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]12, [ − | ℜ (12ω) + 2ℜ (12ω′) | ]1/8, |
| ℜ (12ω) − ℜ (12)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
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(12, 13, 0, J). If we introduce also the function of τ given by
| λ = | 2ℜ (12ω′) + ℜ (12ω) | , |
| ℜ (12ω′) − ℜ (12ω) |
we similarly have τ = s(0, 0, 0, λ); this function λ is a single valued function of τ, which is also a modular function, being unaltered by a group of integral substitutions also of the form τ′ = (p′ + q′τ)/(p + qτ), with pq′ − p′q = 1, but with the restriction that p′ and q are even integers, and therefore p and q′ are odd integers. This group is thus a subgroup of the general modular group, and is in fact of the kind called a self-conjugate subgroup. As in the general case this subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the fundamental region, by a particular one of the substitutions of the subgroup. This fundamental region, putting τ = ρ + iσ, may be taken to be that given by −1 < ρ < 1, (ρ + 12)2 + σ2 > 14, (ρ − 12)2 + σ2 > 14, and is built up of six of the regions which arose for the general modular group associated with J(τ). Within this fundamental region, λ takes every complex value just once, except the values λ = 0, 1, ∞, which arise only at the angular points τ = 0, τ = ∞, τ = − 1 and the equivalent point τ = 1; these angular points are essential singularities for the function λ(τ). For λ(τ) as for J(τ), the region of existence is the upper half plane of τ, there being an essential singularity in every length of the real axis, however short.
If, beside the plane of τ, we take a plane to represent the values of λ, the function τ = s(0, 0, 0, λ) being considered thereon, the values of τ belonging to the interior of the fundamental region of the τ-plane considered above, will require the consideration of the whole of the λ-plane taken once with the exception of the portions of the real axis lying between −∞ and 0 and between 1 and +∞, the two sides of the first portion corresponding to the circumferences of the τ-plane expressed by (ρ + 12)2 + σ2 = 14, (ρ − 12)2 + σ2 = 14, while the two sides of the latter portion, for which λ is real and > 1, correspond to the lines of the τ-plane expressed by ρ = ±1. The line for which λ is real, positive and less than unity corresponds to the imaginary axis of the τ-plane, lying in the interior of the fundamental region. All the values of τ = s(0, 0, 0, λ) may then be derived from those belonging to the fundamental region of the τ-plane by making λ describe a proper succession of circuits about the points λ = 0, λ = 1; any such circuit subjects τ to a linear substitution of the subgroup of τ considered, and corresponds to a change of τ from a point of the fundamental region to a corresponding point of one of the other regions.
§ 22. A Property of Integral Functions deduced from the Theory of Modular Functions.—Consider now the function exp(z), for finite values of z; for such values of z, exp(z) never vanishes, and it is impossible to assign a closed circuit for z in the finite part of the plane of z which will make the function λ = exp(z) pass through a closed succession of values in the plane of λ having λ = 0 in its interior; the function s[0, 0, 0, exp(z)], however z vary in the finite part of the plane, will therefore never be subjected to those linear substitutions imposed upon s(0, 0, 0, λ) by a circuit of λ about λ = 0; more generally, if φ(z) be an integral function of z, never becoming either zero or unity for finite values of z, the function λ = φ(z), however z vary in the finite part of the plane, will never make, in the plane of λ, a circuit about either λ = 0 or λ = 1, and s(0, 0, 0, λ), that is s[0, 0, 0, φ(z)], will be single valued for all finite values of z; it will moreover remain finite, and be monogenic. In other words, s[0, 0, 0, φ(z)] is also an integral function—whose imaginary part, moreover, by the property of s(0, 0, 0, λ), remains positive for all finite values of z. In that case, however, exp {is[0, 0, 0, φ(z)]} would also be an integral function of z with modulus less than unity for all finite values of z. If, however, we describe a circle of radius R in the z plane, and consider the greatest value of the modulus of an integral function upon this circle, this certainly increases indefinitely as R increases. We can infer therefore that an integral function φ(z) which does not vanish for any finite value of z, takes the value unity and hence (by considering the function A−1φ(z)) takes every other value for some definite value of z; or, an integral function for which both the equations φ(z) = A, φ(z) = B are unsatisfied by definite values of z, does not exist, A and B being arbitrary constants.
A similar theorem can be proved in regard to the values assumed by the function φ(z) for points z of modulus greater than R, however great R may be, also with the help of modular functions. In general terms it may be stated that it is a very exceptional thing for an integral function not to assume every complex value an infinite number of times.
Another application of modular functions is to prove that the function s(α, β, γ, λ) is a single valued function of τ = s(0, 0, 0, λ); for, putting τ′ = (τ − i)/(τ + i), the values of τ′ which correspond to the singular points λ = 0, 1, ∞ of s(α, β, γ, λ), though infinite in number, all lie on the circumference of the circle |τ′| = 1, within which therefore s(α, β, γ, x) is expressible in a form anτ′n. More generally any monogenic function of λ which is single valued save for circuits of the points λ = 0, 1, ∞, is a single valued function of τ = s(0, 0, 0, λ). Identifying λ with the square of the modulus in Legendre’s form of the elliptical integral, we have τ = iK′/K, where
| K = | dt | , K′ = | dt | ; |
| √[1 − t2] [1 − λt2] | √[1 − t2] [1 − (1 − λ) t2] |
functions such as λ1/4, (1 − λ)1/4, [λ(1 − λ)]1/4, which have only λ = 0, 1, ∞ as singular points, were expressed by Jacobi as power series in q = eiπτ, and therefore, at least for a limited range of values of τ, as single valued functions of τ; it follows by the theorem given that any product of a root of λ and a root of 1 − λ is a single valued function of τ. More generally the differential equation
| x(1 − x) | d2y | + [γ − (α + β + 1)x] | dy | − αβγ = 0 |
| dx2 | dx |
may be solved by expressing both the independent and dependent variables as single valued functions of a single variable τ, the expression for the independent variable being x = λ(τ).
§ 23. Geometrical Applications of Elliptic Functions.—Consider any irreducible algebraic equation rational in x, y , ƒ(x, y ) = 0, of such a form that the equation represents a plane curve of order n with 12n(n − 3) double points; taking upon this curve n − 3 arbitrary fixed points, draw through these and the double points the most general curve of order n − 2; this will intersect
