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

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 $\frac{1}{M\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0}^{\pi/2}\frac{\mathrm% {d}\theta}{\sqrt{a_{0}^{2}{\cos}^{2}\theta+g_{0}^{2}{\sin}^{2}\theta}}=\frac{1% }{\pi}\int_{0}^{\infty}\frac{\mathrm{d}t}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}.$

The first of these shows that

 19.8.5 $K\left(k\right)=\frac{\pi}{2M\left(1,k^{\prime}\right)},$ $-\infty.

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, $a_{0}=1$, $g_{0}=k^{\prime}$,

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, $-\infty<\alpha^{2}<1$,

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, $1<\alpha^{2}<\infty$,

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

Descending Landen Transformation

Let

 19.8.11 $\displaystyle k_{1}$ $\displaystyle=\frac{1-k^{\prime}}{1+k^{\prime}},$ $\displaystyle\phi_{1}$ $\displaystyle=\phi+\operatorname{arctan}\left(k^{\prime}\tan\phi\right)=% \operatorname{arcsin}\left((1+k^{\prime})\frac{\sin\phi\cos\phi}{\sqrt{1-k^{2}% {\sin}^{2}\phi}}\right).$ ⓘ 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 and $0<\phi<\pi/2$ imply $k_{1} and $\phi<\phi_{1}<2\phi$, and also that $\phi=\pi/2$ implies $\phi_{1}=\pi$.) Then

 19.8.12 $\displaystyle K\left(k\right)$ $\displaystyle=(1+k_{1})K\left(k_{1}\right),$ $\displaystyle E\left(k\right)$ $\displaystyle=(1+k^{\prime})E\left(k_{1}\right)-k^{\prime}K\left(k\right).$
 19.8.13 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=\tfrac{1}{2}(1+k_{1})F\left(\phi_{1},k_{1}\right),$ $\displaystyle E\left(\phi,k\right)$ $\displaystyle=\tfrac{1}{2}(1+k^{\prime})E\left(\phi_{1},k_{1}\right)-k^{\prime% }F\left(\phi,k\right)+\tfrac{1}{2}(1-k^{\prime})\sin\phi_{1}.$
 19.8.14 $2(k^{2}-\alpha^{2})\Pi\left(\phi,\alpha^{2},k\right)=\frac{\omega^{2}-\alpha^{% 2}}{1+k^{\prime}}\Pi\left(\phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}F\left(% \phi,k\right)-{(1+k^{\prime})\alpha_{1}^{2}R_{C}\left(c_{1},c_{1}-\alpha_{1}^{% 2}\right)},$

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

Let

 19.8.16 $\displaystyle k_{2}$ $\displaystyle=2\sqrt{k}/(1+k),$ $\displaystyle 2\phi_{2}$ $\displaystyle=\phi+\operatorname{arcsin}\left(k\sin\phi\right).$ ⓘ 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 and $0<\phi\leq\pi/2$ imply $k and $\phi_{2}<\phi$.) Then

 19.8.17 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=\frac{2}{1+k}F\left(\phi_{2},k_{2}\right),$ $\displaystyle E\left(\phi,k\right)$ $\displaystyle=(1+k)E\left(\phi_{2},k_{2}\right)+(1-k)F\left(\phi_{2},k_{2}% \right)-k\sin\phi.$

§19.8(iii) Gauss Transformation

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

 19.8.19 $\displaystyle F\left(\phi,k\right)$ $\displaystyle=(1+k_{1})F\left(\psi_{1},k_{1}\right),$ $\displaystyle E\left(\phi,k\right)$ $\displaystyle=(1+k^{\prime})E\left(\psi_{1},k_{1}\right)-k^{\prime}F\left(\phi% ,k\right)+(1-\Delta)\cot\phi,$
 19.8.20 $\rho\Pi\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{\prime}}\Pi\left(\psi_{1},% \alpha_{1}^{2},k_{1}\right)+(\rho-1)F\left(\phi,k\right)-R_{C}\left(c-1,c-% \alpha^{2}\right),$

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}, then $\rho$ is pure imaginary.