Number Theory: An Introduction to Mathematics

514 XII Elliptic Functions

it follows thatT≤K(λ),where

K(λ):=

∫ 1

0

dx/gλ(x)^1 /^2.

HenceS′(t)vanishes for sometsuch that 0<t≤K(λ)and we can now takeTto be
the leastt>0forwhichS′(t)=0. ThenS′(T)=0,S(T)=1 and by lettingt→T
we obtainT=K(λ).
This shows thatS(u)maps the interval [0,K(λ)] bijectively onto [0,1], and if

u(ξ)=

∫ξ

0

dx/gλ(x)^1 /^2 ( 0 ≤ξ≤ 1 ),

thenS[u(ξ)]=ξ. Thus, in the real domain, the elliptic integral of the first kind is
invertedby the functionS(u).
Sinceλ=1, it follows thatS(t)=S(t,λ)has period 2K(λ).Sinceλ=0, it
follows from (15) thatS(t,λ)also has period 2iK( 1 −λ). ThusS(t,λ)is adoubly-
periodicfunction, with a real period and a pure imaginary period. We will show that
all periods are given by

2 mK(λ)+ 2 ni K( 1 −λ) (m,n∈Z).

The periods of a nonconstant meromorphic function f form a discrete additive
subgroup ofC.Iffhas two periods whose ratio is not real then, by the simple case
n=2 of Proposition VIII.7, it has periodsω 1 ,ω 2 such that all periods are given by

mω 1 +nω 2 (m,n∈Z).

In the present case we can takeω 1 = 2 K(λ),ω 2 = 2 iK( 1 −λ)since, by construction,
2 K(λ)is the least positive period.
Suppose next thatλ∈Rand eitherλ>1orλ<0. Then, by (14) and (15),S(t,λ)
is again a doubly-periodic function with a real period and a pure imaginary period.
Suppose finally thatλ∈C\R. Without loss of generality,we assumeIλ>0.
Thengλ(z)does not vanish in the upper half-planeH. It follows that there exists a
unique functionhλ(z), holomorphic forz∈H withRhλ(z)>0forznear 0, such
that

hλ(z)^2 =gλ(z). (22)

Moreover, we may extend the definition so thathλ(z)is continuous and (22) continues
to hold forz∈H∪R.
We can writeS(t)=ψ(t^2 ),where

ψ(w)=w+a 2 w^2 +···

is holomorphic at the origin. By inversion of series, there exists a function

φ(z)=z+b 2 z^2 +···,

which is holomorphic at the origin, such thatψ[φ(z)]=z.Forz∈Hnear 0, put