19 Elliptic Integrals Symmetric Integrals 19.21 Connection Formulas 19.23 Integral Representations

§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 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 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 $\realpart{x}>0$ , $\realpart{y}>0$ , , 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
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If , then (19.22.7) reduces to (19.22.3), but if or , then both sides of (19.22.4) are 0 by (19.20.9). If or , 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, : 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, : 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:: : 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, : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Referenced by:: §19.8(i)
Permalink:: http://dlmf.nist.gov/19.22.E12
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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:: : 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, : nonnegative integer, $Q_{n}$ , $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.22.E14
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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 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 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 , , and have positive real parts, assume $p\neq 0$ , and retain (19.22.5) and (19.22.6). Define

19.22.16

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

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

If or , then (19.22.20) reduces to by (19.20.13), and if or then (19.22.19) reduces to by (19.20.20) and (19.22.22). If or , then $z_{{+}}$ and $z_{{-}}$ are complex conjugates. However, if and are complex conjugates and and 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

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

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

Permalink:: http://dlmf.nist.gov/19.22.E23
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and the corresponding equations with , $z_{{+}}$ , and $z_{{-}}$ replaced by , $p_{{+}}$ , and $p_{{-}}$ , respectively. These relations need to be used with caution because is negative when $0<a<z_{{+}}z_{{-}}\left(z_{{+}}^{2}+z_{{-}}^{2}\right)^{{-1/2}}$ .

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 19.21 Connection Formulas 19.23 Integral Representations

§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