502 XII Elliptic Functions
If we puty={(x+d 1 )/(x+d 2 )}^2 ,then
R(x)=R 1 (y)+R 2 (y)y^1 /^2 ,where again the rational functionsR 1 ,R 2 are determined by the rational functionR,
and
dx/g(x)^1 /^2 =±dy/ 2 |d 2 −d 1 |[y(A 1 y+B 1 )(A 2 y+B 2 )]^1 /^2.Thus we are again reduced to the case of a cubic with 3 distinct real roots.
The preceding argument may be applied also whenghas only real roots, provided
the factorsQ 1 andQ 2 are chosen so that their zeros do not interlace. Suppose (without
loss of generality) thatg=gλis in Riemann’s normal form and take
Q 1 =( 1 −x)( 1 −λx), Q 2 = 4 x.In this case we can write
Q 1 ={( 1 +√
λ)^2 (x− 1 /√
λ)^2 −( 1 −√
λ)^2 (x+ 1 /√
λ)^2 }√
λ/ 4 ,
Q 2 =−√
λ{(x− 1 /√
λ)^2 −(x+ 1 /√
λ)^2 }.If we put
1 − 4√
λy/( 1 +√
λ)^2 ={(x− 1 /√
λ)/(x+ 1 /√
λ)}^2 ,we obtain
dx/gλ(x)^1 /^2 =dy/μgρ(y)^1 /^2 ,where
μ= 1 +√
λ, ρ= 4√
λ/( 1 +√
λ)^2.The usefulness of this change of variables will be seen in the next section.
2 TheArithmetic-GeometricMean
Letaandbbe positive real numbers, witha>b,andlet
a 1 =(a+b)/ 2 , b 1 =(ab)^1 /^2be respectively their arithmetic and geometric means. Then
a 1 <(a+a)/ 2 =a, b 1 >(bb)^1 /^2 =b,and
a 1 −b 1 =(a^1 /^2 −b^1 /^2 )^2 / 2 > 0.