Part 32
then [xi] = [phi](v) and [eta] = -[phi]'(v). Thus this equation determines ([xi], [eta]) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function [phi](u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be _ _ / [oo] dx / [oo] dx [omega] = 2| --, [omega]' = 2 | --; _/ e1 y _/ e3 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), ([oo], [oo]), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function [phi](u) agrees with the function RN(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under S 14, _Doubly Periodic Functions_.
S 21. _Modular Functions._--One result of the previous theory is the remarkable fact that if _ _ / [oo] dx / [oo] dx [omega] = 2| --, [omega]' = 2 | --, _/ e1 y _/ e3 y
where y^2 = 4(x - e1) (x - e2) (x - e3), then we have
e1 = (1/2[omega])^(-2) + [Sigma]' {[(m + 1/2)[omega] + m'[omega]']^(-2) - [m[omega] + m'[omega]']^(-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 [omega]'/[omega].
It is easy to see that the series for RN(u), u^(-2) + [Sigma] [(u + m[omega] + m'[omega]')^2-(m[omega] + m'[omega]')^2], is unaffected by replacing [omega], [omega]' by two quantities [Omega], [Omega]' equal respectively to p[omega] + q[omega]', p'[omega]' + q'[omega]', 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 [Omega] = p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', wherein pq' - p'q = 1, can be built up by repetitions of the two particular substitutions ([Omega] = -[omega]', [Omega]' = [omega]), ([Omega] = [omega], [Omega]' = [omega] + [omega]'). Consider the function of the ratio [omega]'/[omega] expressed by
h = -RN (1/2[omega]') / RN(1/2[omega]);
it is at once seen from the properties of the function RN(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 [Omega] = p[omega] + q[omega]', [Omega]' = p'[omega] + q'[omega]', 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 [theta],
(1 - [theta] + [theta]^2)^3 (1 - h + h^2)^3 --------------------------- = ---------------; [theta]^2(1 - [theta])^2 h^2(1 - h)^2
the function of [tau], = [omega]'/[omega], expressed by the right side, is thus unaltered by every one of the substitutions [tau]' = (p' + q'[tau] / p + q[tau]), wherein p, q, p', q' are integers having pq' - p'q = 1. If the imaginary part [sigma], of [tau], which we may write [tau] = [rho] + i[sigma], is positive, the imaginary part of [tau]', which is equal to [sigma](pq' - p'q)/[(p + q[rho])^2 + q^2[sigma]^2], is also positive; suppose [sigma] to be positive; it can be shown that the upper half of the infinite plane of the complex variable [tau] 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, [tau]' = (p' + q'[tau])/(p + q[tau]) leads from an arbitrary point [tau], of one of these regions, to a point [tau]' of another; taking [tau] = [rho] + i[sigma], one of these regions may be taken to be that for which -1/2 < [rho] < 1/2, [rho]^2 + [sigma]^2 > 1, together with the points for which [rho] 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 [tau] = (p' + q'[tau])/(p + q[tau]), and hence by a definite combination of the substitutions [tau]' = -1/[tau], [tau]' = 1 + [tau]. Upon the infinite half plane of [tau], the function considered above,
4 [RN^2(1/2[omega]) + RN(z(1/2[omega]) RN(1/2[omega]') + RN^2 (1/2[omega]')]^3 z([tau]) = -- ---------------------------------------------------------------------------- 27 RN^2(1/2[omega]) RN^2(1/2[omega]') [RN(1/2[omega]) + RN(1/2[omega]')]^2
is a single valued monogenic function, whose only essential singularities are the points [tau]' = (p' + q'[tau])/(p + q[tau]) for which [tau] = [oo], namely those for which [tau]' is any real rational value; the real axis is thus a line over which the function z([tau]) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, z([tau]) has thus only the single essential singularity, r = [rho] + i[sigma], where [sigma] = [oo]; in this fundamental region z([tau]) takes any assigned complex value just once, the relation z([tau]') = z([tau]) requiring, as can be shown, that [tau]' is of the form (p' + q'[tau])/(p + q[tau]), in which p, q, p', q' are integers with pq' - p'q = 1; the function z([tau]) 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 [tau], J([tau]) = z([tau])[z([tau]) - 1], it can be shown that J([tau]) = 0 for [tau] = exp (2/3[pi]i), that J([tau]) = 1 for [tau] = i, these being values of [tau] on the boundary of the fundamental region; like z([tau]) it has an essential singularity for [tau] = [rho] + i[sigma], [sigma] = + [oo]. In the theory of linear differential equations it is important to consider the inverse function [tau](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 [tau] of period 3, such as [tau]' = -(1 + [tau])^(-1)), having a cycle of two values about J = 1 (the circuit leading to a linear substitution for [tau] of period 2, such as [tau]' = -[tau]^(-1)), and having a cycle of infinitely many values about J = [oo] (the circuit leading to a linear substitution for [tau] which is not periodic, such as [tau]' = 1 + [tau]). These are the only singularities for the function [tau](J). Each of the functions _ _ | RN(1/2[omega]) + 2RN(1/2[omega]') |^(1/8) [J([tau])]^(1/3), [J([tau])-1]^1/2, | - --------------------------------- | , |_ RN(1/2[omega]) - RN(1/2)[omega]') _|
beside many others (see below), is a single valued function of [tau], and is expressible without ambiguity in terms of the single valued function of [tau],
/i[pi][tau]\ [oo] [eta]([tau]) = exp( ---------- ) [Pi] [1-exp (2i[pi]n[tau])], \ 12 / n=1
/i[pi][tau]\ _[oo] = exp(----------- ) \ (-1)^m exp [(3m^2 + m) i[pi][tau]]. \ 12 / /_ m = -[oo]
It should be remarked, however, that [eta]([tau]) is not unaltered by all the substitutions we have considered; in fact
[eta](-[tau]^(-1)) = (-i[tau])1/2[eta]([tau]), [eta](1 + [tau]) = exp (1/12 i[pi]) [eta]([tau]).
The aggregate of the substitutions [tau]' = (p' + q'[tau])/(p + q[tau]), wherein p, q, p', q' are integers with pq' - p'q = 1, represents a _Group_; the function J([tau]), 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 + h^2)^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 y^2 = 4x^3 - g2x - g3 (and in fact many other algebraic equations, see below, under S 23, _Geometrical Applications of Elliptic Functions_) in terms of single valued functions x = RN(u), y = -RN'(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([tau]) considered above, unaltered by the group of linear substitutions [tau]' = (p' + q'[tau]) / (p + q[tau]), 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 - a^2 1 - [beta]^2 [alpha]^2 + [beta]^2 - [gamma]^2 - 1 ---- - -- ( --- ) = ---------- + ------------ + ------------------------------------, s' 2 \ s'/ 2(x - 1)^2 2x^2 2x(x - 1)
where s' = ds/dx, &c., of which the dependent variable s is equal to [tau]. 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([alpha], [beta], [gamma], x), we have in fact [tau] = s(1/2, 1/3, 0, J). If we introduce also the function of [tau] given by
2RN (1/2[omega]') + f V(1/2[omega]) [lambda] = -----------------------------------, RN(1/2[omega]') - RN(1/2[omega])
we similarly have [tau] = s(0, 0, 0, [lambda]); this function [lambda] is a single valued function of [tau], which is also a modular function, being unaltered by a group of integral substitutions also of the form [tau]' = (p' + q'[tau])/(p + q[tau]), 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 [tau] = [rho] + i[sigma], may be taken to be that given by -1 < [rho] < 1, ([rho] + 1/2)^2 + [sigma]^2 > 1/4, ([rho] - 1/2)^2 + [sigma]^2 > 1/4, and is built up of six of the regions which arose for the general modular group associated with J([tau]). Within this fundamental region, [lambda] takes every complex value just once, except the values [lambda] = 0, 1, [oo], which arise only at the angular points [tau] = 0, [tau] = [oo], [tau] = -1 and the equivalent point [tau] = 1; these angular points are essential singularities for the function [lambda]([tau]). For [lambda]([tau]) as for J([tau]), the region of existence is the upper half plane of [tau], there being an essential singularity in every length of the real axis, however short.
If, beside the plane of [tau], we take a plane to represent the values of [lambda], the function [tau] = s(0, 0, 0, [lambda]) being considered thereon, the values of [tau] belonging to the interior of the fundamental region of the [tau]-plane considered above, will require the consideration of the whole of the [lambda]-plane taken once with the exception of the portions of the real axis lying between -[oo] and 0 and between 1 and + [oo], the two sides of the first portion corresponding to the circumferences of the [tau]-plane expressed by ([rho] + 1/2)^2 + [sigma]^2 = 1/4, ([rho] - 1/2)^2 + [sigma]^2 = 1/4, while the two sides of the latter portion, for which [lambda] is real and > 1, correspond to the lines of the [tau]-plane expressed by [rho] = [+-]1. The line for which [lambda] is real, positive and less than unity corresponds to the imaginary axis of the [tau]-plane, lying in the interior of the fundamental region. All the values of [tau] = s(0, 0, 0, [lambda]) may then be derived from those belonging to the fundamental region of the [tau]-plane by making [lambda] describe a proper succession of circuits about the points [lambda] = 0, [lambda] = 1; any such circuit subjects [tau] to a linear substitution of the subgroup of [tau] considered, and corresponds to a change of [tau] from a point of the fundamental region to a corresponding point of one of the other regions.
S 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 [lambda] = exp(z) pass through a closed succession of values in the plane of [lambda] having [lambda] = 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, [lambda]) by a circuit of [lambda] about [lambda] = 0; more generally, if [phi](z) be an integral function of z, never becoming either zero or unity for finite values of z, the function [lambda] = [phi](z), however z vary in the finite part of the plane, will never make, in the plane of [lambda], a circuit about either [lambda] = 0 or [lambda] = 1, and s(0, 0, 0, [lambda]), that is s[0, 0, 0, [phi](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, [phi](z)] is also an integral function--whose imaginary part, moreover, by the property of s(0, 0, 0, [lambda]), remains positive for all finite values of z. In that case, however, exp{is[0, 0, 0, [phi](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 [phi](z) which does not vanish for any finite value of z, takes the value unity and hence_ (by considering the function A^(-1)[phi](z)) _takes every other value for some definite value of z_; or, an integral function for which both the equations [phi](z) = A, [phi](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 [phi](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([alpha], [beta], [gamma], [lambda]) is a single valued function of [tau] = s(0, 0, 0, [lambda]); for, putting [tau]' = ([tau] - i)/([tau] + i), the values of [tau]' which correspond to the singular points [lambda] = 0, 1, [oo] of s([alpha], [beta], [gamma], [lambda]), though infinite in number, all lie on the circumference of the circle |[tau]'| = 1, within which therefore s([alpha], [beta], [gamma], x) is expressible in a form [Sigma] [n = 0 to [oo]] a_n[tau]'^n. More generally any monogenic function of [lambda] which is single valued save for circuits of the points [lambda] = 0, 1, [oo], is a single valued function of [tau] = s(0, 0, 0, [lambda]). Identifying [lambda] with the square of the modulus in Legendre's form of the elliptical integral, we have [tau] = iK'/K, where
_ _ /1 dt /1 dt K = | ----------------------------------, K' = | ---------------------------------------; _/0 [root][1 - t^2] [1 - [lambda]t^2] _/0 [root][1 - t^2] [1 - (1 - [lambda])t^2]
functions such as [lambda]^1/4, (1 - [lambda])^1/4, [[lambda](1 - [lambda])]^1/4, which have only [lambda] = 0, 1, [oo] as singular points, were expressed by Jacobi as power series in q = e^(i[pi][tau]), and therefore, at least for a limited range of values of [tau], as single valued functions of [tau]; it follows by the theorem given that any product of a root of [lambda] and a root of 1 - [lambda] is a single valued function of [tau]. More generally the differential equation
d^2y dy x(1 - x) ---- + [[gamma] - ([alpha] + [beta] + 1)x] -- -[alpha][beta][gamma] = 0 dx^2 dx
may be solved by expressing both the independent and dependent variables as single valued functions of a single variable [tau], the expression for the independent variable being x = [lambda]([tau]).
S 23. _Geometrical Applications of Elliptic Functions._--Consider any irreducible algebraic equation rational in x, y, f(x, y) = 0, of such a form that the equation represents a plane curve of order n with 1/2n(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 [f] in n(n - 2) - n(n - 3) - (n - 3) = 3 other points, and will contain homogeneously at least 1/2(n - 1)n - 1/2n(n -3) - (n - 3) = 3 arbitrary constants, and so will be of the form [lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 + ... = 0, wherein [lambda]3, [lambda]4, ... are in general zero. Put now [xi] = [phi]1/[phi], [eta] = [phi]2/[phi] and eliminate x, y between these equations and [f](x, y) = 0, so obtaining a rational irreducible equation F([xi], [eta]) = 0, representing a further plane curve. To any point (x, y) of [f] will then correspond a definite point ([xi], [eta]) of F.
For a general position of (x, y) upon [f] the equations [phi]1(x', y')/[phi](x', y') = [phi]1(x, y)/[phi](x, y), [phi]2(x', y')/[phi](x', y') = [phi]2(x, y)/[phi](x, y), subject to [f](x', y') = 0, will have the same number of solutions (x', y'); if their only solution is x' = x, y' = y, then to any position ([xi],[eta]) of F will conversely correspond only one position (x, y) of [f]. If these equations have another solution beside (x, y), then any curve [lambda][phi] + [lambda]1[phi]1 + [lambda]2[phi]2 = 0 which passes (through the double points of [f] and) through the n - 2 points of [f] constituted by the fixed n-3 points and a point (x0, y0), will necessarily pass through a further point, say (x0', y0'), and will have only one further intersection with [f]; such a curve, with the n - 2 assigned points, beside the double points, of [f], will be of the form [mu][psi] + [mu]1[psi]1 + ... = 0, where [mu]2, [mu]3, ... are generally zero; considering the curves [psi] + t[psi]1 = 0, for variable t, one of these passes through a further arbitrary point of [f], by choosing t properly, and conversely an arbitrary value of t determines a single further point of [f]; the co-ordinates of the points of [f] are thus rational functions of a parameter t, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not 1/2(n - 3)n but 1/2(n - 3)n + 1 double points. We may therefore assume that to every point of F corresponds only one point of [f], and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n-3 fixed points taken upon [f], that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of [f], and will not be rational in the coefficients of [f] unless we can specify a point of [f] whose co-ordinates are rational in these. The curve F is intersected by a straight line a[xi] + b[eta] + c = 0 in as many points as the number of unspecified intersections of [f] with a[phi] + b[phi]1 + c[phi]2 = 0, that is, 3; or F will be a cubic curve, without double points.
Such a cubic curve has at least one point of inflection Y, and if a variable line YPQ be drawn through Y to cut the curve again in P and Q, the locus of a point R such that YR is the harmonic mean of YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation of the cubic curve will then be of the form
ZY^2 = aX^3 + bX^2Z + cXZ^2 + dZ^3;
by putting X equal to [lambda]X + [mu]Z, that is, choosing a suitable line through Y to be X = 0, and choosing [lambda] properly, this is reduced to the form
ZY^2 = 4X^3 - g2XZ^2 -g3Z^3,
of which a representation is given, valid for every point, in terms of the elliptic functions RN(u), RN'(u), by taking X = ZRN(u), Y = ZRN'(u). The value of u belonging to any point is definite save for sums of integral multiples of the periods of the elliptic functions, being given by _ / (x) ZdX - XdZ u = | ---------, _/ ([oo]) ZY
where ([oo]) denotes the point of inflection.
It thus appears that the co-ordinates of any point of a plane curve, [f], of order n with 1/2(n - 3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, [phi], of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of [phi] with [f], have a sum which is unaffected by variation of the coefficients of [phi], save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y^2 = 4x^3 - g2x - g3, in points (u1), (u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general theorem is a corollary from Abel's theorem proved under S 17, _Integrals of Algebraic Functions_. To prove the result directly for the cubic we remark that the variation of one of the intersections (x, y) of the cubic with the straight line y = mx + n, due to a variation [delta]m, [delta]n in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equation
F(x) = 4x^3 - g2x - g3 - (mx + n)^2 = 0,