If and are nonzero real or complex numbers such that , then the set of points , with , constitutes a lattice with and lattice generators.
The generators of a given lattice are not unique. For example, if
then , are generators, as are , . In general, if
where are integers, then , are generators of iff
The double series and double product are absolutely and uniformly convergent in compact sets in that do not include lattice points. Hence the order of the terms or factors is immaterial.
When the functions are related by
and are meromorphic functions with poles at the lattice points. is even and is odd. The poles of are double with residue 0; the poles of are simple with residue 1. The function 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 , , and , respectively.
If , is any pair of generators of , and is defined by (23.2.1), then
Hence is an elliptic function, that is, is meromorphic and periodic on a lattice; equivalently, is meromorphic and has two periods whose ratio is not real. We also have
The function is quasi-periodic: for ,
For , the function satisfies
More generally, if , , , and , then
For further quasi-periodic properties of the -function see Lawden (1989, §6.2).