§23.2 Definitions and Periodic Properties

If ${\omega }_1$ and ${\omega }_3$ are nonzero real or complex numbers such that $\mathrm{\Im }\left({\omega }_3/{\omega }_1\right)>0$, then the set of points $2m{\omega }_1+2n{\omega }_3$, with $m,n\in \mathbb{Z}$, constitutes a lattice $𝕃$ with $2{\omega }_1$ and $2{\omega }_3$ lattice generators.

The generators of a given lattice $𝕃$ are not unique. For example, if

then $2{\omega }_2$, $2{\omega }_3$ are generators, as are $2{\omega }_2$, $2{\omega }_1$. In general, if

where $a,b,c,d$ are integers, then $2{\chi }_1$, $2{\chi }_3$ are generators of $𝕃$ iff

The double series and double product are absolutely and uniformly convergent in compact sets in $ℂ$ that do not include lattice points. Hence the order of the terms or factors is immaterial.

When $z\notin 𝕃$ the functions are related by

$\mathrm{\wp }\left(z\right)$ and $\zeta \left(z\right)$ are meromorphic functions with poles at the lattice points. $\mathrm{\wp }\left(z\right)$ is even and $\zeta \left(z\right)$ is odd. The poles of $\mathrm{\wp }\left(z\right)$ are double with residue $0$; the poles of $\zeta \left(z\right)$ are simple with residue $1$. The function $\sigma \left(z\right)$ is entire and odd, with simple zeros at the lattice points. When it is important to display the lattice with the functions they are denoted by $\mathrm{\wp }\left(z|𝕃\right)$, $\zeta \left(z|𝕃\right)$, and $\sigma \left(z|𝕃\right)$, respectively.

If $2{\omega }_1$, $2{\omega }_3$ is any pair of generators of $𝕃$, and ${\omega }_2$ is defined by (23.2.1), then

Hence $\mathrm{\wp }\left(z\right)$ is an elliptic function, that is, $\mathrm{\wp }\left(z\right)$ is meromorphic and periodic on a lattice; equivalently, $\mathrm{\wp }\left(z\right)$ is meromorphic and has two periods whose ratio is not real. We also have

The function $\zeta \left(z\right)$ is quasi-periodic: for $j=1,2,3$,

where

Also,

For $j=1,2,3$, the function $\sigma \left(z\right)$ satisfies

More generally, if $j=1,2,3$, $k=1,2,3$, $j\ne k$, and $m,n\in \mathbb{Z}$, then

For further quasi-periodic properties of the $\sigma$-function see Lawden (1989, §6.2).