1838 CHAPTER 58. ELLIPTIC FUNCTIONS
and
℘
(12
)−℘
(τ
2
)= ∑
m
π2
sin2 (π( 1
2 +mτ)) − π2
sin2 (π(m+ 1
2
)τ)
= ∑m
π2
cos2 (πmτ)− π2
sin2 (π(m+ 1
2
)τ) .
The following interesting formula for λ results.
λ (τ) =∑m
1cos2(π(m+ 1
2 )τ)− 1
sin2(π(m+ 12 )τ)
∑m1
cos2(πmτ)− 1
sin2(π(m+ 12 )τ)
. (58.1.23)
From this it is obvious λ (−τ) = λ (τ) . Therefore, from 58.1.18,
−λ (τ)+1 = λ
(1τ
)= λ
(−1τ
)(58.1.24)
(It is good to recall that λ has been defined for τ /∈ R.)
58.1.6 Mapping Properties Of λ
The two functional equations, 58.1.24 and 58.1.17 along with some other properties pre-sented above are of fundamental importance. For convenience, they are summarized herein the following lemma.
Lemma 58.1.18 The following functional equations hold for λ .
λ (1+ τ) =λ (τ)
λ (τ)−1,1 = λ (τ)+λ
(−1τ
)(58.1.25)
λ (τ +2) = λ (τ) , (58.1.26)
λ (z) = λ (w) if and only if there exists a unimodular matrix,(a bc d
)∼(
1 00 1
)mod2
such that
w =az+bcz+d
(58.1.27)
Consider the following picture.