Number Theory: An Introduction to Mathematics

522 XII Elliptic Functions

By differentiating with respect tovand then puttingv =0, we obtain in addition
θ 11 ′( 0 )= 2 πiq^1 /^4 Q^30 .But

Q 0 =

∏∞

n= 1

( 1 −qn)( 1 +qn)

=

∏∞

n= 1

( 1 −q^2 n)( 1 −q^2 n−^1 )( 1 +q^2 n)( 1 +q^2 n−^1 ),

which implies

∏∞

n= 1

( 1 −q^2 n−^1 )( 1 +q^2 n)( 1 +q^2 n−^1 )= 1.

It follows that

θ 00 ( 0 )θ 01 ( 0 )θ 10 ( 0 )= 2 q^1 /^4 Q^30

and hence

θ 11 ′( 0 )=πiθ 00 ( 0 )θ 01 ( 0 )θ 10 ( 0 ). (36)

It is evident from their series definitions that, whenqis replaced by−q, the func-
tionsθ 00 andθ 01 are interchanged, whereas the functionsq−^1 /^4 θ 10 andq−^1 /^4 θ 11 are
unaltered. Hence

θ 00 (v;τ+ 1 )=θ 01 (v;τ), θ 10 (v;τ+ 1 )=eπi/^4 θ 10 (v;τ),

θ 01 (v;τ+ 1 )=θ 00 (v;τ), θ 11 (v;τ+ 1 )=eπi/^4 θ 11 (v;τ).

(37)

From Proposition 3 we obtain also the transformation formulas

θ 00 (v;− 1 /τ)=(τ/i)^1 /^2 eπiτv

2

θ 00 (τv;τ),

θ 10 (v;− 1 /τ)=(τ/i)^1 /^2 eπiτv

2

θ 01 (τv;τ),

θ 01 (v;− 1 /τ)=(τ/i)^1 /^2 eπiτv

2

θ 10 (τv;τ),

θ 11 (v;− 1 /τ)=−i(τ/i)^1 /^2 eπiτv

2

θ 11 (τv;τ).

(38)

Up to this point we have used Hermite’s notation just to dress up old results in new
clothes. The next result breaks fresh ground.

Proposition 4Fo r a l lv,w∈Candτ∈H,

θ 00 (v;τ)θ 00 (w;τ)=θ 00 (v+w; 2 τ)θ 00 (v−w; 2 τ)+θ 10 (v+w; 2 τ)θ 10 (v−w; 2 τ),

θ 10 (v;τ)θ 10 (w;τ)=θ 10 (v+w; 2 τ)θ 00 (v−w; 2 τ)+θ 00 (v+w; 2 τ)θ 10 (v−w; 2 τ),

θ 00 (v;τ)θ 01 (w;τ)=θ 01 (v+w; 2 τ)θ 01 (v−w; 2 τ)+θ 11 (v+w; 2 τ)θ 11 (v−w; 2 τ),

θ 01 (v;τ)θ 01 (w;τ)=θ 00 (v+w; 2 τ)θ 00 (v−w; 2 τ)−θ 10 (v+w; 2 τ)θ 10 (v−w; 2 τ),

θ 10 (v;τ)θ 11 (w;τ)=θ 11 (v+w; 2 τ)θ 01 (v−w; 2 τ)−θ 01 (v+w; 2 τ)θ 11 (v−w; 2 τ),

θ 11 (v;τ)θ 11 (w;τ)=θ 10 (v+w; 2 τ)θ 00 (v−w; 2 τ)−θ 00 (v+w; 2 τ)θ 10 (v−w; 2 τ).