Number Theory: An Introduction to Mathematics

534 XII Elliptic Functions

is holomorphic at every pointzinsideT. Hence, by Cauchy’s theorem,
∫

T

f(z)dz= 0.

But, since logλ(z)=log|λ(z)|+iargλ(z),
∫

T

λ′(z)dz/[λ(z)−w 0 ]=i∆.

Similarly, sinceζjis insideT,

∫

T

dz/(z−ζj)= 2 πi.

It follows that

∆= 2 π

∑p

j= 1

mj.

Ifw 0 is outsideT′,then∆=0 and we have a contradiction. Henceλ(z)is never
outsideT′ifzis insideT.Ifw 0 is insideT′,then∆= 2 π. Henceλ(z)assumes
each value insideT′at exactly one pointzinsideT, and at this pointλ′(z)=0.
Finally, ifλ(z)assumed a valuew 0 onT′at a pointz 0 insideT, then it would
assume all values nearw 0 in the neighbourhood ofz 0. In particular, it would assume
values outsideT′, which we have shown to be impossible. It follows thatw=λ(z)
maps the region insideT bijectively onto the region insideT′,andλ′(z)=0forall
zinsideT.
We must also haveλ′(z)=0forallz=0onT.Otherwise,ifλ(z 0 )=w 0 and
λ′(z 0 )=0forsomez 0 ∈T∩Hthen, for somem>1andc=0,

λ(z)−w 0 ∼c(z−z 0 )masz→z 0.

But this implies thatλ(z)takes values outsideT′for someznearz 0 insideT.
By putting together the preceding results we see thatw=λ(τ)maps the domain

D={τ∈H:0<Rτ< 1 ,|τ− 1 / 2 |> 1 / 2 }

bijectively onto the upper half-planeH, with the subdomainkofDmapped onto the
subdomaink′ofH(k= 1 ,..., 6 ), as illustrated in Figure 3. Moreover, the boundary
inHofDis mapped bijectively onto the real axis, with the points 0 and 1 omitted.
If we denote byD ̄the closure ofDinH and byD∗the reflection ofDin the
imaginary axis, then it follows from (59) thatw=λ(τ)maps the region

D ̄∪D∗={τ∈H:0≤Rτ≤ 1 ,|τ− 1 / 2 |≥ 1 / 2 }

∪{τ∈H:− 1 <Rτ< 0 ,|τ+ 1 / 2 |> 1 / 2 }

bijectively onto the whole complex planeC, with the points 0 and 1 omitted.This
answers the question raised in§5.