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
F(z) / Φ(z);
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
F(z) = AΦ(z),
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) = ¼ [ | φ(z) − φ(t) | ] 2. |
| ƒ(z) − ƒ(t) |