A pole is a point (u(0)1, ... u(0)p) in the neighbourhood of which the function is expressible as a quotient of converging power series in u1 − u(0)1 ... up − u(0)p; of these the denominator series D must vanish at (u(0)1, ... u(0)p), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u(0)1, ... u(0)p), or if it does, it must be possible to write D = MD0, N = MN0, where M is a converging power series vanishing at (u(0)1, ...u(0)p), and N0 is a converging power series, in (u1 − u(0)1 ... up − u(0)p), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax + By)/ (ax + by), about x = 0, y = 0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables u1, ... up approach to u(0)1, ... u(0)p by a proper path. It is the necessary existence of such polar points of indetermination, which in case p > 2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which any one of the variables u1, ... up is indefinitely great; such points are brought under the preceding definitions by means of the convention that for u(0)i = ∞, the difference ui − u(0)i is to be understood to stand for u−1i. This being so, a single valued function of u1, ... up without essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p = 1, necessarily a rational function of the variables. A function having no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for p = 1 such a function can be expressed as a quotient of two integral functions having no common zero point other than the points of indetermination of the function; but the proof of this theorem is difficult.
The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u1, ... up are simultaneously increased each by a proper constant, these being the additive constants of indeterminateness for the p integrals ∫ Ri(x, y)dx arising when (x, y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable developments of which we cannot speak, to the theory of a single integral function of u1, ... up, called the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c1u1), exp (c2u2), ... exp (cpup), wherein c1, ... cp are proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14, Doubly Periodic Functions), the function σ(u). In the case of p = 1, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic equation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of p + 1 such periodic functions connected by a single algebraic equation. Let x1, ... xp, y denote p + 1 such functions; then each of the partial derivatives dxi/∂ui will equally be a meromorphic function of the same periods, and so expressible rationally in terms of x1, ... xp, y; thus there will exist p equations of the form
dxi = R1du1 + ... + Rpdup,
and hence p equations of the form
dui = Hi, 1dx1 + ... + Hi, pdxp,
wherein Hi, j are rational functions of x1, ... xp, y, these being connected by a fundamental algebraic (rational) equation, say ƒ(x1, ... xp, y) = 0. This then is the generalized form of the corresponding equation for p = 1.
§ 26. Multiply-Periodic Functions and the Theory of Surfaces.—The theory of algebraic integrals ∫ R(x, y)dx, wherein x, y are connected by a rational equation ƒ(x, y) = 0, has developed concurrently with the theory of algebraic curves; in particular the existence of the number p invariant by all birational transformations is one result of an extensive theory in which curves capable of birational correspondence are regarded as equivalent; this point of view has made possible a general theory of what might otherwise have remained a collection of isolated theorems.
In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indeed for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side, E. Picard has considered the possibility of classifying integrals of the form ∫(Rds + Sdy), belonging to a surface ƒ(x, y, z) = 0, wherein R and S are rational functions of x, y, z, according as they are (1) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n − 4, having properties in relation thereto analogous to those of curves of order n − 3 for a plane curve of order n; if such a surface ƒ(x, y, z) = 0 have a double curve with triple points triple also for the surface, and φ(x, y, z) = 0 be a surface of order n − 4 passing through the double curve, the double integral
| ∫ ∫ | φ dx dy |
| ∂f/∂z |
is everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface ƒ, the theorem being capable of generalization to algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n − 4, possessing the requisite particularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for m = n − 4 is not in all cases equal to the actual number of surfaces of order n − 4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m = n − 4 is also found to be invariant for birational transformation. This number, now denoted by pa, is then a second invariant of birational transformation. The former number, of actual surfaces of order n − 4 with the assigned particularity in regard to the singularities of the surface, is now denoted by pg. The difference pg − pa, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular.