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

§19.8 Quadratic Transformations

Keywords:: Legendre’s elliptic integrals
Permalink:: http://dlmf.nist.gov/19.8

§19.8(i) Gauss’s Arithmetic-Geometric Mean (AGM)
§19.8(ii) Landen Transformations
§19.8(iii) Gauss Transformation

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

Notes:: See Cox (1984, 1985), Borwein and Borwein (1987, Chapter 1). To prove the second equality in (19.8.4), put $\mathop{\tan\/}\nolimits\theta=\sqrt{t}/g_{0}$ . (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).
Defines:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean
Keywords:: Legendre’s elliptic integrals, arithmetic-geometric mean
Referenced by:: §19.22(ii), §19.24(i), §19.36(ii), (a)
Permalink:: http://dlmf.nist.gov/19.8.i

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

19.8.1

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

$g_{{n+1}}=\sqrt{a_{n}g_{n}}$ , $n=0,1,2,\dots$ .

Symbols:: : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E1
Encodings:: TeX, TeX, pMML, pMML, png, png

As $n\to\infty$ , $a_{n}$ and $g_{n}$ converge to a common limit $\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)$ 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:: : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E2
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Then

19.8.3 $c_{{n+1}}=\frac{a_{n}-g_{n}}{2}=\frac{c_{n}^{2}}{4a_{{n+1}}},$

Symbols:: : nonnegative integer, $a_{n}$ : iterate and $g_{n}$ : iterate
Permalink:: http://dlmf.nist.gov/19.8.E3
Encodings:: TeX, pMML, png

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

The AGM has the integral representations

19.8.4 $\frac{1}{\mathop{M\/}\nolimits\!\left(a_{0},g_{0}\right)}=\frac{2}{\pi}\int_{0% }^{{\pi/2}}\frac{d\theta}{\sqrt{a_{0}^{2}{\mathop{\cos\/}\nolimits^{{2}}}% \theta+g_{0}^{2}{\mathop{\sin\/}\nolimits^{{2}}}\theta}}=\frac{1}{\pi}\int_{0}% ^{{\infty}}\frac{dt}{\sqrt{t(t+a_{0}^{2})(t+g_{0}^{2})}}.$

Symbols:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $\mathop{\cos\/}\nolimits z$ : cosine function, : differential of , $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : 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

The first of these shows that

19.8.5 $\mathop{K\/}\nolimits\!\left(k\right)=\frac{\pi}{2\!\mathop{M\/}\nolimits\!% \left(1,k^{{\prime}}\right)},$ $-\infty<k^{2}<1$ .

Symbols:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Permalink:: http://dlmf.nist.gov/19.8.E5
Encodings:: TeX, pMML, png

The AGM appears in

19.8.6 $\mathop{E\/}\nolimits\!\left(k\right)=\frac{\pi}{2\!\mathop{M\/}\nolimits\!% \left(1,k^{{\prime}}\right)}\left(a_{0}^{2}-\sum_{{n=0}}^{{\infty}}2^{{n-1}}c_% {n}^{2}\right)=\mathop{K\/}\nolimits\!\left(k\right)\left(a_{1}^{2}-\sum_{{n=2% }}^{{\infty}}2^{{n-1}}c_{n}^{2}\right),$ $-\infty<k^{2}<1$ , $a_{0}=1$ , $g_{0}=k^{{\prime}}$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, : nonnegative integer, : 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

and in

19.8.7 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\frac{\pi}{4\!\mathop{M\/}% \nolimits\!\left(1,k^{{\prime}}\right)}\left(2+\frac{\alpha^{2}}{1-\alpha^{2}}% \sum_{{n=0}}^{{\infty}}Q_{n}\right),$ $-\infty<k^{2}<1$ , $-\infty<\alpha^{2}<1$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : nonnegative integer, : 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

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

19.8.8

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

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

$Q_{{n+1}}=\tfrac{1}{2}Q_{n}\varepsilon_{n}$ , $n=0,1,\dots$ .

Symbols:: : 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

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

19.8.9 $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)=\frac{\pi}{4\!\mathop{M\/}% \nolimits\!\left(1,k^{{\prime}}\right)}\frac{k^{2}}{k^{2}-\alpha^{2}}\sum_{{n=% 0}}^{{\infty}}Q_{n},$ $-\infty<k^{2}<1$ , $1<\alpha^{2}<\infty$ ,

Symbols:: $\mathop{M\/}\nolimits\!\left(a,g\right)$ : arithmetic-geometric mean, $\mathop{\Pi\/}\nolimits\!\left(\alpha^{2},k\right)$ : Legendre’s complete elliptic integral of the third kind, : nonnegative integer, : 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

where (19.8.8) still applies, but with

19.8.10 $p_{0}^{2}=1-(k^{2}/\alpha^{2}).$

Symbols:: : real or complex modulus and $\alpha^{2}$ : real or complex parameter
Permalink:: http://dlmf.nist.gov/19.8.E10
Encodings:: TeX, pMML, png

§19.8(ii) Landen Transformations

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

¶ Descending Landen Transformation

Keywords:: Landen transformations, Legendre’s elliptic integrals

Let

19.8.11

$k_{1}=\frac{1-k^{{\prime}}}{1+k^{{\prime}}},$

$\phi_{1}=\phi+\mathop{\mathrm{arctan}\/}\nolimits\!\left(k^{{\prime}}\mathop{% \tan\/}\nolimits\phi\right)=\mathop{\mathrm{arcsin}\/}\nolimits\!\left((1+k^{{% \prime}})\frac{\mathop{\sin\/}\nolimits\phi\mathop{\cos\/}\nolimits\phi}{\sqrt% {1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}}\right).$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\mathrm{arctan}\/}\nolimits z$ : arctangent function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\tan\/}\nolimits z$ : tangent function, $\phi$ : real or complex argument, : 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

(Note that 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

$\mathop{K\/}\nolimits\!\left(k\right)=(1+k_{1})\mathop{K\/}\nolimits\!\left(k_% {1}\right),$

$\mathop{E\/}\nolimits\!\left(k\right)=(1+k^{{\prime}})\mathop{E\/}\nolimits\!% \left(k_{1}\right)-k^{{\prime}}\mathop{K\/}\nolimits\!\left(k\right).$

Symbols:: $\mathop{K\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(k\right)$ : Legendre’s complete elliptic integral of the second kind, : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §19.36(iv), §19.5
Permalink:: http://dlmf.nist.gov/19.8.E12
Encodings:: TeX, TeX, pMML, pMML, png, png

19.8.13

$\mathop{F\/}\nolimits\!\left(\phi,k\right)=\tfrac{1}{2}(1+k_{1})\mathop{F\/}% \nolimits\!\left(\phi_{1},k_{1}\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k\right)=\tfrac{1}{2}(1+k^{{\prime}})\mathop% {E\/}\nolimits\!\left(\phi_{1},k_{1}\right)-k^{{\prime}}\mathop{F\/}\nolimits% \!\left(\phi,k\right)+\tfrac{1}{2}(1-k^{{\prime}})\mathop{\sin\/}\nolimits\phi% _{1}.$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus and $k^{{\prime}}$ : complementary modulus
Referenced by:: §19.8(ii)
Permalink:: http://dlmf.nist.gov/19.8.E13
Encodings:: TeX, TeX, pMML, pMML, png, png

19.8.14 $2(k^{2}-\alpha^{2})\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=% \frac{\omega^{2}-\alpha^{2}}{1+k^{{\prime}}}\mathop{\Pi\/}\nolimits\!\left(% \phi_{1},\alpha_{1}^{2},k_{1}\right)+k^{2}\mathop{F\/}\nolimits\!\left(\phi,k% \right)-{(1+k^{{\prime}})\alpha_{1}^{2}\mathop{R_{C}\/}\nolimits\!\left(c_{1},% c_{1}-\alpha_{1}^{2}\right)},$

Symbols:: $\mathop{R_{C}\/}\nolimits\!\left(x,y\right)$ : Carlson’s combination of inverse circular and inverse hyperbolic functions, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)$ : Legendre’s incomplete elliptic integral of the third kind, $\phi$ : real or complex argument, : real or complex modulus, $k^{{\prime}}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $\omega$
Permalink:: http://dlmf.nist.gov/19.8.E14
Encodings:: TeX, pMML, png

where

19.8.15

$\omega^{2}=\frac{k^{2}-\alpha^{2}}{1-\alpha^{2}},$

$\alpha_{1}^{2}=\frac{\alpha^{2}\omega^{2}}{(1+k^{{\prime}})^{2}},$

$c_{1}={\mathop{\csc\/}\nolimits^{{2}}}\phi_{1}.$

Symbols:: $\mathop{\csc\/}\nolimits z$ : cosecant function, $\phi$ : real or complex argument, : real or complex modulus, $k^{{\prime}}$ : complementary modulus, $\alpha^{2}$ : real or complex parameter and $\omega$
Permalink:: http://dlmf.nist.gov/19.8.E15
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

¶ Ascending Landen Transformation

Keywords:: Landen transformations, Legendre’s elliptic integrals

Let

19.8.16

$k_{2}=2\sqrt{k}/(1+k),$

$2\phi_{2}=\phi+\mathop{\mathrm{arcsin}\/}\nolimits\!\left(k\mathop{\sin\/}% \nolimits\phi\right).$

Symbols:: $\mathop{\mathrm{arcsin}\/}\nolimits z$ : arcsine function, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and : real or complex modulus
Referenced by:: §19.8(ii)
Permalink:: http://dlmf.nist.gov/19.8.E16
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(Note that and $0<\phi\leq\pi/2$ imply $k<k_{2}<1$ and $\phi_{2}<\phi$ .) Then

19.8.17

$\mathop{F\/}\nolimits\!\left(\phi,k\right)=\frac{2}{1+k}\mathop{F\/}\nolimits% \!\left(\phi_{2},k_{2}\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k\right)=(1+k)\mathop{E\/}\nolimits\!\left(% \phi_{2},k_{2}\right)+(1-k)\mathop{F\/}\nolimits\!\left(\phi_{2},k_{2}\right)-% k\mathop{\sin\/}\nolimits\phi.$

Symbols:: $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument and : real or complex modulus
Referenced by:: §19.8(ii)
Permalink:: http://dlmf.nist.gov/19.8.E17
Encodings:: TeX, TeX, pMML, pMML, png, png

§19.8(iii) Gauss Transformation

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

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

19.8.18

$k_{1}=(1-k^{{\prime}})/(1+k^{{\prime}}),$

$\mathop{\sin\/}\nolimits\psi_{1}=\frac{(1+k^{{\prime}})\mathop{\sin\/}% \nolimits\phi}{1+\Delta},$

$\Delta=\sqrt{1-k^{2}{\mathop{\sin\/}\nolimits^{{2}}}\phi}.$

Symbols:: $\mathop{\sin\/}\nolimits z$ : sine function, $\phi$ : real or complex argument, : real or complex modulus, $k^{{\prime}}$ : complementary modulus, $\Delta$ and $\psi_{j}$ : angle
Permalink:: http://dlmf.nist.gov/19.8.E18
Encodings:: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png

(Note that 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

$\mathop{F\/}\nolimits\!\left(\phi,k\right)=(1+k_{1})\mathop{F\/}\nolimits\!% \left(\psi_{1},k_{1}\right),$

$\mathop{E\/}\nolimits\!\left(\phi,k\right)=(1+k^{{\prime}})\mathop{E\/}% \nolimits\!\left(\psi_{1},k_{1}\right)-k^{{\prime}}\mathop{F\/}\nolimits\!% \left(\phi,k\right)+(1-\Delta)\mathop{\cot\/}\nolimits\phi,$

Symbols:: $\mathop{\cot\/}\nolimits z$ : cotangent function, $\mathop{F\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the first kind, $\mathop{E\/}\nolimits\!\left(\phi,k\right)$ : Legendre’s incomplete elliptic integral of the second kind, $\phi$ : real or complex argument, : real or complex modulus, $k^{{\prime}}$ : complementary modulus, $\Delta$ and $\psi_{j}$ : angle
Permalink:: http://dlmf.nist.gov/19.8.E19
Encodings:: TeX, TeX, pMML, pMML, png, png

19.8.20 $\rho\mathop{\Pi\/}\nolimits\!\left(\phi,\alpha^{2},k\right)=\frac{4}{1+k^{{% \prime}}}\mathop{\Pi\/}\nolimits\!\left(\psi_{1},\alpha^{2}_{1},k_{1}\right)+(% \rho-1)\mathop{F\/}\nolimits\!\left(\phi,k\right)-\mathop{R_{C}\/}\nolimits\!% \left(c-1,c-\alpha^{2}\right),$