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 (2⁄3πi), 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 theory of linear differential equations it is important to consider the inverse function τ(J); this is infinitely many valued, having a cycle of three values for circulation of J about J = 0 (the circuit of this point leading to a linear substitution for τ of period 3, such as τ′ = −(1 + τ)−1), having a cycle of two values about J = 1 (the circuit leading to a linear substitution for τ of period 2, such as τ′ = −τ−1), and having a cycle of infinitely many values about J = ∞ (the circuit leading to a linear substitution for τ which is not periodic, such as τ′ = 1 + τ). These are the only singularities for the function τ(J). Each of the functions
| [J(τ)]1/3, [J(τ) − 1]1/2, [ − | ℜ (½ω) + 2ℜ (½ω′) | ]1/8, |
| ℜ (½ω) − ℜ (½)ω′) |
beside many others (see below), is a single valued function of τ, and is expressible without ambiguity in terms of the single valued function of τ,
| η(τ) = exp ( | iπτ | ) Π ∞n=1 [1 − exp (2iπnτ)] = exp ( | iπτ | ) Σ ∞m=−∞ (−1)m exp [(3m2 + m) iπτ]. |
| 12 | 12 |
It should be remarked, however, that η(τ) is not unaltered by all the substitutions we have considered; in fact
η(−τ−1) = (−iτ) ½η (τ), η(1 + τ) = exp (1⁄12 iπ) η(τ).
The aggregate of the substitutions τ′ = (p′ + q′τ)/(p + qτ), wherein p, q, p′, q′ are integers with pq′ − p′q = 1, represents a Group; the function J(τ), unaltered by all these substitutions, is called a Modular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called an Automorphic Function. A rational function, of its variable h, of this character, is the function (1 − h + h2)3 h−2(1 − h)−2 presenting itself incidentally above; and there are other rational functions with a similar property, the group of substitutions belonging to any one of these being, what is a very curious fact, associable with that of the rotations of one of the regular solids, about an axis through its centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already discussed; these, as we have seen, enable us to solve the algebraic equation y2 = 4x3 − g2x − g3 (and in fact many other algebraic equations, see below, under § 23, Geometrical Applications of Elliptic Functions) in terms of single valued functions x = ℜ(u), y = −ℜ′(u). A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.
The modular function J(τ) considered above, unaltered by the group of linear substitutions τ′ = (p′ + q′τ) / (p + qτ), where p, q, p′, q′ are integers with pq′ − p′q = 1, may be taken as the independent variable x of a differential equation of the third order, of the form
| s″′ | − | 3 | ( | s″ | )2 = | 1 − α2 | + | 1 − β2 | + | α2 + β2 − γ2 − 1 | , |
| s′ | 2 | s′ | 2(x − 1)2 | 2x2 | 2x (x − 1) |
where s′ = ds/dx, &c., of which the dependent variable s is equal to τ. A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(α,β,γ, x), we have in fact τ = s(½, 1⁄3, 0, J). If we introduce also the function of τ given by