Number Theory: An Introduction to Mathematics

3 Elliptic Functions 515

u(z)=φ(z)^1 /^2 ,

where the square root is chosen so thatRu(z)>0. ThenS[u(z)]=z. Differentiating
and then squaring, we obtain

S′[u(z)]u′(z)= 1 , u′(z)^2 = 1 /gλ(z).

Butu′(z)also has positive real part, sinceS′[u(z)]∼ 2 u(z)forz→0. Consequently
u′(z)= 1 /hλ(z).Sinceu(z)→0asz→0, we conclude that

u(z)=

∫z

0

dζ/hλ(ζ), (23)

where the path of integration is (say) a straight line segment. However, the function on
the right is holomorphic for allz∈H. Consequently, if we defineu(z)by (23) then,
by analytic continuation, the relationS[u(z)]=zcontinues to hold for allz∈H.
Lettingz→1, we now obtainS(h)=1forh=K(λ),where

K(λ):=

∫ 1

0

dx/gλ(x)^1 /^2

and the square root is chosen so thatgλ(x)^1 /^2 is continuous and has positive real part
for smallx>0 and actually, as we will see in a moment, for 0<x<1. HenceS(t)
has period 2K(λ). Furthermore, by (15),S(t)also has period 2iK( 1 −λ).
For 0<x<1wehave

1 /gλ(x)^1 /^2 =( 1 −λ ̄x)^1 /^2 /[4x( 1 −x)]^1 /^2 | 1 −λx|.

Ifλ=μ+iv,wherev>0, then 1−λ ̄x=γ+iδ,whereγ= 1 −μxandδ=vx> 0
for 0<x<1. Hence

( 1 −λ ̄x)^1 /^2 =α+iβ,

where

α={γ+(γ^2 +δ^2 )^1 /^2 }^1 /^2 /

√

2 , 2 αβ=δ,

first for smallx>0 and then, by continuity, for 0<x<1. Thusαandβare positive
for 0<x<1. ConsequentlyRgλ(x)^1 /^2 >0for0<x<1and

K(λ)=A+iB,

whereA>0,B>0.
Similarly, for 0<y<1wehave

1 /g 1 −λ(y)^1 /^2 =( 1 −( 1 −λ) ̄ y)^1 /^2 /[4y( 1 −y)]^1 /^2 | 1 −( 1 −λ)y|

and 1−( 1 − ̄λ)y=γ′−iδ′,whereγ′= 1 −( 1 −μ)yandδ′=vy>0for0<y<1.
Hence

( 1 −( 1 −λ) ̄ y)^1 /^2 =α′−iβ′,