| ½ω = ∫ ∞1 | dt | , ½ω′ = ∫ ∞e3 | dt | = ½εω; |
| 2(t³ − 1)1/2 | 2(t³ − 1)1/2 |
drawing the equilateral triangle whose vertices are O, of argument O, A of argument ω, and B of argument ω + ω′ = −ε²ω, and the equilateral triangle whose angular points are O, B and C, of argument ω′, let E, of argument 1⁄3(2ω + ω′), and D, of argument 1⁄3(ω + 2ω′), be the centroids of these triangles respectively, and let BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut OC, BC, OB in F, G, H respectively; then if u = ξ + iη be any point of the interior of the triangle OEH and v = εu0 = ε(ξ − iη) be any point of the interior of the triangle OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB, EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by −εv, ω + εu, ω − η²v, ω + ω′ + ε²u, ω + ω′ − v, ω + ω′ − u, ω + ω′ + εv, ω′ − εu, ω′ + ε²v, −ε²u. Further, when u is real, since the term − 2(u + mω + m′ε²ω)−3, which is the conjugate complex of −2(u + mω + m′ε²ω)3, arises in the infinite sum which expresses ℜ′(u), namely as −2(u + μω + μ′εω)−3, where μ = m − m′, μ′ = −m′, it follows that ℜ′(u) is real; in a similar way we prove that ℜ′(u) is pure imaginary when u is pure imaginary, and that ℜ′(u) = ℜ′(εu) = ℜ′(ε²u), as also that for v = εu0, ℜ′(v) is the conjugate complex of ℜ′(u). Hence it follows that the variable
t = ½ iℜ′(u)
takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively ∞, 1, 0, − 1 at O, D, H, E, and takes every complex value of imaginary part positive once in the interior of this triangle. This leads to
u = 1⁄3 i ∫ ∞t (t2 − 1)−2/3dt
in accordance with the general theory.
It can be deduced that τ = t2 represents the triangle ODH on the upper half plane of τ, and ζ = (i − τ−1)1/2 represents similarly the triangle OBD.
§ 16. Multiple valued Functions. Algebraic Functions.—The explanations and definitions of a monogenic function hitherto given have been framed for the most part with a view to single valued functions. But starting from a power series, say in z − c, which represents a single value at all points of its circle of convergence, suppose that, by means of a derived series in z − c′, where c′ is interior to the circle of convergence, we can continue the function beyond this, and then by means of a series derived from the first derived series we can make a further continuation, and so on; it may well be that when, after a closed circuit, we again consider points in the first circle of convergence, the value represented may not agree with the original value. One example is the case z1/2, for which two values exist for any value of z; another is the generalized logarithm λ(z), for which there is an infinite number of values. In such cases, as before, the region of existence of the function consists of all points which can be reached by such continuations with power series, and the singular points, which are the limiting points of the point-aggregate constituting the region of existence, are those points in whose neighbourhood the radii of convergence of derived series have zero for limit. In this description the point z = ∞ does not occupy an exceptional position, a power series in z − c being transformed to a series in 1/z when z is near enough to c by means of z − c = c(1 − cz−1) [1 − (1 − cz−1)]−1, and a series in 1/z to a series in z − c, when z is near enough to c, by means of 1/z = 1/c [1 + (z − c / c)]−1.
The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation ƒ(s, z) = p0sn + p1sn−1 + ... + pn = 0, wherein p0, p1, ... pn are integral polynomials in z. Assuming ƒ(s, z) incapable of being written as a product of polynomials rational in s and z, and excepting values of z for which the polynomial coefficient of sn vanishes, as also the values of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also in general the point z = ∞, the roots of this equation about any point z=c are given by n power series in z − c. About a finite point z = c for which the equation ∂f(s, z)/∂s = 0 is satisfied by one or more of the roots s of ƒ(s, z) = 0, the n roots break up into a certain number of cycles, the r roots of a cycle being given by a set of power series in a radical (z − c)1/r, these series of the cycle being obtainable from one another by replacing (z − c)1/r by ω(z − r)1/r, where ω, equal to exp (2πih/r), is one of the rth roots of unity. Putting then z − c = tr we may say that the r roots of a cycle are given by a single power series in t, an increase of 2π in the phase of t giving an increase of 2πr in the phase of z − c. This single series in t, giving the values of s belonging to one cycle in the neighbourhood of z = c when the phase of z − c varies through 2πr, is to be looked upon as defining a single place among the aggregate of values of z and s which satisfy ƒ(s, z) = 0; two such places may be at the same point (z = c, s = d) without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of z, z = c, there are n places for which the neighbouring values of s are given by n power series in z − c; for a value of z for which ∂f(s, z)/∂s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = ∞; there may be n places whose neighbourhood is given by n power series in z− 1 or fewer, one of these being associated with a series in t, where t = (z−1)1/r; the sum of the values of r which thus arise is always n. In general, then, we may say, with t of one of the forms (z − c), (z − c)1/r, z−1, (z−1)1/r. that the neighbourhood of any place (c, d) for which ƒ(c, d) = 0 is given by a pair of expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular case of a) power series vanishing for t = 0, and Q(t) is a power series vanishing for t = 0, and t vanishes at (c, d), the expression z − c being replaced by z−1 when c is infinite, and similarly the expression s − d by s−1 when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of sn vanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0 + λ1τ + λ2τ² + ..., where τ is a new variable and λ1 is not zero; in particular for an ordinary finite place this equation simply becomes t = t0 + τ. It can be shown that all the pairs of power series z = c + P(t), s = d + Q(t) which are necessary to represent all pairs of values of z, s satisfying the equation ƒ(s, z) = 0 can be obtained from one of them by this process of continuation, a fact which we express by saying that the equation ƒ(s, z) = 0 defines a monogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation ƒ(s, z) = 0 determines a single monogenic function s of z.