Number Theory: An Introduction to Mathematics

532 XII Elliptic Functions

where the bar denotes complex conjugation, and hence

λ(− ̄τ)=λ(τ). (59)

We next note that, by takingτ = iin the relationλ(− 1 /τ) = 1 −λ(τ),we
obtainλ(i)= 1 /2. We have already seen in§5thatλ(τ)is real on the imaginary
axisτ = iy(y > 0 ), and decreases from 1 to 0 asy increases from 0 to∞.
Sinceλ(τ+ 1 )=λ(τ)/[λ(τ)−1], it follows thatλ(τ)is real also on the half-line
τ= 1 +iy(y> 0 ), and increases from−∞to0asyincreases from 0 to∞.More-
over,λ( 1 +i)=−1.
The linear fractional mapτ=(τ′− 1 )/τ′maps the half-lineRτ′= 1 ,Iτ′> 0
onto the semi-circle|τ− 1 / 2 |= 1 / 2 ,Iτ>0, andτ′ = 1 +iis mapped to
τ=( 1 +i)/2. Since

λ((τ′− 1 )/τ′)=[λ(τ′)−1]/λ(τ′),

it follows from what we have just proved that, asτtraverses this semi-circle from
0to1,λ(τ)is real and increases from 1 to∞. Moreover,λ(( 1 +i)/ 2 )=2.
IfRτ= 1 /2, thenτ ̄= 1 −τand hence, by (59),

λ(τ)=λ(τ− 1 )=λ(τ)/[λ(τ)−1],

which implies

|λ(τ)− 1 |^2 = 1.

Thusw=λ(τ)maps the half-lineRτ= 1 /2,Iτ>0 into the circle|w− 1 |=1.
Furthermore, the map is injective. For ifλ(τ 1 )=λ(τ 2 ),thenλ( 2 τ 1 )=λ( 2 τ 2 ),by
Proposition 11, and the map is injective on the half-lineRτ = 1,Iτ>0. If
τ= 1 / 2 +iy,wherey→+∞,then

θ 00 ( 0 ;τ)→ 1 ,θ 10 ( 0 ;τ)∼ 2 eπiτ/^4

and hence

λ(τ)∼ 16 ie−πy.

In particular,λ(τ)∈ H andλ(τ) →0. Sinceλ(( 1 +i)/ 2 )=2, it follows that
w=λ(τ)maps the half-lineτ= 1 / 2 +iy(y> 1 / 2 )bijectively onto the semi-circle
|w− 1 |=1,Iw>0.
If|τ|=1,Iτ>0andτ′ =τ/( 1 +τ),thenRτ′ = 1 /2,Iτ′ >0and
λ(τ′) = 1 /λ(τ). Consequently, by what we have just proved,w = λ(τ)maps
the semi-circle|τ|=1,Iτ>0 bijectively onto the half-lineRw = 1 /2,
Iw>0.
The pointeπi/^3 =( 1 +i

√

3 )/2isinHand lies on both the lineRτ= 1 /2and
the circle|τ|=1. Henceλ(eπi/^3 )lies on both the semi-circle|w− 1 |=1,Iw> 0
and the lineRw= 1 /2, which implies that

λ(eπi/^3 )=eπi/^3.