The nome is given in terms of the modulus by
where , are defined in §19.2(ii). Inversely,
where and the theta functions are defined in §20.2(i).
As a function of , with fixed , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. Each is meromorphic in for fixed , with simple poles and simple zeros, and each is meromorphic in for fixed . For , all functions are real for .
The Jacobian functions are related in the following way. Let , , be any three of the letters , , , . Then
with the convention that functions with the same two letters are replaced by unity; e.g. .
The six functions containing the letter in their two-letter name are odd in ; the other six are even in .
In terms of Neville’s theta functions (§20.1)
and , are any pair of the letters , , , .