whereby a definite power series in u, valid for sufficiently small value
of u, is found for t, and hence a definite power series for x, of the form
Let this expression be valid for 0 < |u| < R, and the function defined thereby, which has a pole of the second order for u=0, be denoted by φ(u). In the range in question it is single valued and satisfies the differential equation
in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an odd function, the sign attached to y in the original integral for x = ∞ is immaterial. Now for any two values u, v in the range in question consider the function
| F(u, v) = 14 [ | φ′(u) − φ′(v) | ]2 − φ(u) − φ(v); |
| φ(u) − φ(v) |
it is at once seen, from the differential equation, to be such that ∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing |u + v| < R we infer therefore, by putting v = 0, that
| φ(u + v) = 14 [ | φ′(u) − φ′(v) | ]2 − φ(u) − φ(v). |
| φ(u) − φ(v) |
By repetition of this equation we infer that if u1, ... un be any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then φ(u1 + ... + un) is a rational function of the 2n functions φ(us), φ′(us); and hence, if |u| < R, that
| φ(u) = H [ φ ( | u | ), φ′ ( | u | ) ], |
| n | n |
where H is some rational function of the arguments φ(u/n), φ′(u/n). In fact, however, so long as |u/n| < R, each of the functions φ(u/n), φ′(u/n) is single valued and without singularity save for the pole at u=0; and a rational function of single valued functions, each of which has no singularities other than poles in a certain region, is also a single valued function without singularities other than poles in this region. We infer, therefore, that the function of u expressed by H [φ(u/n), φ′(u/n)] is single valued and without singularities other than poles so long as |u| < nR; it agrees with φ(u) when |u| < R, and hence furnishes a continuation of this function over the extended range |u| < nR. Moreover, from the method of its derivation, it satisfies the differential equation [φ′(u)]2 = 4[φ(u)]3 − g2φ(u) − g3. This equation has therefore one solution which is a single valued monogenic function with no singularities other than poles for any finite part of the plane, having in particular for u = 0, a pole of the second order; and the method adopted for obtaining this near u=0 shows that the differential equation has no other such solution. This, however, is not the only solution which is a single valued meromorphic function, a the functions φ(u + α), wherein α is arbitrary, being such. Taking now any range of values of u, from u = 0, and putting for any value of u, x = φ(u), y = −φ′(u), so that y2=4x3-g2x-g3, we clearly have
| u = ∫ (∞)(x, y ) | dx | ; |
| y |
conversely if x0 = φ(u0), y0 = −φ′(u0) and ξ, η be any values satisfying η2 = 4ξ2 − g2ξ − g3, which are sufficiently near respectively to x0, y0, while v is defined by
| v − u0 = − ∫ (ξ, η)(x0, y0) | dξ | , |
| η |
then ξ, η are respectively φ(v) and −φ′(v); for this equation leads to an expansion for ξ − x0 in terms of v = u0 and only one such expansion, and this is obtained by the same work as would be necessary to expand φ(v) when v is near to u0; the function φ(u) can therefore be continued by the help of this equation, from v = u0, provided the lower limit of |ξ − x0| necessary for the expansions is not zero in the neighbourhood of any value (x0, y0). In fact the function φ(u) can have only a finite number of poles in any finite part of the plane of u; each of these can be surrounded by a small circle, and in the portion of the finite part of the plane of u which is outside these circles, the lower limit of the radii of convergence of the expansions of φ(u) is greater than zero; the same will therefore be the case for the lower limit of the radii |ξ − x0| necessary for the continuations spoken of above provided that the values of (ξ, η) considered do not lead to infinitely increasing values of v; there does not exist, however, any definite point (ξ0, η0) in the neighbourhood of which the integral ∫ (ξ, η)(x0, y0) dξ/η increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if (ξ, η) be any point, where η2 = 4ξ3 − g2ξ − g3, and v be defined by
| v = ∫ (∞)(ξ, η) | dx | , |
| y |
then ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function φ(u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be
| ω = 2 ∫ ∞e1 | dx | , ω′ = 2 ∫ ∞e3 | dx | ; |
| y | y |
these quantities cannot therefore have a real ratio, for else, being periods of a monogenic function, they would, as we have previously seen, be each integral multiples of another period; there would then be a closed path for (x, y ), starting from an arbitrary point (x0, y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3, 0), (∞, ∞), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function φ(u) agrees with the function ℜ(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14, Doubly Periodic Functions.
§ 21. Modular Functions.—One result of the previous theory is the remarkable fact that if
| ω = 2 ∫ ∞e1 | dx | , ω′ = 2 ∫ ∞e3 | dx | ; |
| y | y |
where y2 = 4(x − e1) (x − e2) (x − e3), then we have
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
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
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 −12 < ρ < 12, 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) = 427 | [ℜ2 (12ω) + ℜ (z(12ω) ℜ (12ω′) + ℜ2 (12ω′)]3 | |
| ℜ2 (12ω) ℜ2 (12ω′) [ℜ (12ω) + ℜ (12ω′]2 |
is a single valued monogenic function, whose only essential singularities are the points τ′ = (p′ + q′τ)/(p + qτ) for which τ = ∞, namely those for which τ′ is any real rational value; the real axis is thus a line over which the function z(τ) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, z(τ) has thus only the single essential singularity, r = ρ + iσ, where σ = ∞; in this fundamental region z(τ) takes any assigned complex value just once, the relation z(τ′) = z(τ) requiring, as can be shown, that τ′ is of the form (p′ + q′τ)/(p + qτ), in which p, q, p′, q′ are integers with pq′ − p′q = 1; the function z(τ) has thus a similar behaviour in every other of the regions. The division of the plane into regions is analogous to the division of the plane, in the case of doubly periodic functions, into parallelograms; in that case we considered only functions without essential singularities, and in each of the regions the function assumed every complex value twice, at least. Putting, as another function of τ, J(τ) = z(τ) [z(τ) − 1], it can be shown that J(τ) = 0 for τ = exp (23πi), that J(τ) = 1 for τ = i, these being values of τ on the boundary of the fundamental region; like z(τ) it has an essential singularity for τ = ρ + iσ, σ = + ∞. In the
