58.1. PERIODIC FUNCTIONS 1831

consider (w1 +w2)/2 and choose Pa such that its boundary contains no zeros of the ellipticfunction, z→℘(z)−℘((w1 +w2)/2) and Pa contains no poles of ℘ on its interior otherthan 0. Then if ℘(w2/2) =℘((w1 +w2)/2) , it follows from Theorem 58.1.7 on Page1820 w2/2 and (w1 +w2)/2 are the only two zeros of ℘(z)−℘((w1 +w2)/2) on Pa andby Corollary 58.1.9 on Page 1822

w1 +w1 +w2

2= aw1 +bw2 ∈M

for some integers a,b which leads to the same contradiction as before about w1/w2 notbeing real. The other cases are similar. This proves the lemma.

Lemma 58.1.14 proves the ei are distinct. Number the ei such that

e1 =℘(w1/2) ,e2 =℘(w2/2)

ande3 =℘((w1 +w2)/2) .

To summarize, it has been shown that for complex numbers, w1 and w2 with w2/w1 notreal, an elliptic function, ℘has been defined. Denote this function as℘(z) =℘(z,w1,w2) .This in turn determined numbers, ei as described above. Thus these numbers depend on w1and w2 and as described above,

e1 (w1,w2) = ℘

(w1

2,w1,w2

), e2 (w1,w2) =℘

(w2

2,w1,w2

)e3 (w1,w2) = ℘

(w1 +w2

2,w1,w2

).

Therefore, using the formula for ℘, 58.1.9,

℘(z)≡ 1z2 + ∑

w ̸=0

(1

(z−w)2 −1

w2

)you see that if the two periods w1 and w2 are replaced with tw1 and tw2 respectively, then

ei (tw1, tw2) = t−2ei (w1,w2) .

Let τ denote the complex number which equals the ratio, w2/w1 which was assumed in allthis to not be real. Then

ei (w1,w2) = w−21 ei (1,τ)

Now define the function, λ (τ)

λ (τ)≡ e3 (1,τ)− e2 (1,τ)e1 (1,τ)− e2 (1,τ)

(=

e3 (w1,w2)− e2 (w1,w2)

e1 (w1,w2)− e2 (w1,w2)

). (58.1.10)

This function is meromorphic for Imτ > 0 or for Imτ < 0. However, since the denominatoris never equal to zero the function must actually be analytic on both the upper half planeand the lower half plane. It never is equal to 0 because e3 ̸= e2 and it never equals 1 becausee3 ̸= e1. This is stated as an observation.