Number Theory: An Introduction to Mathematics

6 The Modular Function 535

There remains the practical problem, for a givenw∈C, of determiningτ∈H
such thatλ(τ)=w.If0<w<1, we can calculateτby theAGMalgorithm, using
the formula (4), sinceτ=iK( 1 −w)/K(w).Forcomplexwwe can use an extension
of theAGMalgorithm, or proceed in the following way.
Since

( 1 −λ(τ))^1 /^4 =θ 01 ( 0 ;τ)/θ 00 ( 0 ;τ)

and

θ 00 ( 0 ;τ)= 1 + 2

∑∞

n= 1

qn

2

,θ 01 ( 0 ;τ)= 1 + 2

∑∞

n= 1

(− 1 )nqn

2

,

we have

[1−( 1 −λ(τ))^1 /^4 ]/[1+( 1 −λ(τ))^1 /^4 ]

=[θ 00 ( 0 ;τ)−θ 01 ( 0 ;τ)]/[θ 00 ( 0 ;τ)+θ 01 ( 0 ;τ)]

= 2 (q+q^9 +q^25 +···)/( 1 + 2 q^4 + 2 q^16 +···).

Thus if we put

:=[1−( 1 −w)^1 /^4 ]/[1+( 1 −w)^1 /^4 ],

we have to solve forqthe equation

/ 2 =(q+q^9 +q^25 +···)/( 1 + 2 q^4 + 2 q^16 +···).

Expanding the right side as a power series inqand inverting the relationship, we obtain

q=/ 2 + 2 (/ 2 )^5 + 15 (/ 2 )^9 + 150 (/ 2 )^13 +O(/ 2 )^17.

To ensure rapid convergence we may suppose that, in Figure 3,wis situated in
the region 5′or on its boundary, since the general case may be reduced to this by a
linear fractional transformation. It is not difficult to show that in this region||takes
its maximum value whenw=eπi/^3 ,andthen

0 1/2^10 1/2^12

τ-plane w-plane

–1

5 4

36

21

3' 2'

4' 1'

5' 6'

Fig. 3.w=λ(τ)mapsDontoH.