contains no negative powers of z, being equal to a power series in z^2
beginning with a term in z^2. The function F(z) is, however, doubly
periodic, with periods [omega], [omega]', and can only be infinite
when either [f](z) or [phi](z) is infinite; this follows from its form
in [f](z) and [phi](z); thus in one parallelogram of periods it can be
infinite only when z = 0; we have proved, however, that it is not
infinite, but, on the contrary, vanishes, when z = 0. Being,
therefore, never infinite for finite values of z it is a constant, and
therefore necessarily always zero. Putting therefore [f](z) = [zeta]
and [phi](z) = d[zeta]/dz we see that
dz
------- = (4[zeta]^3 - 60[sigma]2[zeta] - 140[sigma]3)^(-1/2)
d[zeta]
Historically it was in the discussion of integrals such as
_
/
| d[zeta](4[zeta]^3 - 60[sigma]2.[zeta] - 140[sigma]3)^(-1/2),
_/
regarded as a branch of Integral Calculus, that the doubly periodic
functions arose. As in the familiar case
_
/ [zeta]
z = | (1 - [zeta]^2)^(-1/2) d[zeta],
_/ 0
where [zeta] = sin z, it has proved finally to be simpler to regard
[zeta] as a function of z. We shall come to the other point of view
below, under S 20, _Elliptic Integrals_.
To prove that any doubly periodic function F(z) with periods [omega],
[omega]', having poles at the points z = a1, ... z = a_m of a
parallelogram, these being, for simplicity of explanation, supposed to
be all of the first order, is rationally expressible in terms of
[phi](z) and [f](z), and we proceed as follows:--