .
Referenced by:: §19.22(ii), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.i
See also:: Annotations for §19.22 and Ch.19

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

19.22.1	$\displaystyle R_{F}\left(0,x^{2},y^{2}\right)$	$\displaystyle=R_{F}\left(0,xy,a^{2}\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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 and Ch.19
19.22.2	$\displaystyle 2R_{G}\left(0,x^{2},y^{2}\right)$	$\displaystyle=4R_{G}\left(0,xy,a^{2}\right)-xyR_{F}\left(0,xy,a^{2}\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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 and Ch.19
19.22.3	$\displaystyle 2y^{2}R_{D}\left(0,x^{2},y^{2}\right)$	$\displaystyle=\tfrac{1}{4}(y^{2}-x^{2})R_{D}\left(0,xy,a^{2}\right)+3R_{F}% \left(0,xy,a^{2}\right).$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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 and Ch.19

19.22.4		$(p_{\pm}^{2}-p_{\mp}^{2})R_{J}\left(0,x^{2},y^{2},p^{2}\right)=2(p_{\pm}^{2}-a% ^{2})R_{J}\left(0,xy,a^{2},p_{\pm}^{2}\right)-3R_{F}\left(0,xy,a^{2}\right)+3% \pi/(2p),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $\pi$ : the ratio of the circumference of a circle to its diameter and $p_{\pm}$ : modified parameter Referenced by: §19.22(i), §19.22(i) Permalink: http://dlmf.nist.gov/19.22.E4 Encodings: TeX, pMML, png See also: Annotations for §19.22(i), §19.22 and Ch.19

where

19.22.5		$2p_{\pm}=\sqrt{(p+x)(p+y)}\pm\sqrt{(p-x)(p-y)},$
ⓘ Defines: $p_{\pm}$ : modified parameter (locally) 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), §19.22 and Ch.19

and hence

19.22.6	$\displaystyle p_{+}p_{-}$	$\displaystyle=pa,$
	$\displaystyle p_{+}^{2}+p_{-}^{2}$	$\displaystyle=p^{2}+xy,$
	$\displaystyle p_{+}^{2}-p_{-}^{2}$	$\displaystyle=\sqrt{(p^{2}-x^{2})(p^{2}-y^{2})},$
	$\displaystyle 4(p_{\pm}^{2}-a^{2})$	$\displaystyle=(\sqrt{p^{2}-x^{2}}\pm\sqrt{p^{2}-y^{2}})^{2}.$
ⓘ Symbols: $p_{\pm}$ : modified parameter 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), §19.22 and Ch.19

Bartky’s Transformation

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Keywords:: Bartky’s transformation, symmetric elliptic integrals
See also:: Annotations for §19.22(i), §19.22 and Ch.19

19.22.7		$2p^{2}R_{J}\left(0,x^{2},y^{2},p^{2}\right)=v_{+}v_{-}R_{J}\left(0,xy,a^{2},v^% {2}_{+}\right)+3R_{F}\left(0,xy,a^{2}\right),$
$v_{\pm}=(p^{2}\pm xy)/(2p)$ .
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : 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), §19.22(i), §19.22 and Ch.19

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 $x<p<y$ or $y<p<x$ , then $p_{+}$ and $p_{-}$ are complex conjugates.

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

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Keywords:: arithmetic-geometric mean, symmetric elliptic integrals
Notes:: The formulas in §19.22(ii) come from iterating those in §19.22(i). See (19.16.20), (19.16.23), and Carlson (2002, Section 2).
Referenced by:: §15.17(iv), §19.22(ii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.ii
See also:: Annotations for §19.22 and Ch.19

The AGM, $M\left(a_{0},g_{0}\right)$ , 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		$\frac{2}{\pi}R_{F}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)},$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.22.E8 Encodings: TeX, pMML, png See also: Annotations for §19.22(ii), §19.22 and Ch.19

19.22.9		$\frac{4}{\pi}R_{G}\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{M\left(a_{0},g_{% 0}\right)}\left(a_{0}^{2}-\sum_{n=0}^{\infty}2^{n-1}c_{n}^{2}\right)=\frac{1}{% M\left(a_{0},g_{0}\right)}\left(a_{1}^{2}-\sum_{n=2}^{\infty}2^{n-1}c_{n}^{2}% \right),$
ⓘ Symbols: $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of second kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.22.E9 Encodings: TeX, pMML, png See also: Annotations for §19.22(ii), §19.22 and Ch.19

and

19.22.10		$R_{D}\left(0,g_{0}^{2},a_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_{0}\right)% a_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate Referenced by: §19.22(ii) Permalink: http://dlmf.nist.gov/19.22.E10 Encodings: TeX, pMML, png See also: Annotations for §19.22(ii), §19.22 and Ch.19

where

19.22.11	$\displaystyle Q_{0}$	$\displaystyle=1,$
19.22.11	$\displaystyle Q_{n+1}$	$\displaystyle=\tfrac{1}{2}Q_{n}\frac{a_{n}-g_{n}}{a_{n}+g_{n}}.$
ⓘ 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), §19.22 and Ch.19

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

19.22.12		$R_{J}\left(0,g_{0}^{2},a_{0}^{2},p_{0}^{2}\right)=\frac{3\pi}{4M\left(a_{0},g_% {0}\right)p_{0}^{2}}\sum_{n=0}^{\infty}Q_{n},$
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $p_{j}$ : coefficient, $a_{n}$ : iterate and $g_{n}$ : iterate Referenced by: §19.8(i) Permalink: http://dlmf.nist.gov/19.22.E12 Encodings: TeX, pMML, png See also: Annotations for §19.22(ii), §19.22 and Ch.19

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, $Q_{n}$ , $\varepsilon_{n}$ , $p_{j}$ : coefficient, $a_{n}$ : 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), §19.22 and Ch.19

(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 $M\left(a_{0},g_{0}\right)$ and 0, respectively, and $Q_{n}$ converges to 0 faster than quadratically. If the last variable of $R_{J}$ is negative, then the Cauchy principal value is

19.22.14		$R_{J}\left(0,g_{0}^{2},a_{0}^{2},-q_{0}^{2}\right)=\frac{-3\pi}{4M\left(a_{0},% g_{0}\right)(q_{0}^{2}+a_{0}^{2})}\*\left(2+\frac{a_{0}^{2}-g_{0}^{2}}{q_{0}^{% 2}+g_{0}^{2}}\sum_{n=0}^{\infty}Q_{n}\right),$
ⓘ Symbols: $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind, $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.22.E14 Encodings: TeX, pMML, png See also: Annotations for §19.22(ii), §19.22 and Ch.19

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: $p_{j}$ : coefficient, $a_{n}$ : 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 and Ch.19

§19.22(iii) Incomplete Integrals

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Keywords:: Gauss transformations, Landen transformations, symmetric elliptic integrals, transformations replaced by symmetry
Notes:: For (19.22.18) see Carlson (1964, (5.13)). For (19.22.19) put $p=z$ in (19.22.20). For (19.22.20) see Zill and Carlson (1970, (5.7)) and Carlson (1990, (8.5)). For (19.22.21) see Carlson (1964, (5.16)). For (19.22.22) put $z=y$ in (19.22.18). In the ascending Landen case let $k^{2}=(z_{+}^{2}-z_{-}^{2})/(z_{+}^{2}-a^{2})$ and $k_{1}^{2}=(z^{2}-y^{2})/(z^{2}-x^{2})$ to get the second equation in (19.8.11). In the descending Gauss case let $k_{1}^{2}=(a^{2}-z_{-}^{2})/(a^{2}-z_{+}^{2})$ and $k^{2}=(z^{2}-y^{2})/(z^{2}-x^{2})$ to get the first equation in (19.8.11).
Referenced by:: §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.iii
See also:: Annotations for §19.22 and Ch.19

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	$\displaystyle a$	$\displaystyle=(x+y)/2,$
19.22.16	$\displaystyle 2z_{\pm}$	$\displaystyle=\sqrt{(z+x)(z+y)}\pm\sqrt{(z-x)(z-y)},$
ⓘ 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), §19.22 and Ch.19

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), §19.22 and Ch.19

Then

19.22.18		$R_{F}\left(x^{2},y^{2},z^{2}\right)=R_{F}\left(a^{2},z_{-}^{2},z_{+}^{2}\right),$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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 and Ch.19

19.22.19		$(z_{\pm}^{2}-z_{\mp}^{2})R_{D}\left(x^{2},y^{2},z^{2}\right)={2(z_{\pm}^{2}-a^% {2})}R_{D}\left(a^{2},z_{\mp}^{2},z_{\pm}^{2}\right)-3R_{F}\left(x^{2},y^{2},z% ^{2}\right)+(3/z),$
ⓘ Symbols: $R_{D}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : elliptic integral symmetric in only two variables and $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind Referenced by: §19.22(iii), §19.22(iii), §19.36(ii) Permalink: http://dlmf.nist.gov/19.22.E19 Encodings: TeX, pMML, png See also: Annotations for §19.22(iii), §19.22 and Ch.19

19.22.20		$(p_{\pm}^{2}-p_{\mp}^{2})R_{J}\left(x^{2},y^{2},z^{2},p^{2}\right)=2(p_{\pm}^{% 2}-a^{2})R_{J}\left(a^{2},z_{+}^{2},z_{-}^{2},p_{\pm}^{2}\right)-3R_{F}\left(x% ^{2},y^{2},z^{2}\right)+3R_{C}\left(z^{2},p^{2}\right),$
ⓘ 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, $R_{J}\left(\NVar{x},\NVar{y},\NVar{z},\NVar{p}\right)$ : symmetric elliptic integral of third kind and $p_{\pm}$ : modified parameter Referenced by: §19.22(i), §19.22(iii), §19.22(iii), §19.36(ii) Permalink: http://dlmf.nist.gov/19.22.E20 Encodings: TeX, pMML, png See also: Annotations for §19.22(iii), §19.22 and Ch.19

19.22.21		$2R_{G}\left(x^{2},y^{2},z^{2}\right)=4R_{G}\left(a^{2},z_{+}^{2},z_{-}^{2}% \right)-xyR_{F}\left(x^{2},y^{2},z^{2}\right)-z,$
ⓘ Symbols: $R_{F}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : symmetric elliptic integral of first kind and $R_{G}\left(\NVar{x},\NVar{y},\NVar{z}\right)$ : 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 and Ch.19

19.22.22		$R_{C}\left(x^{2},y^{2}\right)=R_{C}\left(a^{2},ay\right).$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : 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), §19.22 and Ch.19

If $x, y, z$ are real and positive, then (19.22.18)–(19.22.21) are ascending Landen transformations when $x,y<z$ (implying $a<z_{-}<z_{+}$ ), and descending Gauss transformations when $z<x,y$ (implying $z_{+}<z_{-}<a$ ). 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 $x<z<y$ or $y<z<x$ , 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), §19.22 and Ch.19

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 $0<a<z_{+}z_{-}\left(z_{+}^{2}+z_{-}^{2}\right)^{-1/2}$ .

§19.22 Quadratic Transformations

Contents

§19.22(i) Complete Integrals

Bartky’s Transformation

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

§19.22(iii) Incomplete Integrals