Editing Arithmetic–geometric mean (section)

== Related concepts ==
The reciprocal of the arithmetic–geometric mean of 1 and the [[square root of 2]] is [[Gauss's constant]].<math display=block>\frac{1}{M(1, \sqrt{2})} = G = 0.8346268\dots</math>In 1799, Gauss proved<ref group="note">By 1799, Gauss had two proofs of the theorem, but neither of them was rigorous from the modern point of view.</ref> that<math display="block">M(1,\sqrt{2})=\frac{\pi}{\varpi}</math>where <math>\varpi</math> is the [[lemniscate constant]].


In 1941, <math>M(1,\sqrt{2})</math> (and hence <math>G</math>) was proved [[Transcendental number|transcendental]] by [[Theodor Schneider]].<ref group="note">In particular, he proved that the [[beta function]] <math>\Beta (a,b)</math> is transcendental for all <math>a,b\in\mathbb{Q}\setminus\mathbb{Z}</math> such that <math>a+b\notin \mathbb{Z}_0^-</math>. The fact that <math>M(1,\sqrt{2})</math> is transcendental follows from <math>M(1,\sqrt{2})=\tfrac{1}{2}\Beta \left(\tfrac{1}{2},\tfrac{3}{4}\right).</math></ref><ref>{{cite journal |first=Theodor |last=Schneider |url=https://www.deepdyve.com/lp/de-gruyter/zur-theorie-der-abelschen-funktionen-und-integrale-mn0U50bvkB |title=Zur Theorie der Abelschen Funktionen und Integrale |year=1941 |journal=Journal für die reine und angewandte Mathematik |volume=183 |number=19 |pages=110–128 |doi=10.1515/crll.1941.183.110 |s2cid=118624331 }}</ref><ref>{{Cite journal |title=The Lemniscate Constants |last=Todd |first=John |journal=[[Communications of the ACM]] |volume=18 |number=1 <!-- |month=January -->|year=1975 |pages=14–19 |doi=10.1145/360569.360580 |s2cid=85873 |doi-access=free }}</ref> The set <math>\{\pi,M(1,1/\sqrt{2})\}</math> is [[Algebraic independence|algebraically independent]] over <math>\mathbb{Q}</math>,<ref>G. V. Choodnovsky: ''Algebraic independence of constants connected with the functions of analysis'', Notices of the AMS 22, 1975, p. A-486</ref><ref>G. V. Chudnovsky: ''Contributions to The Theory of Transcendental Numbers'', American Mathematical Society, 1984, p. 6</ref> but the set <math>\{\pi,M(1,1/\sqrt{2}),M'(1,1/\sqrt{2})\}</math> (where the prime denotes the [[derivative]] with respect to the second variable) is not algebraically independent over <math>\mathbb{Q}</math>. In fact,<ref>{{Cite book |last1=Borwein |first1=Jonathan M. |last2=Borwein| first2=Peter B. |title=Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity |publisher=Wiley-Interscience |year=1987 |edition=First |isbn=0-471-83138-7}} p. 45</ref><math display="block">\pi=2\sqrt{2}\frac{M^3(1,1/\sqrt{2})}{M'(1,1/\sqrt{2})}.</math>The [[geometric–harmonic mean]] GH can be calculated using analogous sequences of geometric and [[harmonic mean|harmonic]] means, and in fact {{math|1= GH(''x'', ''y'') = 1/''M''(1/''x'', 1/''y'') = ''xy''/''M''(''x'', ''y'')}}.<ref>{{cite journal
 | last = Newman | first = D. J.
 | doi = 10.2307/2007804
 | journal = Mathematics of Computation
 | pages = 207–210
 | title = A simplified version of the fast algorithms of Brent and Salamin
 | volume = 44
 | year = 1985| issue = 169
 | jstor = 2007804
 }}</ref>
The arithmetic–harmonic mean [[Geometric mean#Iterative means|is equivalent to]] the [[geometric mean]].

The arithmetic–geometric mean can be used to compute – among others – [[Logarithm#Arithmetic–geometric mean approximation|logarithms]],  [[Elliptic integral|complete and incomplete elliptic integrals of the first and second kind]],<ref>{{AS ref|17|598–599}}</ref> and [[Jacobi elliptic functions]].<ref>{{cite book |first=Louis V. |last=King |author-link=Louis Vessot King |url=https://archive.org/details/onthenumerical032686mbp |title=On the Direct Numerical Calculation of Elliptic Functions and Integrals |publisher=Cambridge University Press |year=1924 }}</ref>