e1 = (½ω)−2 + Σ′ {[(m + ½) ω + m′ω′]−2 − [mω + m′ω′]−2},
and a similar equation for e3, where the summation refers to all integer values of m and m′ other than the one pair m = 0, m′ = 0. This, with similar results, has led to the consideration of functions of the complex ratio ω′/ω.
It is easy to see that the series for ℜ(u), u−2 + Σ′[(u + mω + m′ω′)2 − (mω + m′ω′)2], is unaffected by replacing ω, ω′ by two quantities Ω, Ω′ equal respectively to pω + qω′, p′ω′ + q′ω′, where p, q, p′, q′ are any integers for which pq′ − p′q = ±1; further it can be proved that all substitutions with integer coefficients Ω = pω + qω′, Ω′ = p′ω + q′ω′, wherein pq′ − p′q = 1, can be built up by repetitions of the two particular substitutions (Ω = −ω′, Ω′ = ω), (Ω = ω, Ω′ = ω + ω′). Consider the function of the ratio ω′/ω expressed by
h = −ℜ (½ω′) / ℜ(½ω);
it is at once seen from the properties of the function ℜ(u) that by the two particular substitutions referred to we obtain the corresponding substitutions for h expressed by
h′ = 1/h, h′ = 1 − h;
thus, by all the integer substitutions Ω = pω + qω′, Ω′ = p′ω + q′ω′, in which pq′ − p′q = 1, the function h can only take one of the six values h, 1/h, 1 − h, 1/(1 − h), h/(h − 1), (h − 1)/h, which are the roots of an equation in θ,
| (1 − θ + θ2)3 | = | (1 − h + h2)3 | ; |
| θ2(1 − θ)2 | h2(1 − h)2 |
the function of τ, = ω′/ω, expressed by the right side, is thus unaltered by every one of the substitutions τ′ = (p′ + q′τ / p + qτ), wherein p, q, p′, q′ are integers having pq′ − p′q = 1. If the imaginary part σ, of τ, which we may write τ = ρ + iσ, is positive, the imaginary part of τ′, which is equal to σ(pq′ − p′q)/[(p + qρ)2 + q2σ2], is also positive; suppose σ to be positive; it can be shown that the upper half of the infinite plane of the complex variable τ can be divided into regions, all bounded by arcs of circles (or straight lines), no two of these regions overlapping, such that any substitution of the kind under consideration, τ′ = (p′ + q′τ)/(p + qτ) leads from an arbitrary point τ, of one of these regions, to a point τ′ of another; taking τ = ρ + iσ, one of these regions may be taken to be that for which −½ < ρ < ½, ρ2 + σ2 > 1, together with the points for which ρ is negative on the curves limiting this region; then every other region is obtained from this so-called fundamental region by one and only one of the substitutions τ = (p′ + q′τ)/(p + qτ), and hence by a definite combination of the substitutions τ′ = −1/τ, τ′ = 1 + τ. Upon the infinite half plane of τ, the function considered above,
| z(τ) = 4⁄27 | [ℜ2 (½ω) + ℜ (z(½ω) ℜ (½ω′) + ℜ2 (½ω′)]3 | |
| ℜ2 (½ω) ℜ2 (½ω′) [ℜ (½ω) + ℜ (½ω′]2 |