Part 28
Before passing to this it may be convenient to make here a few remarks as to the periodicity of (single valued) monogenic functions. To say that [f](z) is periodic is to say that there exists a constant [omega] such that for every point z of the interior of the region of existence of [f](z) we have [f](z + [omega]) = [f](z). This involves, considering all existing periods [omega] = [rho] + i[sigma], that there exists a lower limit of [rho]^2 + [sigma]^2 other than zero; for otherwise all the differential coefficients of [f](z) would be zero, and [f](z) a constant; we can then suppose that not both [rho] and [sigma] are numerically less than [epsilon], where [epsilon] > [sigma]. Hence, if g be any real quantity, since the range (-g, ... g) contains only a finite number of intervals of length [epsilon], and there cannot be two periods [omega] = [rho] + i[sigma] such that [mu][epsilon] =< [rho] < ([mu] + 1)[epsilon], [nu][epsilon] =< [sigma] < ([nu] + 1)[epsilon], where [mu], [nu] are integers, it follows that there is only a finite number of periods for which both [rho] and [sigma] are in the interval (-g ... g). Considering then all the periods of the function which are real multiples of one period [omega], and in particular those periods [lambda][omega] wherein 0 < [lambda] =< 1, there is a lower limit for [lambda], greater than zero, and therefore, since there is only a finite number of such periods for which the real and imaginary parts both lie between -g and g, a least value of [lambda], say [lambda]0. If [Omega] = [lambda]0[omega] and [lambda] = M[lambda]0 + [lambda]', where M is an integer and 0 [< = ] [lambda]' < [lambda]0, any period [lambda][omega] is of the form M[Omega] + [lambda]'[omega]; since, however, [Omega], M[Omega] and [lambda][omega] are periods, so also is [lambda]'[omega], and hence, by the construction of [lambda]0, we have [lambda]' = 0; thus all periods which are real multiples of [omega] are expressible in the form M[Omega] where M is an integer, and [Omega] a period.
If beside [omega] the functions have a period [omega]' which is not a real multiple of [omega], consider all existing periods of the form [mu][omega] + [nu][omega]' wherein [mu], [nu] are real, and of these those for which 0 [< = ] [mu] =< 1, 0 < [nu] =< 1; as before there is a least value for [nu], actually occurring in one or more periods, say in the period [Omega]' = [mu]0[omega] + [nu]0[omega]'; now take, if [mu][omega] + [nu][omega]' be a period, [nu] = N'[nu]0 + [nu]', where N' is an integer, and 0 =< [nu]' < [nu]0; thence [mu][omega] + [nu][omega]' = [mu][omega] + N'([Omega]' - [mu]0[omega]) + [nu]'[omega]'; take then [mu] - N[mu]0 = N[lambda]0 + [lambda]', where N is an integer and [lambda]0 is as above, and 0 =< [lambda]' < [lambda]0; we thus have a period N[Omega] + N'[Omega]' + [lambda]'[omega] + [nu]'[omega]', and hence a period [lambda]'[omega] + [nu]'[omega]', wherein [lambda]' < [lambda]0, [nu]' < [nu]0; hence [nu]' = 0 and [lambda]' = 0. All periods of the form [mu][omega] + [nu][omega]' are thus expressible in the form N[Omega] + N'[Omega]', where [Omega], [Omega]' are periods and N, N' are integers. But in fact any complex quantity, P + iQ, and in particular any other possible period of the function, is expressible, with [mu], [nu] real, in the form [mu][omega] + [nu][omega]'; for if [omega] = [rho] + i[sigma], [omega]' = [rho]' + i[sigma]', this requires only P = [mu][rho] + [nu][rho]', Q = [mu][sigma] + [nu][sigma]', equations which, since [omega]'/[omega] is not real, always give finite values for [mu] and [nu].
It thus appears that if a single valued monogenic function of z be periodic, either all its periods are real multiples of one of them, and then all are of the form M[Omega], where [Omega] is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form N[Omega] + N'[Omega]', where [Omega], [Omega]' are periods, and N, N' are integers. In the former case, putting [zeta] = 2[pi]iz/[Omega], and the function [f](z) = [phi]([zeta]), the function [phi]([zeta]) has, like exp ([zeta]), the period 2[pi]i, and if we take t = exp([zeta]) or [zeta] = [lambda](t) the function is a single valued function of t. If then in particular [f](z) is an integral function, regarded as a function of t, it has singularities only for t = 0 and t = [oo], and may be expanded in the form [Sigma](-[oo] to [oo]) a_nt^n.
Taking the case when the single valued monogenic function has two periods [omega], [omega]' whose ratio is not real, we can form a network of parallelograms covering the plane of z whose angular points are the points c + m[omega] + m'[omega]', wherein c is some constant and m, m' are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of which the function is at all capable occur for points of this primary parallelogram, any point, z', of the plane being, as it is called, _congruent_ to a definite point, z, of the primary parallelogram, z' - z being of the form m[omega] + m'[omega]', where m, m' are integers. Such a function cannot be an integral function, since then, if, in the primary parallelogram |[f](z)| < M, it would also be the case, on a circle of centre the origin and radius R, that |[f](z)| < M, and therefore, if [Sigma]a_nz^n be the expansion of the function, which is valid for an integral function for all finite values of z, we should have |a_n| < MR^(-n), which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z.
We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. Supposing the constant c used in naming the corners of the parallelograms so chosen that no pole falls on the perimeter of a parallelogram, it is clear that the integral 1/(2[pi]i) [int][f](z)dz round the perimeter of the primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter points as z, z + [omega] (or as z, z + [omega]') are mutually destructive. This integral is, however, equal to the sum of the residues of [f](z) at the poles interior to the parallelogram. Which sum is therefore zero. There cannot therefore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (z - a)^(-2) + (power series in z - a).
Considering next the function [phi](z) = [[f](z)]^(-1) d[f](z)/dz, it is easily seen that an ordinary point of [f](z) is an ordinary point of [phi](z), that a zero of order m for [f](z) in the neighbourhood of which [f](z) has a form, (z - a)^m multiplied by a power series, is a pole of [phi](z) of residue m, and that a pole of [f](z) of order n is a pole of [phi](z) of residue -n; manifestly [phi](z) has the two periods of [f](z). We thus infer, since the sum of the residues of [phi](z) is zero, that for the function [f](z), the sum of the orders of its vanishing at points belonging to one parallelogram, [Sigma]m, is equal to the sum of the orders of its poles, [Sigma]n; which is briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function [f](z) - A, where A is an arbitrary constant, we have the result, that the function [f](z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is proved above, every conceivable complex value does arise as a value for the doubly periodic function [f](z) in any one of its parallelograms, and in fact at least twice. The number of times it arises is called the _order_ of the function; the result suggests a property of rational functions.
Consider further the integral [int] z [f]'(z)/[f](z) dz, where [f]'(z) = d[f](z)/dz taken round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as z and z + [omega] is of the form -[omega] [int] z [f]'(z)/[f](z) dz, which, as z increases from z0 to z0 + [omega]', gives, if [lambda] denote the generalized logarithm, - [omega]{[lambda][[f](z0 + [omega]')] - [lambda][[f](z0)]}, that is, since [f](z0 + [omega]') = [f](z0), gives 2[pi]iN[omega], where N is an integer; similarly the result of the integration along the other two opposite sides is of the form 2[pi]iN'[omega]', where N' is an integer. The integral, however, is equal to 2[pi]i times the sum of the residues of z[f]'(z)/[f](z) at the poles interior to the parallelogram. For a zero, of order m, of [f](z) at z = a, the contribution to this sum is 2[pi]ima, for a pole of order n at z = b the contribution is -2[pi]inb; we thus infer that [Sigma]ma - [Sigma]nb = N[omega] + N'[omega]'; this we express in words by saying that the sum of the values of z where [f](z) = 0 within any parallelogram is equal to the sum of the values of z where [f](z) = [oo] save for integral multiples of the periods. By considering similarly the function [f](z) - A where A is an arbitrary constant, we prove that each of these sums is equal to the sum of the values of z where the function takes the value A in the parallelogram.
We pass now to the construction of a function having two arbitrary periods [omega], [omega]' of unreal ratio, which has a single pole of the second order in any one of its parallelograms.
For this consider first the network of parallelograms whose corners are the points [Omega] = m[omega] + m'[omega]', where m, m' take all positive and negative integer values; putting a small circle about each corner of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by z0, z0 + [omega], z0 + [omega] + [omega]', z0 + [omega]'; we shall denote z0, z0 + [omega] by A0, B0; this parallelogram [Pi]0 is surrounded by eight other parallelograms, forming with [Pi]0 a larger parallelogram [Pi]1, of which one side, for instance, contains the points z0 - [omega] - [omega]', z0 - [omega]', z0 - [omega]' + [omega], z0 - [omega]' + 2[omega], which we shall denote by A1, B1, C1, D1. This parallelogram [Pi]1 is surrounded by sixteen of the original parallelograms, forming with [Pi]1 a still larger parallelogram [Pi]2 of which one side, for instance, contains the points z0 - 2[omega] - 2[omega]', z0 - [omega] - 2[omega]', z0 - 2[omega]', z0 + [omega] - 2[omega]', z0 + 2[omega] - 2[omega]', z0 + 3[omega] - 2[omega]', which we shall denote by A2, B2, C2, D2, E2, F2. And so on. Now consider the sum of the inverse cubes of the distances of the point P from the corners of all the original parallelograms. The sum will contain the terms
1 / 1 1 1 \ / 1 1 1 \ S0 = ----- + ( ----- + ----- + ----- ) + ( ----- + ----- + ... + ----- ) + ... PA0^3 \PA1^3 PB1^3 PC1^3/ \PA2^3 PB2^3 PE2^3/
and three other sets of terms, each infinite in number, formed in a similar way. If the perpendiculars from P to the sides A0B0, A1B1C1, A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0 is at most equal to
1 3 5 2n + 1 --- + --------- + ---------- + ... + ---------- + ... p^3 (p + q)^3 (p + 2q)^3 (p + nq)^3
of which the general term is ultimately, when n is large, in a ratio of equality with 2q^(-3)n^(-2), so that the series S0 is convergent, as we know the sum [Sigma]n^(-2) to be; this assumes that p[/ = ]0; if P be on A0B0 the proof for the convergence of S0 - 1/PA0^3, is the same. Taking the three other sums analogous to S0 we thus reach the result that the series
[phi](z) = -2[Sigma](z - [Omega])^(-3),
where [Omega] is m[omega] + m'[omega]', and m, m' are to take all positive and negative integer values, and z is any point outside small circles described with the points [Omega] as centres, is _absolutely convergent_. Its sum is therefore independent of the order of its terms. By the nature of the proof, which holds for all positions of z outside the small circles spoken of, the series is also clearly _uniformly convergent_ outside these circles. Each term of the series being a monogenic function of z, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the periods [omega], [omega]'; for [phi](z + [omega]) is the same sum as [phi](z) with the terms in a slightly different order. Thus [phi](z + [omega]) = [phi](z) and [phi](z + [omega]') = [phi](z).
Consider now the function _ _ _ 1 / z | 2 | [f](z) = --- + | | [phi](z) + -- | dz, z^2 _/ 0 |_ z^3 _|
where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this gives
d[f](z) ------- = [phi](z) dz
and _ _ 1 | 1 1 | [f](z) = --- + [Sigma]' | -------------- - -------- |, z^2 |_ (z - [Omega])^2 [Omega]^2 _|
wherein [Sigma]' is a sum excluding the term for which m = 0 and m' = 0. Hence [f](z + [omega]) - [f](z) and [f](z + [omega]') - [f](z) are both independent of z. Noticing, however, that, by its form, [f](z) is an even function of z, and putting z = -1/2[omega], z = -1/2[omega]' respectively, we infer that also [f](z) has the two periods [omega] and [omega]'. In the primary parallelogram [Pi]0, however, [f](z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z^(-2) + (power series in z). Thus [f](z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.
It can be shown that any single valued meromorphic function of z with [omega] and [omega]' as periods can be expressed rationally in terms of [f](z) and [phi](z), and that [[phi](z)]^2 is of the form 4[[f](z)]^3 + A[f](z) + B, where A, B are constants.
To prove the last of these results, we write, for |z| < |[Omega]|,
1 1 2z 3z^2 --------------- - --------- = --------- + --------- + ..., (z - [Omega])^2 [Omega]^2 [Omega]^3 [Omega]^4
and hence, if [Sigma]'[Omega]^(-2n) = [sigma]_n, since [Sigma]'[Omega]^(-(2n - 1)) = 0, we have, for sufficiently small z greater than zero,
[f](z) = z^(-2) + 3[sigma]2.z^2 + 5[sigma]3.z^4 + ...
and
[phi](z) = -2z^(-3) + 6[sigma]2.z + 20[sigma]3.z^3 + ...;
using these series we find that the function
F(z) = [[phi](z)]^2 - 4[[f](z)]^3 + 60[sigma]2[f](z) + 140[sigma]3
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:--
Consider the expression
([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,