| ƒ(z) = | 1 | + ∫ z0 { φ(z) + | 2 | } dz, |
| z2 | z3 |
where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this gives
| dƒ(z) | = φ(z) |
| dz |
and
| ƒ(z) = | 1 | + Σ′ { | 1 | − | 1 | } , |
| z2 | (z − Ω)2 | Ω2 |
wherein Σ′ is a sum excluding the term for which m = 0 and m′ = 0. Hence ƒ(z + ω) − ƒ(z) and ƒ(z + ω′) − ƒ(z) are both independent of z. Noticing, however, that, by its form, ƒ(z) is an even function of z, and putting z = −½ω, z = −½ω′ respectively, we infer that also ƒ(z) has the two periods ω and ω′. In the primary parallelogram Π0, however, ƒ(z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z−2 + (power series in z). Thus ƒ(z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.
It can be shown that any single valued meromorphic function of z with ω and ω′ as periods can be expressed rationally in terms of ƒ(z) and φ(z), and that [φ(z)]2 is of the form 4[ƒ(z)]3 + Aƒ(z) + B, where A, B are constants.
To prove the last of these results, we write, for |z| < |Ω|,
| 1 | − | 1 | = | 2z | + | 3z² | + ..., |
| (z − Ω)² | Ω² | Ω³ | Ω4 |