Number Theory: An Introduction to Mathematics

2 The Arithmetic-Geometric Mean 503

Thusa 1 ,b 1 satisfy the same hypotheses asa,band the procedure can be repeated. If
we define sequences{an},{bn}inductively by

a 0 =a, b 0 =b,

an+ 1 =(an+bn)/ 2 , bn+ 1 =(anbn)^1 /^2 (n= 0 , 1 ,...),

then

0 <b 0 <b 1 <b 2 <···<a 2 <a 1 <a 0.

It follows thatan→λandbn→μasn→∞,whereλ≥μ>0. In factλ=μ,as
one sees by lettingn→∞in the relationan+ 1 =(an+bn)/2. The convergence of
the sequences{an}and{bn}to their common limit is extremely rapid, since

an−bn=(an− 1 −bn− 1 )^2 / 8 an+ 1.

(As an example, ifa=

√

2andb=1, calculation shows thata 4 andb 4 differ by only
one unit in the 20th decimal place.)
The common limit of the sequences{an}and{bn}will be denoted byM(a,b).
The definition can be extended to arbitrary positive real numbersa,bby putting

M(a,a)=a, M(b,a)=M(a,b).

Following Gauss (1818),M(a,b)is known as thearithmetic-geometric meanofa
andb. However, the preceding algorithm, which we will call theAGMalgorithm, was
first introduced by Lagrange (1784/5), who showed that it had a remarkable applica-
tion to the numerical calculation of arbitrary elliptic integrals. The first tables of elliptic
integrals, which made them as accessible as logarithms, were constructed in this way
under the supervision of Legendre (1826). Today the algorithm can be used directly by
electronic computers.
By putting 1−λx=t^2 /a^2 in Riemann’s normal form, it may be seen that any real
elliptic integral may be brought to the form

∫

φ(t)[(a^2 −t^2 )(t^2 −b^2 )]−^1 /^2 dt,

whereφ(t)is a rational function oft^2 with real coefficients,a>b>0andt∈[b,a].
We will restrict attention here to thecompleteelliptic integral

J=

∫a

b

φ(t)[(a^2 −t^2 )(t^2 −b^2 )]−^1 /^2 dt,

but at the cost of some complication the discussion may be extended toincomplete
elliptic integrals (where the interval of integration is a proper subinterval of [b,a]).
If we make the change of variables

t^2 =a^2 sin^2 θ+b^2 cos^2 θ( 0 ≤θ≤π/ 2 ),