58.1. PERIODIC FUNCTIONS 1825

Thus the ratio is not real if and only if(

λ −λ

)̸= 0 if and only if w1w2−w1w2 ̸= 0.

Now to verify w′2/w′1 is not real,

det(

w′1 w′1w′2 w′2

)= det

((a bc d

)(w1 w1w2 w2

))= ±det

(w1 w1w2 w2

)̸= 0

This proves the theorem.

58.1.2 The Search For An Elliptic FunctionBy Theorem 58.1.4 and 58.1.6 if you want to find a nonconstant elliptic function it mustfail to be analytic and also have either no terms in its Laurent expansion which are of theform b1 (z−a)−1 or else these terms must cancel out. It is simplest to look for a functionwhich simply does not have them. Weierstrass looked for a function of the form

℘(z)≡ 1z2 + ∑

w ̸=0

(1

(z−w)2 −1

w2

)(58.1.6)

where w consists of all numbers of the form aw1 +bw2 for a,b integers. Sometimes peoplewrite this as ℘(z,w1,w2) to emphasize its dependence on the periods, w1 and w2 but Iwon’t do so. It is understood there exist these periods, which are given. This is a reasonablething to try. Suppose you formally differentiate the right side. Never mind whether this isjustified for now. This yields

℘′ (z) =

−2z3 − ∑

w̸=0

−2

(z−w)3 = ∑w

−2

(z−w)3

which is clearly periodic having both periods w1 and w2. Therefore, ℘(z+w1)−℘(z) and℘(z+w2)−℘(z) are both constants, c1 and c2 respectively. The reason for this is thatsince ℘′ is periodic with periods w1 and w2, it follows ℘′ (z+wi)−℘′ (z) = 0 as long as zis not a period. From 58.1.6 you can see right away that

℘(z) =℘(−z)

Indeed

℘(−z) =1z2 + ∑

w̸=0

(1

(−z−w)2 −1

w2

)

=1z2 + ∑

w̸=0

(1

(−z+w)2 −1

w2

)=℘(z) .