19 Elliptic Integrals Legendre’s Integrals 19.7 Connection Formulas 19.9 Inequalities

§19.8 Quadratic Transformations

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Keywords:: Legendre’s elliptic integrals, quadratic transformations
Permalink:: http://dlmf.nist.gov/19.8
See also:: Annotations for Ch.19

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)

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Defines:: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean
Keywords:: Legendre’s elliptic integrals, arithmetic-geometric mean, integral representations
Notes:: See Cox (1984, 1985), Borwein and Borwein (1987, Chapter 1). To prove the second equality in (19.8.4), put $\tan\theta=\sqrt{t}/g_{0}$ . (19.8.7) is derived from (19.22.12) and (19.25.14), and (19.8.9) is derived from (19.6.5) and (19.8.7); see also Carlson (2002).
Referenced by:: §19.22(ii), §19.24(i), §19.36(ii), item 1
Permalink:: http://dlmf.nist.gov/19.8.i
See also:: Annotations for §19.8 and Ch.19

When $a_{0}$ and $g_{0}$ are positive numbers, define

19.8.1		$\displaystyle a_{n+1}$	$\displaystyle=\frac{a_{n}+g_{n}}{2},$
		$\displaystyle g_{n+1}$	$\displaystyle=\sqrt{a_{n}g_{n}}$ ,
	$n=0,1,2,\dots$ .
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.8.E1 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(i), §19.8 and Ch.19

As $n\to\infty$ , $a_{n}$ and $g_{n}$ converge to a common limit $M\left(a_{0},g_{0}\right)$ called the AGM (Arithmetic-Geometric Mean) of $a_{0}$ and $g_{0}$ . By symmetry in $a_{0}$ and $g_{0}$ we may assume $a_{0}\geq g_{0}$ and define

19.8.2		$c_{n}=\sqrt{a_{n}^{2}-g_{n}^{2}}.$
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.8.E2 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

Then

19.8.3		$c_{n+1}=\frac{a_{n}-g_{n}}{2}=\frac{c_{n}^{2}}{4a_{n+1}},$
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.8.E3 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

showing that the convergence of $c_{n}$ to 0 and of $a_{n}$ and $g_{n}$ to $M\left(a_{0},g_{0}\right)$ is quadratic in each case.

The AGM has the integral representations

19.8.4		$\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\,% \mathrm{d}\theta}{\sqrt{a_{0}^{2}{\cos}^{2}\theta+g_{0}^{2}{\sin}^{2}\theta}}=% \frac{1}{\pi}\int_{0}^{\infty}\frac{\,\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0% }^{2})}}.$
ⓘ Symbols: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\sin\NVar{z}$ : sine function, $a_{n}$ : iterate and $g_{n}$ : iterate Referenced by: §19.8(i) Permalink: http://dlmf.nist.gov/19.8.E4 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

The first of these shows that

19.8.5		$K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},$
$-\infty<k^{2}<1$ .
ⓘ Symbols: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/19.8.E5 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

The AGM appears in

19.8.6		$E\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)}\left(a_{0}^{2}-\sum_{n% =0}^{\infty}2^{n-1}c_{n}^{2}\right)=K\left(k\right)\left(a_{1}^{2}-\sum_{n=2}^% {\infty}2^{n-1}c_{n}^{2}\right),$
$-\infty<k^{2}<1$ , $a_{0}=1$ , $g_{0}=k^{\prime}$ ,
ⓘ Symbols: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $n$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $a_{n}$ : iterate and $g_{n}$ : iterate Permalink: http://dlmf.nist.gov/19.8.E6 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

and in

19.8.7		$\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\left(2+% \frac{\alpha^{2}}{1-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n}\right),$
$-\infty<k^{2}<1$ , $-\infty<\alpha^{2}<1$ ,
ⓘ Symbols: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $n$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $Q_{n}$ Referenced by: §19.8(i) Permalink: http://dlmf.nist.gov/19.8.E7 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

where $a_{0}=1$ , $g_{0}=k^{\prime}$ , $p_{0}^{2}=1-\alpha^{2}$ , $Q_{0}=1$ , and

19.8.8		$\displaystyle p_{n+1}$	$\displaystyle=\frac{p_{n}^{2}+a_{n}g_{n}}{2p_{n}},$
		$\displaystyle\varepsilon_{n}$	$\displaystyle=\frac{p_{n}^{2}-a_{n}g_{n}}{p_{n}^{2}+a_{n}g_{n}},$
		$\displaystyle Q_{n+1}$	$\displaystyle=\tfrac{1}{2}Q_{n}\varepsilon_{n}$ ,
	$n=0,1,\dots$ .
ⓘ Symbols: $n$ : nonnegative integer, $a_{n}$ : iterate, $g_{n}$ : iterate, $Q_{n}$ and $\varepsilon_{n}$ Referenced by: §19.8(i) Permalink: http://dlmf.nist.gov/19.8.E8 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.8(i), §19.8 and Ch.19

Again, $p_{n}$ 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 $\alpha^{2}>1$ , then the Cauchy principal value is

19.8.9		$\Pi\left(\alpha^{2},k\right)=\frac{\pi}{4M\left(1,k^{\prime}\right)}\frac{k^{2% }}{k^{2}-\alpha^{2}}\sum_{n=0}^{\infty}Q_{n},$
$-\infty<k^{2}<1$ , $1<\alpha^{2}<\infty$ ,
ⓘ Symbols: $M\left(\NVar{a},\NVar{g}\right)$ : arithmetic-geometric mean, $\pi$ : the ratio of the circumference of a circle to its diameter, $\Pi\left(\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s complete elliptic integral of the third kind, $n$ : nonnegative integer, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $Q_{n}$ Referenced by: §19.8(i) Permalink: http://dlmf.nist.gov/19.8.E9 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

where (19.8.8) still applies, but with

19.8.10		$p_{0}^{2}=1-(k^{2}/\alpha^{2}).$
ⓘ Symbols: $k$ : real or complex modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.8.E10 Encodings: TeX, pMML, png See also: Annotations for §19.8(i), §19.8 and Ch.19

§19.8(ii) Landen Transformations

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Notes:: See Cazenave (1969, pp. 119–127). For (19.8.16) and (19.8.17) replace $(\phi,k)$ by $(\phi_{2},k_{2})$ , and then $(\phi_{1},k_{1})$ by $(\phi,k)$ in (19.8.11) and (19.8.13). See also Hancock (1958, pp. 74–77) for proof of (19.8.13) and (19.8.17).
Referenced by:: §19.15
Permalink:: http://dlmf.nist.gov/19.8.ii
See also:: Annotations for §19.8 and Ch.19

Descending Landen Transformation

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Keywords:: Landen transformations, Legendre’s elliptic integrals, descending
See also:: Annotations for §19.8(ii), §19.8 and Ch.19

Let

19.8.11	$\displaystyle k_{1}$	$\displaystyle=\frac{1-k^{\prime}}{1+k^{\prime}},$
19.8.11	$\displaystyle\phi_{1}$	$\displaystyle=\phi+\operatorname{arctan}\left(k^{\prime}\tan\phi\right)=% \operatorname{arcsin}\left((1+k^{\prime})\frac{\sin\phi\cos\phi}{\sqrt{1-k^{2}% {\sin}^{2}\phi}}\right).$
ⓘ Defines: $k_{1}$ : change of variable (locally) and $\phi_{1}$ : change of variable (locally) Symbols: $\cos\NVar{z}$ : cosine function, $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\sin\NVar{z}$ : sine function, $\tan\NVar{z}$ : tangent function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Referenced by: §19.22(iii), §19.8(ii) Permalink: http://dlmf.nist.gov/19.8.E11 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

(Note that $0<k<1$ and $0<\phi<\pi/2$ imply $k_{1}<k$ and $\phi<\phi_{1}<2\phi$ , and also that $\phi=\pi/2$ implies $\phi_{1}=\pi$ .) Then

19.8.12	$\displaystyle K\left(k\right)$	$\displaystyle=(1+k_{1})K\left(k_{1}\right),$
19.8.12	$\displaystyle E\left(k\right)$	$\displaystyle=(1+k^{\prime})E\left(k_{1}\right)-k^{\prime}K\left(k\right).$
ⓘ Symbols: $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $E\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the second kind, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $k_{1}$ : change of variable Referenced by: §19.36(iv), §19.5 Permalink: http://dlmf.nist.gov/19.8.E12 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

19.8.13	$\displaystyle F\left(\phi,k\right)$	$\displaystyle=\tfrac{1}{2}(1+k_{1})F\left(\phi_{1},k_{1}\right),$
19.8.13	$\displaystyle E\left(\phi,k\right)$	$\displaystyle=\tfrac{1}{2}(1+k^{\prime})E\left(\phi_{1},k_{1}\right)-k^{\prime% }F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{\prime})\sin\phi_{1}.$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $k_{1}$ : change of variable and $\phi_{1}$ : change of variable Referenced by: §19.8(ii) Permalink: http://dlmf.nist.gov/19.8.E13 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

19.8.14		$2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)=\frac{\omega^{2}-\alpha^{% 2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}F\left(% \phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}\left(c_{1},c_{1}-\alpha_{1}^{% 2}\right)},$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter, $k_{1}$ : change of variable, $\phi_{1}$ : change of variable, $\omega$ : change of variable, $\alpha_{1}$ : change of variable and $c_{1}$ : change of variable Permalink: http://dlmf.nist.gov/19.8.E14 Encodings: TeX, pMML, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

where

19.8.15	$\displaystyle\omega^{2}$	$\displaystyle=\frac{k^{2}-\alpha^{2}}{1-\alpha^{2}},$
	$\displaystyle\alpha_{1}^{2}$	$\displaystyle=\frac{\alpha^{2}\omega^{2}}{(1+k^{\prime})^{2}},$
	$\displaystyle c_{1}$	$\displaystyle={\csc}^{2}\phi_{1}.$
ⓘ Defines: $\omega$ : change of variable (locally), $\alpha_{1}$ : change of variable (locally) and $c_{1}$ : change of variable (locally) Symbols: $\csc\NVar{z}$ : cosecant function, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $\phi_{1}$ : change of variable Permalink: http://dlmf.nist.gov/19.8.E15 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

Ascending Landen Transformation

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Keywords:: Landen transformations, Legendre’s elliptic integrals, ascending
See also:: Annotations for §19.8(ii), §19.8 and Ch.19

Let

19.8.16	$\displaystyle k_{2}$	$\displaystyle=2\sqrt{k}/(1+k),$
19.8.16	$\displaystyle 2\phi_{2}$	$\displaystyle=\phi+\operatorname{arcsin}\left(k\sin\phi\right).$
ⓘ Defines: $k_{2}$ : change of variable (locally) and $\phi_{2}$ : change of variable (locally) Symbols: $\operatorname{arcsin}\NVar{z}$ : arcsine function, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument and $k$ : real or complex modulus Referenced by: §19.8(ii) Permalink: http://dlmf.nist.gov/19.8.E16 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

(Note that $0<k<1$ and $0<\phi\leq\pi/2$ imply $k<k_{2}<1$ and $\phi_{2}<\phi$ .) Then

19.8.17	$\displaystyle F\left(\phi,k\right)$	$\displaystyle=\frac{2}{1+k}F\left(\phi_{2},k_{2}\right),$
19.8.17	$\displaystyle E\left(\phi,k\right)$	$\displaystyle=(1+k)E\left(\phi_{2},k_{2}\right)+(1-k)F\left(\phi_{2},k_{2}% \right)-k\sin\phi.$
ⓘ Symbols: $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k_{2}$ : change of variable and $\phi_{2}$ : change of variable Referenced by: §19.8(ii) Permalink: http://dlmf.nist.gov/19.8.E17 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(ii), §19.8(ii), §19.8 and Ch.19

§19.8(iii) Gauss Transformation

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Keywords:: Gauss transformation, Legendre’s elliptic integrals
Notes:: See Cazenave (1969, pp. 114–119 and 126–127).
Referenced by:: §19.15
Permalink:: http://dlmf.nist.gov/19.8.iii
See also:: Annotations for §19.8 and Ch.19

We consider only the descending Gauss transformation because its (ascending) inverse moves $F\left(\phi,k\right)$ closer to the singularity at $k=\sin\phi=1$ . Let

19.8.18	$\displaystyle k_{1}$	$\displaystyle=(1-k^{\prime})/(1+k^{\prime}),$
	$\displaystyle\sin\psi_{1}$	$\displaystyle=\frac{(1+k^{\prime})\sin\phi}{1+\Delta},$
	$\displaystyle\Delta$	$\displaystyle=\sqrt{1-k^{2}{\sin}^{2}\phi}.$
ⓘ Defines: $k_{1}$ : change of variable (locally), $\psi_{1}$ : change of variable (locally) and $\Delta$ : change of variable (locally) Symbols: $\sin\NVar{z}$ : sine function, $\phi$ : real or complex argument, $k$ : real or complex modulus and $k^{\prime}$ : complementary modulus Permalink: http://dlmf.nist.gov/19.8.E18 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.8(iii), §19.8 and Ch.19

(Note that $0<k<1$ and $0<\phi<\pi/2$ imply $k_{1}<k$ and $\psi_{1}<\phi$ , and also that $\phi=\pi/2$ implies $\psi_{1}=\pi/2$ , thus preserving completeness.) Then

19.8.19	$\displaystyle F\left(\phi,k\right)$	$\displaystyle=(1+k_{1})F\left(\psi_{1},k_{1}\right),$
19.8.19	$\displaystyle E\left(\phi,k\right)$	$\displaystyle=(1+k^{\prime})E\left(\psi_{1},k_{1}\right)-k^{\prime}F\left(\phi% ,k\right)+(1-\Delta)\cot\phi,$
ⓘ Symbols: $\cot\NVar{z}$ : cotangent function, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $E\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $k_{1}$ : change of variable, $\psi_{1}$ : change of variable and $\Delta$ : change of variable Permalink: http://dlmf.nist.gov/19.8.E19 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for §19.8(iii), §19.8 and Ch.19

19.8.20		$\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{\prime}}\Pi\left(\psi_{1},% \alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k\right)-R_{C}\left(c-1,c-% \alpha^{2}\right),$
ⓘ Symbols: $R_{C}\left(\NVar{x},\NVar{y}\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $F\left(\NVar{\phi},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\Pi\left(\NVar{\phi},\NVar{\alpha}^{2},\NVar{k}\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter, $k_{1}$ : change of variable, $\psi_{1}$ : change of variable, $\rho$ : change of variable, $\alpha_{1}^{2}$ : change of variable and $c$ : change of variable Referenced by: §19.36(ii) Permalink: http://dlmf.nist.gov/19.8.E20 Encodings: TeX, pMML, png See also: Annotations for §19.8(iii), §19.8 and Ch.19

where

19.8.21	$\displaystyle\rho$	$\displaystyle=\sqrt{1-(k^{2}/\alpha^{2})},$
	$\displaystyle\alpha_{1}^{2}$	$\displaystyle=\alpha^{2}(1+\rho)^{2}/(1+k^{\prime})^{2},$
	$\displaystyle c$	$\displaystyle={\csc}^{2}\phi.$
ⓘ Defines: $\rho$ : change of variable (locally), $\alpha_{1}^{2}$ : change of variable (locally) and $c$ : change of variable (locally) Symbols: $\csc\NVar{z}$ : cosecant function, $\phi$ : real or complex argument, $k$ : real or complex modulus, $k^{\prime}$ : complementary modulus and $\alpha^{2}$ : real or complex parameter Permalink: http://dlmf.nist.gov/19.8.E21 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §19.8(iii), §19.8 and Ch.19

If $0<\alpha^{2}<k^{2}$ , then $\rho$ is pure imaginary.

19.7 Connection Formulas 19.9 Inequalities

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