§19.22 Quadratic Transformations

Permalink:: http://dlmf.nist.gov/19.22

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

§19.22(i) Complete Integrals

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 $(\pi/4)R_{{L}}(y,z,p)=\mathop{R_{F}\/}\nolimits\!\left(0,y,z\right)-(p/3)\mathop{R_{J}\/}\nolimits\!\left(0,y,z,p\right)$ .
Referenced by:: §19.22(ii), §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.i

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

19.22.1 $\mathop{R_{F}\/}\nolimits\!\left(0,x^{2},y^{2}\right)=\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.22.2 $2\!\mathop{R_{G}\/}\nolimits\!\left(0,x^{2},y^{2}\right)=4\!\mathop{R_{G}\/}\nolimits\!\left(0,xy,a^{2}\right)-xy\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind
Referenced by:: §19.22(i)
Permalink:: http://dlmf.nist.gov/19.22.E2
Encodings:: TeX, pMML, png

19.22.3 $2y^{2}\mathop{R_{D}\/}\nolimits\!\left(0,x^{2},y^{2}\right)=\tfrac{1}{4}(y^{2}-x^{2})\mathop{R_{D}\/}\nolimits\!\left(0,xy,a^{2}\right)+3\!\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right).$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.22.4 $(p_{{\pm}}^{2}-p_{{\mp}}^{2})\mathop{R_{J}\/}\nolimits\!\left(0,x^{2},y^{2},p^{2}\right)=2(p_{{\pm}}^{2}-a^{2})\mathop{R_{J}\/}\nolimits\!\left(0,xy,a^{2},p_{{\pm}}^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(0,xy,a^{2}\right)+3\pi/(2p),$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
Referenced by:: ¶ ‣ §19.22(i), §19.22(i)
Permalink:: http://dlmf.nist.gov/19.22.E4
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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
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and hence

19.22.6

$p_{{+}}p_{{-}}=pa,$

$p_{{+}}^{2}+p_{{-}}^{2}=p^{2}+xy,$

$p_{{+}}^{2}-p_{{-}}^{2}=\sqrt{(p^{2}-x^{2})(p^{2}-y^{2})},$

$4(p_{{\pm}}^{2}-a^{2})=(\sqrt{p^{2}-x^{2}}\pm\sqrt{p^{2}-y^{2}})^{2}.$

Referenced by:: §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.E6
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¶ Bartky’s Transformation

Keywords:: Bartky’s transformation, symmetric elliptic integrals

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:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : 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

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)

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).
Keywords:: arithmetic-geometric mean, symmetric elliptic integrals
Referenced by:: §15.17(iv), §19.22(ii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.ii

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 $\frac{2}{\pi}\mathop{R_{F}\/}\nolimits\!\left(0,a_{0}^{2},g_{0}^{2}\right)=\frac{1}{\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)},$

Symbols:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind, $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E8
Encodings:: TeX, pMML, png

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

Symbols:: $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of second kind, $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E9
Encodings:: TeX, pMML, png

and

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

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables, $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.22(ii)
Permalink:: http://dlmf.nist.gov/19.22.E10
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where

19.22.11

$Q_{0}=1,$

$Q_{{n+1}}=\tfrac{1}{2}Q_{n}\frac{a_{n}-g_{n}}{a_{n}+g_{n}}.$

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

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

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.8(i)
Permalink:: http://dlmf.nist.gov/19.22.E12
Encodings:: TeX, pMML, png

where $p_{0}>0$ and

19.22.13

$p_{{n+1}}=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}},$

$\varepsilon _{n}=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}},$

$Q_{0}=1,$

$Q_{{n+1}}=\tfrac{1}{2}Q_{n}\varepsilon _{n}.$

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

Symbols:: $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind, $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $n$ : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E14
Encodings:: TeX, pMML, png

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

§19.22(iii) Incomplete Integrals

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).
Keywords:: Landen transformations, symmetric elliptic integrals
Referenced by:: §19.22(iii)
Permalink:: http://dlmf.nist.gov/19.22.iii

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

$a=(x+y)/2,$

$2z_{{\pm}}=\sqrt{(z+x)(z+y)}\pm\sqrt{(z-x)(z-y)},$

Referenced by:: §19.22(iii), §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.E16
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so that

19.22.17

$z_{{+}}z_{{-}}=za,$

$z_{{+}}^{2}+z_{{-}}^{2}=z^{2}+xy,$

$z_{{+}}^{2}-z_{{-}}^{2}=\sqrt{(z^{2}-x^{2})(z^{2}-y^{2})},$

$4(z_{{\pm}}^{2}-a^{2})=(\sqrt{z^{2}-x^{2}}\pm\sqrt{z^{2}-y^{2}})^{2}.$

Referenced by:: §19.36(ii)
Permalink:: http://dlmf.nist.gov/19.22.E17
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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:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.22.19 $(z_{{\pm}}^{2}-z_{{\mp}}^{2})\mathop{R_{D}\/}\nolimits\!\left(x^{2},y^{2},z^{2}\right)={2(z_{{\pm}}^{2}-a^{2})}\mathop{R_{D}\/}\nolimits\!\left(a^{2},z_{{\mp}}^{2},z_{{\pm}}^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(x^{2},y^{2},z^{2}\right)+(3/z),$

Symbols:: $\mathop{R_{D}\/}\nolimits\!\left(x,y,z\right)$ : elliptic integral symmetric in only two variables and $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.22.20 $(p_{{\pm}}^{2}-p_{{\mp}}^{2})\mathop{R_{J}\/}\nolimits\!\left(x^{2},y^{2},z^{2},p^{2}\right)=2(p_{{\pm}}^{2}-a^{2})\mathop{R_{J}\/}\nolimits\!\left(a^{2},z_{{+}}^{2},z_{{-}}^{2},p_{{\pm}}^{2}\right)-3\!\mathop{R_{F}\/}\nolimits\!\left(x^{2},y^{2},z^{2}\right)+3\!\mathop{R_{C}\/}\nolimits\!\left(z^{2},p^{2}\right),$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{J}\/}\nolimits\!\left(x,y,z,p\right)$ : symmetric elliptic integral of third kind
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

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:: $\mathop{R_{F}\/}\nolimits\!\left(x,y,z\right)$ : symmetric elliptic integral of first kind and $\mathop{R_{G}\/}\nolimits\!\left(x,y,z\right)$ : 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

19.22.22 $\mathop{R_{C}\/}\nolimits\!\left(x^{2},y^{2}\right)=\mathop{R_{C}\/}\nolimits\!\left(a^{2},ay\right).$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : 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

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

$x+y=2a,$

$x-y=(\ifrac{2}{a})\sqrt{(a^{2}-z_{{+}}^{2})(a^{2}-z_{{-}}^{2})},$

$z=\ifrac{z_{{+}}z_{{-}}}{a},$

Permalink:: http://dlmf.nist.gov/19.22.E23
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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