Number Theory: An Introduction to Mathematics

528 XII Elliptic Functions

The addition formulas show that the evaluation of the Jacobian elliptic functions for
arbitrary complex argument may be reduced to their evaluation for real and pure imag-
inary arguments.
The usual addition formulas for the sine and cosine functions may be regarded as
limiting cases of (55). For ifτ=iyandy→∞, the product expansions (35) show
that

θ 00 (v)→ 1 ,θ 01 (v)→ 1 ,

θ 10 (v)∼ 2 eπiτ/^4 cosπv, θ 11 (v)∼ 2 ieπiτ/^4 sinπv,

and hence

λ→ 0 , u→πv,

snu→sinu, cnu→cosu, dnu→ 1.

The definitions (46) of the Jacobian elliptic functions and the transformation
formulas (37)–(38) for the theta functions imply alsotransformation formulasfor the
Jacobian functions:

Proposition 9Fo r a l l u∈Candτ∈H,

sn(u;τ+ 1 )=( 1 −λ(τ))^1 /^2 sn(u′;τ)/dn(u′;τ),

cn(u;τ+ 1 )=cn(u′;τ)/dn(u′;τ),

dn(u;τ+ 1 )= 1 /dn(u′;τ),

where

u′=u/( 1 −λ(τ))^1 /^2

and

( 1 −λ(τ))^1 /^2 =θ 012 ( 0 ;τ)/θ 002 ( 0 ;τ).

Furthermore,

λ(τ+ 1 )=λ(τ)/[λ(τ)−1],

K(τ+ 1 )=( 1 −λ(τ))^1 /^2 K(τ).

Proof Withv=u/πθ 002 ( 0 ;τ+ 1 )we have, by (37),

dn(u;τ+ 1 )=θ 00 ( 0 ;τ)θ 01 (v;τ)/θ 01 ( 0 ;τ)θ 00 (v;τ)= 1 /dn(u′;τ),

where

u′=πθ 002 ( 0 ;τ)v=θ 002 ( 0 ;τ)u/θ 012 ( 0 ;τ)=u/( 1 −λ(τ))^1 /^2.

Similarly, from (37) and (48)-(49), we obtain

λ(τ+ 1 )=−θ 104 ( 0 ;τ)/θ 014 ( 0 ;τ)=λ(τ)/[λ(τ)−1].

The other relations are established in the same way.