round the perimeter of a parallelogram of sides ω, ω′ containing z = 0 in its interior, we find ηω′ − η′ω = 1, so that neither of η, η′ is zero. We have ζ′(z) =−ℜ(z). From ζ(z) by means of the equation
| σ(z) | = exp { ∫ z0 [ ζ(x) − | 1 | ] dz } = Π′ [ ( 1 − | z | ) exp ( | z | + | z2 | ) ], |
| z | z | Ω | Ω | 2Ω2 |
we determine an integral function σ(z), termed the Sigma-function, having a zero of the first order at each of the points z = Ω; it can be seen to satisfy the equations
| σ(z + ω) | = −exp [2πiη(z + ½ω)], | σ(z + ω′) | = −exp [2πiη′ (z + ½ω′)]. |
| σ(z) | σ(z) |
By means of these equations, if a1 + a2 + ... + am = a′1 + a′2 + ... + a′m, it is readily shown that
| σ(z − a′1) σ(z − a′2) ... σ(z − a′m) |
| σ(z − a1) σ(z − a2) ... σ(z − am) |
is a doubly periodic function having a1, ... am as its simple poles, and a′1, ... a′m as its simple zeros. Thus the function σ(z) has the important property of enabling us to write any meromorphic doubly periodic function as a product of factors each having one zero in the parallelogram of periods; these form a generalization of the simple factors, z − a, which have the same utility for rational functions of z. We have ζ(z) = σ′(z)/σ(z).
The functions ζ(z), ℜ(z) may be used to write any meromorphic doubly periodic function F(z) as a sum of terms having each only one pole; for if in the expansion of F(z) near a pole z = a the terms with negative powers of z − a be
A1(z − a)−1 + A2(z − a)−2 + ... + Am+1(z − a)−(m+1),
then the difference