Number Theory: An Introduction to Mathematics

1 Elliptic Integrals 499

The range ofλcan be further restricted by linear changes of variables. The trans-
formationy=( 1 −λx)/( 1 −λ)replaces Riemann’s normal form by one of the same
type withλreplaced byUλ= 1 −λ. Similarly, the transformationy= 1 −λxreplaces
Riemann’s normal form by one of the same type withλreplaced byVλ= 1 /( 1 −λ).
The transformationsUandVtogether generate a groupGof order 6 (isomorphic to
the symmetric groupS 3 of all permutations of three letters), since

U^2 =V^3 =(UV)^2 =I.

The values ofλcorresponding to the elementsI,V,V^2 ,U,UV,UV^2 ofGare

λ, 1 /( 1 −λ), (λ− 1 )/λ, 1 −λ, λ/(λ− 1 ), 1 /λ.

The regionFof the complex planeCdefined by the inequalities

|λ− 1 |< 1 , 0 <Rλ< 1 / 2 ,

is afundamental domainfor the groupG; i.e., no point ofFis mapped to a different
point ofFby an element ofGand each point ofCis mapped to a point ofFor its
boundary∂Fby some element ofG. Consequently the sets{G(F):G∈G}form a
tilingofC; i.e.,

C= ∪

G∈G

G(F∪∂F), G(F)∩G′(F)=∅ ifG,G′∈GandG=G′.

This is illustrated in Figure 1, where the setG(F)is represented simply by the group
elementGand, in particular,Fis represented byI. It follows that in Riemann’s
normal form we may supposeλ∈F∪∂F.
The changes of variable in the preceding reduction to Riemann’s normal form may
be complex, even though the original integrand was real. It will now be shown that any
real elliptic integral can be reduced by a real change of variables to one in Riemann’s
normal form, where 0<λ<1 and the independent variable is restricted to the interval
0 ≤x≤1.

VUUV I U V V

(^22)
–1 0 1/2 12
Fig. 1.Fundamental domain forλ.