Number Theory: An Introduction to Mathematics

6 The Modular Function 533

A

B

C

A' B '

C'

0 1/2 1 0 1/2 1 2

τ-plane w-plane

'

Fig. 2.w=λ(τ)mapsTontoT′.

Again, sinceλ(τ− 1 ) = λ(τ)/[λ(τ)−1],w = λ(τ)maps the semi-circle
|τ− 1 |=1,Iτ>0 bijectively onto the semi-circle|w|=1,Iw>0.
In particular, we have the behaviour illustrated in Figure 2:w=λ(τ)maps the
boundary of the (non-Euclidean) ‘triangle’T with verticesA=0,B=( 1 +i)/2,
C=eπi/^3 bijectively onto the boundary of the ‘triangle’T′with verticesA′=1,
B′=2,C′=eπi/^3. We are going to deduce from this that the region insideT is
mapped bijectively onto the region insideT′. The reasoning here does not depend on
special properties of the function or the domain, but is quite general (the ‘principle
of the argument’). To emphasize this, we will temporarily denote the independent
variable byz, instead ofτ.
Choose anyw 0 ∈Cwhich is either inside or outside the ‘triangle’T′,andlet
∆denote the change in the argument ofw−w 0 aswtraversesT′in the direction
A′B′C′. Thus∆= 2 πor 0 according asw 0 is inside or outsideT′.But∆is also the
change in the argument ofλ(z)−w 0 asztraversesT in the directionABC.Since
λ(z)is a nonconstant holomorphic function, the number of times that it assumes the
valuew 0 insideT is either zero or a positive integerp.
Suppose the latter, and letz = ζ 1 ,...,ζpbe the points insideT for which
λ(z)=w 0. In the neighbourhood ofζjwe have, for some positive integermjand
somea 0 j=0,

λ(z)−w 0 =a 0 j(z−ζj)mj+a 1 j(z−ζj)mj+^1 +···

and

λ′(z)=mja 0 j(z−ζj)mj−^1 +(mj+ 1 )a 1 j(z−ζj)mj+···.

Hence

λ′(z)/[λ(z)−w 0 ]=mj/(z−ζj)+fj(z),

wherefj(z)is holomorphic atζj. Consequently

f(z):=λ′(z)/[λ(z)−w 0 ]−

∑p

j= 1

mj/(z−ζj)