Page:EB1911 - Volume 11.djvu/332

Rh To prove the last of these results, we write, for |z| < ||,

and hence, if ′ −2n = n, since ′ −(2n−1) = 0, we have, for sufficiently small z greater than zero,

ƒ(z) = z−2 + 32·z2 + 53·z4 + ...

and

(z) = −2z−3 + 62·z + 203·z3 + ...;

using these series we find that the function

F(z) = [(z)]2 − 4[ƒ(z)]3 + 602ƒ(z) + 1403

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, ′, and can only be infinite when either ƒ(z) or (z) is infinite; this follows from its form in ƒ(z) and (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) = and (z) = d/dz we see that

Historically it was in the discussion of integrals such as

∫ d (43 − 602· − 1403)−1/2,

regarded as a branch of Integral Calculus, that the doubly periodic functions arose. As in the familiar case

z = ∫ 0 (1 − 2)−1/2 d,

where = sin z, it has proved finally to be simpler to regard  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 , ′, 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 (z) and ƒ(z), and we proceed as follows:—

Consider the expression

where As = ƒ(as), is an abbreviation for ƒ(z) and  for (z), and (, 1) m, (, 1) m−2 , denote integral polynomials in , 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 + m′′ where ƒ(z) = ∞. The function (z) is a monogenic function of z with the periods, ′, becoming infinite (and having singularities) only when (1) = ∞ or (2) one of the factors -As is zero. In a period parallelogram including z = 0 the first arises only for z = 0; since for = ∞,  is in a finite ratio to undefined; the function (z) for  = ∞ is not infinite provided the coefficient of m in (, 1) m is not zero; thus (z) is regular about z = 0. When − As = 0, that is ƒ(z) = ƒ(as), we have z = ±as + m + m′′, 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 (as) = Bs, and therefore (−as) = −Bs, so that we have the m relations

(As, 1) m − Bs(As, 1) m−2 = 0;

then the function (z) will only have the m poles a1, ... a m. Denoting further the m zeros of F(z) by a1′, ... a m ′, putting ƒ(as′) = As′, (as′) = Bs′, suppose the coefficients of the numerator of (z) to satisfy the further m − 1 conditions

(As′, 1) m + Bs′ (As′, 1) m−2 = 0

for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the numerator of (z) can always be chosen so that the m + (m − 1) linear conditions are all satisfied. Consider then the ratio

F(z) / (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 (z) vanishing spontaneously in a m ′. We have

F(z) = A(z),

where A is a constant; by which F(z) is expressed rationally in terms of ƒ(z) and (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

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 ƒ(z + t), (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 (z), and therefore, so far as they depend on z and t, rationally in terms of ƒ(z), ƒ(t), (z) and (t). It can in fact be shown, by reasoning analogous to that given above, that

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), (z) above are usually denoted by ℜ(z), ℜ′(z); further the fundamental differential equation is usually written

(ℜ′z)2 = 4(ℜz)3 − g2ℜz − g3,

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 −and, ℜ′ = −ℜ′ (−) = −ℜ′ , and hence ℜ′  = 0; hence we can take e1 = ℜ, e2 = ℜ ( + ′), e3 = ℜ. It can then be proved that [ℜ(z) − e1] [ℜ (z + ) − 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 (, ′). In fact near its zero its expansion is (x − ) ℜ′  + (z − )2 ℜ″  + ...; we have seen that ℜ′ = 0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [ℜ(z) − e1]undefined, 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 + ) = ±ƒ1(z), ƒ1(z + ′) = ±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 (, ′); from the expansion of ƒ1(z) about z = 0, namely z − 1 (1 − e1z2 + ...), it follows that ƒ1(z) is an odd function, and hence ƒ1 (−′) = −ƒ1 (′), which is not zero since [ƒ1 (′)]2 = e3 − e1, so that we have ƒ1 (z + ′) = −ƒ1(z); an equation f1(z + ) = −ƒ1(z) would then give ƒ1(z + + ′) = ƒ1(z), and hence ƒ1( + ′) = ƒ1(− − ′), of which the latter is −ƒ1( + ′); this would give ƒ1( + ′) = 0, while [ƒ1( + ′)]2 = e2 − e1. We thus infer that ƒ1(z + ) = ƒ1(z), ƒ1(z + ′) = −ƒ1(z), ƒ1(z + + ′) = −ƒ1(z). The function ƒ1(z) is thus doubly periodic with the periods and 2′; in a parallelogram of which two sides are and 2′ it has poles at z = 0, z = ′ each of the first order, and zeros of the first order at z =, z = + ′; it is thus a doubly periodic function of the second order with two different poles of the first order in its parallelogram (, 2′). We may similarly consider the functions ƒ2(z) = [ℜ(z) − e2]undefined, ƒ3(z) = [ℜ(z) − e3]undefined; they give

Taking u = z (e1 − e3)undefined, with a definite determination of the constant (e1 − e3)undefined, 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

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

if we put x = sin we have

and if we call the amplitude of u, we may write  = 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 am(u), where meant (1 − k2sin2)undefined. The addition equation for each of the functions ƒ1(z), ƒ2(z), ƒ3(z) is very simple, being

where ƒ1′(z) means dƒ1(z)/dz, which is equal to −ƒ2(z)·ƒ3(z), and ƒ2(z)