Number Theory: An Introduction to Mathematics

2 The Arithmetic-Geometric Mean 507

Using (1)–(3), complete elliptic integrals of all three kinds can be calculated by
theAGMalgorithm. We now consider another application, the utility of which will be
seen in§6.
By puttingt 1 =( 1 / 2 )(t+ab/t)again, one sees that
∫∞

a

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

∫∞

a 1

[(t 12 −a^21 )(t 12 −b^21 )]−^1 /^2 dt 1.

But the change of variablesu=a( 1 −b^2 /t^2 )^1 /^2 shows that
∫∞

a

[(t^2 −a^2 )(t^2 −b^2 )]−^1 /^2 dt=K(a,c),

where c=(a^2 −b^2 )^1 /^2. It follows that

K(a,c)=K(a 1 ,c 1 )/ 2 =···=K(an,cn)/ 2 n,

where cn=(an^2 −bn^2 )^1 /^2. The asymptotic behaviour ofK(an,cn) may be determined
in the following way.
If we puts=ac/t,thensdecreases fromatocastincreases fromctoa,and

ds/dt=−[(a^2 −s^2 )(s^2 −c^2 )]^1 /^2 /[(a^2 −t^2 )(t^2 −c^2 )]^1 /^2.

Sinces=twhent=h:=(ac)^1 /^2 , it follows that

K(a,c)= 2

∫h

c

[(a^2 −t^2 )(t^2 −c^2 )]−^1 /^2 dt.

But, forc≤t≤h,

b−^1 =(a^2 −c^2 )−^1 /^2 ≤(a^2 −t^2 )−^1 /^2 ≤(a^2 −h^2 )−^1 /^2 =a−^1 ( 1 −c/a)−^1 /^2.

Hence

2 b−^1 L≤K(a,c)≤ 2 a−^1 ( 1 −c/a)−^1 /^2 L,

where

L:=

∫h

c

(t^2 −c^2 )−^1 /^2 dt=log{(a/c)^1 /^2 +(a/c− 1 )^1 /^2 }.

If we now replacea,b,cbyan,bn,cnthen, sincean/cn →∞and moreover
an,bn→M(a,b), we deduce that

2 nK(a,c)/log( 4 an/cn)→ 1 /M(a,b)= 2 K(a,b)/π.

But 4an/cn=( 4 an/cn− 1 )^2 ,sincecn=(an− 1 −bn− 1 )/2, and hence

2 −nlog( 4 an/cn)

= 21 −nlog( 4 an− 1 /cn− 1 )− 21 −nlog(an− 1 /an)

=···

=log( 4 a 0 /c 0 )−log(a 0 /a 1 )− 2 −^1 log(a 1 /a 2 )−···− 21 −nlog(an− 1 /an).