1836 CHAPTER 58. ELLIPTIC FUNCTIONS

Therefore, τ ′ = w′2/w′1 = w1/w2 = 1/τ. Then

λ(τ′) = λ (1/τ)

=℘

(w′1+w′2

2

)−℘

(w′22

)℘

(w′12

)−℘

(w′22

)=

℘(w1+w2

2

)−℘

(w12

)℘(w2

2

)−℘

(w12

)=

e3− e1

e2− e1=−e3− e2 + e2− e1

e1− e2

= −(λ (τ)−1) =−λ (τ)+1. (58.1.18)

You could try other unimodular matrices and attempt to find other functional equationsif you like but this much will suffice here.

58.1.5 A Formula For λ

Recall the formula of Mittag-Leffler for cot(πα) given in 57.2.15. For convenience, hereit is.

1α+

∞

∑n=1

2α

α2−n2 = π cotπα.

As explained in the derivation of this formula it can also be written as

∞

∑n=−∞

α

α2−n2 = π cotπα.

Differentiating both sides yields

π2 csc2 (πα) =

∞

∑n=−∞

α2 +n2

(α2−n2)2

=∞

∑n=−∞

(α +n)2−2αn

(α +n)2 (α−n)2

=∞

∑n=−∞

(α +n)2

(α +n)2 (α−n)2 −

=0︷ ︸︸ ︷∞

∑n=−∞

2αn

(α2−n2)2

=∞

∑n=−∞

1

(α−n)2 . (58.1.19)

Now this formula can be used to obtain a formula for λ (τ) . As pointed out above, λ

depends only on the ratio w2/w1 and so it suffices to take w1 = 1 and w2 = τ. Thus

λ (τ) =℘( 1+τ

2

)−℘

(τ

2

)℘( 1

2

)−℘

(τ

2

) . (58.1.20)