To prove the last of these results, we write, for |z| < |Ω|,
| 1 | − | 1 | = | 2z | + | 3z2 | + ..., |
| (z − Ω)2 | Ω2 | Ω3 | Ω4 |
and hence, if Σ′Ω−2n = σn, since Σ′Ω−(2n−1) = 0, we have, for sufficiently small z greater than zero,
and
using these series we find that the function
contains no negative powers of z, being equal to a power series in z2 beginning with a term in z2. The function F(z) is, however, doubly periodic, with periods ω, ω′, and can only be infinite when either ƒ(z) or φ(z) is infinite; this follows from its form in ƒ(z) and φ(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 ƒ(z) = ζ and φ(z) = dζ/dz we see that
| dz | = (4ζ3 − 60σ2ζ − 140σ3)−1/2. |
| dζ |
Historically it was in the discussion of integrals such as
regarded as a branch of Integral Calculus, that the doubly periodic functions arose. As in the familiar case
where ζ = sin z, it has proved finally to be simpler to regard ζ as a function of z. We shall come to the other point of view below, under § 20, Elliptic Integrals.
To prove that any doubly periodic function F(z) with periods ω, ω′, having poles at the points z = a1, ... z = am of a parallelogram, these being, for simplicity of explanation, supposed to be all of the first order, is rationally expressible in terms of φ(z) and ƒ(z), and we proceed as follows:—
Consider the expression
| Φ(z) = | (ζ, 1)m + η(ζ, 1)m−2 |
| (ζ − A1) (ζ − A2)...(ζ − Am) |
where As = ƒ(as), ζ is an abbreviation for ƒ(z) and η for φ(z), and (ζ, 1)m, (ζ, 1)m−2, denote integral polynomials in ζ, 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, ... am is one of the points mω + m′ω′ where ƒ(z) = ∞. The function Φ(z) is a monogenic function of z with the periods ω, ω′, becoming infinite (and having singularities) only when (1) ζ = ∞ or (2) one of the factors ζ-As is zero. In a period parallelogram including z = 0 the first arises only for z = 0; since for ζ = ∞, η is in a finite ratio to ζ32; the function Φ(z) for ζ = ∞ is not infinite provided the coefficient of ζm in (ζ, 1)m is not zero; thus Φ(z) is regular about z = 0. When ζ − As = 0, that is ƒ(z) = ƒ(as), we have z = ±as + mω + m′ω′, 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, ... −am; that is, if φ(as) = Bs, and therefore φ(−as) = −Bs, so that we have the m relations
then the function Φ(z) will only have the m poles a1, ... am. Denoting further the m zeros of F(z) by a1′, ... am′, putting ƒ(as′) = As′, φ(as′) = Bs′, suppose the coefficients of the numerator of Φ(z) to satisfy the further m − 1 conditions
for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the numerator of Φ(z) can always be chosen so that the m + (m − 1) linear conditions are all satisfied. Consider then the ratio
it is a doubly periodic function with no singularity other than the one pole am′. It is therefore a constant, the numerator of Φ(z) vanishing spontaneously in am′. We have
where A is a constant; by which F(z) is expressed rationally in terms of ƒ(z) and φ(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, ... am, similar reasoning can be applied to a function
| (ζ, 1)h + η(ζ, 1)k | , |
| (ζ − A1) ... (ζ − Am) |
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, ... am 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 ƒ(z + t), φ(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 ƒ(z) and φ(z), and therefore, so far as they depend on z and t, rationally in terms of ƒ(z), ƒ(t), φ(z) and φ(t). It can in fact be shown, by reasoning analogous to that given above, that
| ƒ(z + t) + ƒ(z) + ƒ(t) = 14 [ | φ(z) − φ(t) | ] 2. |
| ƒ(z) − ƒ(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 ƒ(z), φ(z) above are usually denoted by ℜ(z), ℜ′(z); further the fundamental differential equation is usually written
and the roots of the cubic on the right are denoted by e1, e2, e3; for the odd function, ℜ′z, we have, for the congruent arguments −12ωand 12ω, ℜ′ (12ω) = −ℜ′ (−12ω) = −ℜ′ (12ω), and hence ℜ′ (12ω) = 0; hence we can take e1 = ℜ (12ω), e2 = ℜ (12ω + 12ω′), e3 = ℜ (12ω). It can then be proved that [ℜ(z) − e1] [ℜ (z + 12ω) − e1] = (e1 − e2) (e1 − e3), with similar equations for the other half periods. Consider more particularly the function ℜ(z) − e1; like ℜ(z) it has a pole of the second order at z = 0, its expansion in its neighbourhood being of the form z−2 (1 − e1z2 + Az4 + ...); having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (ω, ω′). In fact near its zero 12ω its expansion is (x − 12ω) ℜ′ (12ω) + 12(z − 12ω)2 ℜ″ (12ω) + ...; we have seen that ℜ′ (12ω) = 0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [ℜ(z) − e1]12, 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 ƒ1(z), we have ƒ1(z + ω) = ±ƒ1(z), ƒ1(z + ω′) = ±ƒ1(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 (ω, ω′); from the expansion of ƒ1(z) about z = 0, namely z− 1 (1 − 12e1z2 + ...), it follows that ƒ1(z) is an odd function, and hence ƒ1 (−12ω′) = −ƒ1 (12ω′), which is not zero since [ƒ1 (12ω′)]2 = e3 − e1, so that we have ƒ1 (z + ω′) = −ƒ1(z); an equation f1(z + ω) = −ƒ1(z) would then give ƒ1(z + ω + ω′) = ƒ1(z), and hence ƒ1(12ω + 12ω′) = ƒ1(−12ω − 12ω′), of which the latter is −ƒ1(12ω + 12ω′); this would give ƒ1(12ω + 12ω′) = 0, while [ƒ1(12ω + 12ω′)]2 = e2 − e1. We thus infer that ƒ1(z + ω) = ƒ1(z), ƒ1(z + ω′) = −ƒ1(z), ƒ1(z + ω + ω′) = −ƒ1(z). The function ƒ1(z) is thus doubly periodic with the periods ω and 2ω′; in a parallelogram of which two sides are ω and 2ω′ it has poles at z = 0, z = ω′ each of the first order, and zeros of the first order at z = 12ω, z = 12ω + ω′; it is thus a doubly periodic function of the second order with two different poles of the first order in its parallelogram (ω, 2ω′). We may similarly consider the functions ƒ2(z) = [ℜ(z) − e2]12, ƒ3(z) = [ℜ(z) − e3]12; they give
| ƒ2(z + ω + ω′) = ƒ2(z), ƒ2(z + ω) = −ƒ2(z), ƒ2(z + ω′) = −ƒ2(z), | ƒ3(z + ω′) = ƒ3z, ƒ3(z + ω) = −ƒ3(z), ƒ3(z + ω + ω′) = −ƒ3(z). |
Taking u = z (e1 − e3)12, with a definite determination of the constant (e1 − e3)12, it is usual, taking the preliminary signs so that for z = 0 each of zƒ1(z), zƒ2(z), zƒ3(z) is equal to +1, to put
| sn(u) = | (e1 − e3)12 | , cn(u) = | ƒ1(z) | , dn(u) = | f2(z) | , |
| ƒ3(z) | ƒ3(z) | ƒ3(z) |
| k2 = (e2 − e3) / (e1 − e3), K = 12ω (e1 − e3)12, iK′ = 12ω′ (e1 − e3)12; |
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 sn2(u) + cn2(u) = 1, k2sn2(u) + dn2(u) = 1; if x = sn(u) we at once find, from the relations given here, that
| du | = [(1 − x2) (1 − k2x2)]−1/2; |
| dx |
if we put x = sinφ we have
| du | = [1 − k2sin2φ]−1/2, |
| dφ |
and if we call φ the amplitude of u, we may write φ = 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 Δam(u), where Δ(φ) meant (1 − k2sin2φ)12. The addition equation for each of the functions ƒ1(z), ƒ2(z), ƒ3(z) is very simple, being
| ƒ(z + t) = 12 ( | ∂ | + | ∂ | ) log | ƒ(z) + ƒ(t) | = | ƒ(z)ƒ′(t) − ƒ(t)ƒ′(z) | , |
| ∂z | ∂i | ƒ(z) − ƒ(t) | ƒ2(z) − ƒ2(t) |
where ƒ1′(z) means dƒ1(z)/dz, which is equal to −ƒ2(z)·ƒ3(z), and ƒ2(z)
