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
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)12 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 z12, 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.
Any rational function of z and s, where ƒ(s, z) = 0, may be considered in the neighbourhood of any place (c, d) by substituting therein z = c + P(t), s = d + Q(t); the result is necessarily of the form tmH(t), where H(t) is a power series in t not vanishing for t=0 and m is an integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative, = −μ, the function is infinite to order μ at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) −A is of the form tmH(t), where m is positive, we say that R(s, z) becomes m times equal to A at the place; if R(s, z) is infinite of order μ at the place, so also is R(s, z) − A. It can be shown that the sum of the values of m at all the places, including the places z = ∞, where R(s, z) vanishes, which we call the number of zeros of R(s, z) on the algebraic construct, is finite, and equal to the sum of the values of μ where R(s, z) is infinite, and more generally equal to the sum of the values of m where R(s, z) = A; this we express by saying that a rational function R(s, z) takes any value (including ∞) the same number of times on the algebraic construct; this number is called the order of the rational function.
That the total number of zeros of R(s, z) is finite is at once obvious, these values being obtainable by rational elimination of s between ƒ(s, z) = 0, R(s, z) = 0. That the number is equal to the total number of infinities is best deduced by means of a theorem which is also of more general utility. Let R(s, z) be any rational function of s, z, which are connected by ƒ(s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the product
| R(s, z) | dz |
| dt |
in powers of t and pick out the coefficient of t−1. There is only a finite number of places of this kind. The theorem is that the sum of these coefficients of t−1 is zero. This we express by
| [ R(s, z) | dz | ]t−1 = 0. |
| dt |
The theorem holds for the case n=1, that is, for rational functions of one variable z; in that case, about any finite point we have z − c = t, and about z = ∞ we have z−1 = t, and therefore dz/dt = −t−2; in that case, then, the theorem is that in any rational function of z,
| Σ ( | A1 | + | A2 | + ... + | Am | ) + Pzh + Qzh−1 + ... + R, |
| z − a | (z − a)2 | (z − a)m |
the sum ΣA1 of the sum of the residues at the finite poles is equal to the coefficient of 1/z in the expansion, in ascending powers of 1/z, about z = ∞; an obvious result. In general, if for a finite place of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood is given by z = c + tr, s = d + Q(t), there be a coefficient of t−1 in R(s, z) dz/dt, this will be r times the coefficient of t−r in R(s, z) or R[d + Q(t), c + tr], namely will be the coefficient of t−r in the sum of the r series obtainable from R [d + Q(t), c + tr] by replacing t by ωt, where ω is an rth root of unity; thus the sum of the coefficients of t−1 in R(s, z) dz/dt for all the places which arise for z = c, and the corresponding values of s, is equal to the coefficient of (z − c)−1 in R(s1, z) + R(s2z) + ... + R(sn, z), where s1, ... sn are the n values of s for a value of z near to z = c; this latter sum Σ R(si , z) is, however, a rational function of z only. Similarly, near z = ∞, for a place given by z−1=tr, s = d + Q(t), or s−1 = Q(t), the coefficient of t−1 in R(s, z) dz/dt is equal to −r times the coefficient of tr in R[d + Q(t), t−r], that is equal to the negative coefficient of z−l in the sum of the r series R[d + Q(ωt), t−r], so that, as before, the sum of the coefficients of t−1 in R(s, z) dz/dt at the various places which arise for z = ∞ is equal to the negative coefficient of z− 1 in the same rational function of z, Σ R(si , z). Thus, from the corresponding theorem for rational functions of one variable, the general theorem now being proved is seen to follow.
Apply this theorem now to the rational function of s and z,
| 1 | dR(s, z) | ; | |
| R(s, z) | dz |
at a zero of R(s, z) near which R(s, z) = tmH(t), we have
| 1 | dR(s, z) | dz | = | d | {λ [R(s, z)] }, | ||
| R(s, z) | dz | dt | dt |
where λ denotes the generalized logarithmic function, that is equal to
similarly at a place for which R(s, z) = t−μK(t); the theorem
| [ | 1 | dR(s, z) | dz | ]t−1 = 0 | ||
| R(s, z) | dz | dt |
thus gives Σm = Σμ, or, in words, the total number of zeros of R(s, z) on the algebraic construct is equal to the total number of its poles. The same is therefore true of the function R(s, z) − A, where A is an arbitrary constant; thus the number in question, being equal to the number of poles of R(s, z) − A, is equal also to the number of times that R(s, z) = A on the algebraic construct.
