I haven't read Borwein & Borwein, and I don't know much about the AGM beyond its definition.
(Or one def. at least: for a,b > 0 define F(a,b) := ((a+b)/2, sqrt(ab)) and iterate; the limit is then (AGM(a,b), AGM(a,b)) if memory serves.)*
Is much known about when AGM(a,b) is rational, algebraic, or transcendental?
--Dan
Sort of. That formula (a) on p15 of B&B effectively says (AGM(a,b) = AGM(b,a)) = %pi*b/(2*\k(sqrt(1-a^2/b^2))) %pi b agm(a, b) = agm(b, a) = ----------------- 2 a 2 K(sqrt(1 - --)) 2 b where K is the complete elliptic integral. Unfortunately, if you try this with Macsyma (elliptic_kc) or Mma (EllipticK, ArithmeticGeometricMean) (but not Maple (Elliptick)), you'll lose until you lose the sqrt, because the first two expect the "parameter" (:=k^2) instead of the modulus (:=k). K then takes on special ("singular") values, of the form algebraic*powers(Gammas), which are algebraic multiples (of sqrt) of those etas I'm flaming about. These are fractional Gammas over sqrt pi, so they're all transcendental, and nobody can prove it. Btw, A great virtue of AGM is that it converges quadratically (a'-b' ~ (a-b)^2), providing the fastest numerical methods for Gamma of n/12 and n/8, except that for Gamma(1/2) the advantage over series methods may be purely theoretical. --rwg ___________________________________
* Btw, is there a natural continuum of "means" with the AM & GM at the extremes, and the AGM halfway between them?
Beaucoup. See if you can locate a paper by DH Lehmer on iterated means.