The arithmetic-geometric mean of Gauss. (English) Zbl 0583.33002

Cox, David A.

Enseign. Math., II. Sér. 30, 275-330 (1984).

This paper is an expository account of the arithmetic-geometric mean M(a,b) of two numbers a,b. For \(a,b>0\) define \(a_ 0=a\), \(b_ 0=b\) and \(a_{n+1}=(a_ n+b_ n)/2,\quad b_{n+1}=(a_ nb_ n)^{1/2},\quad n=0,1,2,\ldots.\) It follows by elementary methods that the two sequences \(a_ n\), \(b_ n\) have a common limit M(a,b). M(a,b) is connected with elliptic integrals by the formula \[ (1/M(1,\sqrt{1-k^ 2}))=(2/\pi)\int^{\pi /2}_{0}(1-k^ 2 \sin^ 2 \gamma)^{-1/2} d\gamma. \] One also has \(M(\sqrt{2,1})=\pi /\tilde w\) where \(\tilde w=2\int^{1}_{0}(1-z^ 4)^{-1/2} dz\) is half the arc length of the limniscate \(r^ 2=\cos 2\theta\). The genesis of the entire subject lay in Gauß’ discovery of this equation.
If a,b take complex values the mathematics becomes much deeper. There is a choice involved in the square root: call a choice of \(b_ 1=\sqrt{a_ 0b_ 0}\) right if \(| a_ 1-b_ 1| \leq | a_ 1+b_ 1|\) and if in addition Im (b\({}_ 1/a_ 1)>0\) when equality holds. Then any pair of sequences \(a_ n\), \(b_ n\) converge to a common limit which is non-zero only if the sequence contains at most finitely many wrong choices. The value of M(a,b) obtained when all choices are right is called the simplest. One has: Let \(a,b\in {\mathbb{C}}^*\), \(a\neq \pm b\), \(| a| \geq | b|\), and let \(\mu\), \(\lambda\) be the simplest values of M(a,b) and \(M(a+b,a-b),\) respectively. Then all values \(\mu\) ’ of M(a,b) are given by \((1/\mu ')=(d/\mu)+(ic/\lambda)\), where d,c\(\in {\mathbb{Z}}\) are relatively prime and \(d\equiv 1 mod 4\), \(c\equiv 0 mod 4.\)
This remarkable result was known to Gauß. Its proof involves ”uniformising” M(a,b) using quotients of the classical Jacobian theta functions, which are modular functions for certain congruence subgroups of level four in SL(2,\({\mathbb{Z}})\). The author gives a complete modern exposition of the proof of an extensive discussion of the history, in particular the chain of reasoning which led Gauß to its discovery.

Reviewer: C.Series

MSC:

33-03	History of special functions
11F03	Modular and automorphic functions
01A55	History of mathematics in the 19th century
32N10	Automorphic forms in several complex variables
33E05	Elliptic functions and integrals

Keywords:

arithmetic-geometric mean; elliptic integrals; Jacobian theta functions; congruence subgroups; Gauß