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Pi: A Source Book pp 481–536Cite as

The Arithmetic-Geometric Mean of Gauss

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Abstract

The arithmetic-geometric mean of two numbers a and b is defined to be the common limit of the two sequences EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGHbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!3E85!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{a_n}} \right\}_{n = 0}^\infty $$, and EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaiWaaeaaca % WGIbWaaSbaaSqaaiaad6gaaeqaaaGccaGL7bGaayzFaaWaa0baaSqa % aiaad6gacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!3E86!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\left\{ {{b_n}} \right\}_{n = 0}^\infty $$, determined by the algorithm

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGHb % WaaSbaaSqaaiaaicdaaeqaaOGaeyypa0JaamyyaiaacYcacaaMc8Ua % amOyamaaBaaaleaacaaIWaaabeaakiabg2da9iaadkgacaGGSaaaba % GaamyyamaaBaaaleaacaWGUbGaey4kaSIaaGymaaqabaGccqGH9aqp % daqadaqaaiaadggadaWgaaWcbaGaamOBaaqabaGccqGHRaWkcaWGIb % WaaSbaaSqaaiaad6gaaeqaaaGccaGLOaGaayzkaaGaai4laiaaikda % caGGSaGaaGPaVlaadkgadaWgaaWcbaGaamOBaiabgUcaRiaaigdaae % qaaOGaeyypa0ZaaeWaaeaacaWGHbWaaSbaaSqaaiaad6gaaeqaaOGa % amOyamaaBaaaleaacaWGUbaabeaaaOGaayjkaiaawMcaamaaCaaale % qabaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGccaGGSaGaaGPaVlaa % d6gacqGH9aqpcaaIWaGaaiilaiaaigdacaGGSaGaaGOmaiaacYcacq % WIMaYsaaaa!6508!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$\begin{gathered} {a_0} = a,\,{b_0} = b, \hfill \\ {a_{n + 1}} = \left( {{a_n} + {b_n}} \right)/2,\,{b_{n + 1}} = {\left( {{a_n}{b_n}} \right)^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}},\,n = 0,1,2, \ldots \hfill \\ \end{gathered} $$
(0.1)

. Note that a 1 and b 1 are the respective arithmetic and geometric means of a and b, a 2 and b 2 the corresponding means of a 1 and b 1, etc. Thus the limit

EquationSource% MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaabm % aabaGaamyyaiaacYcacaWGIbaacaGLOaGaayzkaaGaeyypa0ZaaCbe % aeaaciGGSbGaaiyAaiaac2gaaSqaaiaad6gacqGHsgIRcqGHEisPae % qaaOGaaGPaVlaadggadaWgaaWcbaGaamOBaaqabaGccqGH9aqpdaWf % qaqaaiGacYgacaGGPbGaaiyBaaWcbaGaamOBaiabgkziUkabg6HiLc % qabaGccaaMc8UaamOyamaaBaaaleaacaWGUbaabeaaaaa!52CB!]]</EquationSource><EquationSource Format="TEX"><![CDATA[$$M\left( {a,b} \right) = \mathop {\lim }\limits_{n \to \infty } \,{a_n} = \mathop {\lim }\limits_{n \to \infty } \,{b_n}$$
(0.2)

really does deserve to be called the arithmetic-geometric mean of a and b. This algorithm first appeared in a paper of Lagrange, but it was Gauss who really discovered the amazing depth of this subject. Unfortunately, Gauss published little on the agM (his abbreviation for the arithmetic-geometric mean) during his lifetime. It was only with the publication of his collected works [12] between 1868 and 1927 that the full extent of his work became apparent. Immediately after the last volume appeared, several papers (see [15] and [35]) were written to bring this material to a wider mathematical audience. Since then, little has been done, and only the more elementary properties of the agM are widely known today.

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  1. Department of Mathematics, Amherst College, Amherst, MA, 01002, USA

    David A. Cox

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Cox, D.A. (1997). The Arithmetic-Geometric Mean of Gauss. In: Pi: A Source Book. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-2736-4_55

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