([zeta], 1)_m + [eta]([zeta], 1)_(m - 2)
[Phi](z) = ------------------------------------------
([zeta]- A1)([zeta] - A2)...([zeta] - A_m)
where A_s = [f](a_s), [zeta] is an abbreviation for [f](z) and [eta]
for [phi](z), and ([zeta], 1)_m, ([zeta], 1)_(m - 2), denote integral
polynomials in [zeta], of respective orders m and m - 2, so that there
are 2m unspecified, homogeneously entering, constants in the
numerator. It is supposed that no one of the points a1, ... a_m is one
of the points m[omega] + m'[omega]' where f(z) = [oo]. The function
[Phi](z) is a monogenic function of z with the periods [omega],
[omega]', becoming infinite (and having singularities) only when (1)
[zeta] = [oo] or (2) one of the factors [zeta] - A_s is zero. In a
period parallelogram including z = 0 the first arises only for z = 0;
since for [zeta] = [oo], [eta] is in a finite ratio to [zeta]^(3/2);
the function [Phi](z) for [zeta] = [oo] is not infinite provided the
coefficient of [zeta]^m in ([zeta], 1)_m is not zero; thus [Phi](z) is
regular about z = 0. When [zeta] - A_s = 0, that is [f](z) = f(a_s),
we have z = [+-]a_s + m[omega] + m'[omega]', and no other values of z, m
and m' being integers; suppose the unspecified coefficients in the
numerator so taken that the numerator vanished to the first order in
each of the m points -a1, -a2, ... -a_m; that is, if [phi](a_s) = B_s,
and therefore [phi](-a_s) = -B_s, so that we have the m relations
(A_s, 1)_m - B_s(A_s, 1)_(m - 2) = 0;
then the function [Phi](z) will only have the m poles a1, ... a_m.
Denoting further the m zeros of F(z) by a1', ... a_m', putting
[f](a_s') = A_s', [phi](a_s') = B_s', suppose the coefficients of the
numerator of [Phi](z) to satisfy the further m-1 conditions
(A_s', 1)_m + B_s'(A_s',1)_(m - 2) = 0
for s = 1, 2, ... (m - 1). The ratios of the 2m coefficients in the
numerator of [Phi](z) can always be chosen so that the m + (m - 1)
linear conditions are all satisfied. Consider then the ratio
F(z)/[Phi](z);
it is a doubly periodic function with no singularity other than the
one pole a_m'. It is therefore a constant, the numerator of [Phi](z)
vanishing spontaneously in a_m'. We have
F(z) = A[Phi](z),
where A is a constant; by which F(z) is expressed rationally in terms
of [f](z) and [phi](z), as was desired.
When z = 0 is a pole of F(z), say of order r, the other poles, each of
the first order, being a1, ... a_m, similar reasoning can be applied
to a function
([zeta], 1)_h + [eta]([zeta], 1)_k
----------------------------------,
([zeta] - A1)...([zeta] - A_m)
where h, k are such that the greater of 2h - 2m, 2k + 3 - 2m is equal
to r; the case where some of the poles a1, ... a_m are multiple is to
be met by introducing corresponding multiple factors in the
denominator and taking a corresponding numerator. We give a solution
of the general problem below, of a different form.
One important application of the result is the theorem that the
functions [f](z + t), [phi](z + t), which are such doubly periodic
function of z as have been discussed, can each be expressed, so far as
they depend on z, rationally in terms of [f](z) and [phi](z), and
therefore, so far as they depend on z and t, rationally in terms of
[f](z), [f](t), [phi](z) and [phi](t). It can in fact be shown, by
reasoning analogous to that given above, that
_ _
| [phi](z) - [phi](t) |^2
[f](z + t) + [f](z) + [f](t) = 1/4 | ------------------- |.
|_ [f](z) - [f](t) _|
This shows that if F(z) be any single valued monogenic function which
is doubly periodic and of meromorphic character, then F(z + t) is an
algebraic function of F(z) and F(t). Conversely any single valued
monogenic function of meromorphic character, F(z), which is such that
F(z + t) is an algebraic function of F(z) and F(t), can be shown to be
a doubly periodic function, or a function obtained from such by
degeneration (in virtue of special relations connecting the
fundamental constants).
The functions [f](z), [phi](z) above are usually denoted by RN(z),
RN'(z); further the fundamental differential equation is usually
written
(RN'z)^2 = 4(RNz)^3 - g2RNz - g3,
and the roots of the cubic on the right are denoted by e1, e2, e3; for
the odd function, RN'z, we have, for the congruent arguments
-1/2[omega]and 1/2[omega], RN'(1/2[omega]) = -RN'(-1/2[omega]) =
-RN'(1/2[omega]), and hence RN'(1/2[omega]) = 0; hence we can take e1
= RN(1/2[omega]), e2 = RN(1/2[omega] + 1/2[omega]'), e3 =
RN(1/2[omega]). It can then be proved that [RN(z) - e1][RN(z +
1/2[omega]) - e1] = (e1 - e2)(e1 - e3), with similar equations for the
other half periods. Consider more particularly the function RN(z) -
e1; like RN(z) it has a pole of the second order at z = 0, its
expansion in its neighbourhood being of the form z^(-2)(1 - e1z^2 +
Az^4 + ...); having no other pole, it has therefore either two zeros,
or a double zero in a period parallelogram ([omega], [omega]'). In
fact near its zero 1/2[omega] its expansion is (x - 1/2[omega])
RN'(1/2[omega]) + 1/2(z - 1/2[omega])^2 RN"(1/2[omega]) + ...; we have
seen that RN'(1/2[omega]) = 0; thus it has a zero of the second order
wherever it vanishes. Thus it appears that the square root [RN(z) -
e1]^1/2, if we attach a definite sign to it for some particular value
of z, is a single valued function of z; for it can at most have two
values, and the only small circuits in the plane which could lead to
an interchange of these values are those about either a pole or a
zero, neither of which, as we have seen, has this effect; the function
is therefore single valued for any circuit. Denoting the function, for
a moment, by [f]1(z), we have [f]1(z + [omega]) = [+-][f]1(z), [f]1(z
+ [omega]') = [+-][f]1(z); it can be seen by considerations of
continuity that the right sign in either of these equations does not
vary with z; not both these signs can be positive, since the function
has only one pole, of the first order, in a parallelogram ([omega],
[omega]'); from the expansion of [f]1(z) about z = 0, namely z^(-1) (1
- 1/2e1z^2 + ...), it follows that [f]1(z) is an odd function, and
hence [f]1(-1/2[omega]') = -[f]1(1/2[omega]'), which is not zero since
[[f]1(1/2[omega]')]^2 = e3 - e1, so that we have [f]1(z + [omega]') =
-[f]1(z); an equation f1(z + [omega]) = -[f]1(z) would then give
[f]1(z + [omega] + [omega]') = [f]1(z), and hence [f]1(1/2[omega] +
1/2[omega]') = [f]1(-1/2[omega] - 1/2[omega]'), of which the latter is
-[f]1(1/2[omega] + 1/2[omega]'); this would give [f]1(1/2[omega] +
1/2[omega]') = 0, while [[f]1 (1/2[omega] + 1/2[omega]')]^2 = e2 - e1.
We thus infer that [f]1(z + [omega]) = [f]1(z), [f]1(z + [omega]') =
-[f]1(z), [f]1(z + [omega] + [omega]') = -[f]1(z). The function
[f]1(z) is thus doubly periodic with the periods [omega] and
2[omega]'; in a parallelogram of which two sides are [omega] and
2[omega]' it has poles at z = 0, z = [omega]' each of the first order,
and zeros of the first order at z = 1/2[omega], z = 1/2[omega] +
[omega]'; it is thus a doubly periodic function of the second order
with two different poles of the first order in its parallelogram
([omega], 2[omega]'). We may similarly consider the functions [f]2(z)
= [RN(z) - e2]^1/2, [f]3(z) = [RN(z) - e3]^1/2; they give
[f]2(z + [omega] + [omega]') = [f]2(z), [f]2(z + [omega]) = -[f]2(z), [f]2(z + [omega]') = -[f]2(z),
[f]3(z + [omega]') = [f]3z, [f]3(z + [omega]) = -[f]3(z), [f]3(z + [omega] + [omega]') = -[f]3(z).
Taking u = z(e1 - e3)^1/2, with a definite determination of the
constant (e1 - e3)^1/2, it is usual, taking the preliminary signs so
that for z = 0 each of z[f]1(z), z[f]2(z), z[f]3(z) is equal to + 1,
to put
(e1 - e3)^1/2 [f]1(z) f2(z)
sn(u) = -------------, cn(u) = -------, dn(u) = -----,
[f]3(z) [f]3(z) f3(z)
k^2 = (e2 - e3)/(e1 - e3), K = 1/2[omega](e1 - e3)^1/2, iK' = 1/2[omega]'(e1 - e3)^1/2;
thus sn(u) is an odd doubly periodic function of the second order with
the periods 4K, 2iK, having poles of the first order at u = iK', u =
2K + iK', and zeros of the first order at u = 0, u = 2K; similarly
cn(u), dn(u) are even doubly periodic functions whose periods can be
written down, and sn^2(u) + cn^2(u) = 1, k^2sn^2(u) + dn^2(u) = 1; if
x = sn(u) we at once find, from the relations given here, that
du
-- = [(1 - x^2) (1 - k^2x^2)]^(-1/2);
dx
if we put x = sin[phi] we have
du
------ = [1 - k^2sin^2 [phi]]^(-1/2),
d[phi]
and if we call [phi] the amplitude of u, we may write [phi] = am(u), x
= sin.am(u), which explains the origin of the notation sn(u).
Similarly cn(u) is an abbreviation of cos.am(u), and dn(u) of
[Delta]am(u), where [Delta]([phi]) meant (1 - k^2sin^2 [phi])^1/2. The
addition equation for each of the functions [f]1(z), [f]2(z), [f]3(z)
is very simple, being
/(Pd) (Pd) \ [f](z) + [f](t) [f](z)[f]'(t) - [f](t)[f]'(z)
[f](z + t) = 1/2( ----- + ----- ) log --------------- = -----------------------------,
\(Pd)z (Pd)i/ [f](z) - [f](t) [f]^2(z) - [f]^2(t)
where f1'(z) means d[f]1(z)/dz, which is equal to -[f]2(z).[f]3(z),
and [f]^2(z) means [[f](z)]^2. This may be verified directly by
showing, if R denote the right side of the equation, that (Pd)R/(Pd)z
= (Pd)R/(Pd)t; this will require the use of the differential equation
[[f]1'^(z)]^2 = [[f]1^2(z) + e1 - e2] [[f]1^2(z) + e1 - e3],
and in fact we find
/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) + [f](t)] = [f]^2(z) - [f]^2(t) =
\(Pd)z^2 dt^2 /
/ (Pd)^2 (Pd)^2\
( ------- - ------ ) log [[f](z) - [f](t)];
\(Pd)z^2 dt^2 /
hence it will follow that R is a function of z + t, and R is at once
seen to reduce to [f](z) when t = 0. From this the addition equation
for each of the functions sn(u), cn(u), dn(u) can be deduced at once;
if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1),
sn(u2), cn(u2), dn(u2), they can be put into the forms
sn(u1 + u2) = (s1c2d2 + s2c1d1)/D,
cn(u1 + u2) = (c1c2 - s1s2d1d2)/D,
dn(u1 + u2) = (d1d2 - k^2s1s2c1c2)/D,
where