Augustus E. H. Love and Henry F. Baker

“ Singular Points.—The region of existence of a differentiable function of z is an unclosed aggregate of points, each of which is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called a singular point of the function. ”

“ 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. ”

“ We thus infer, since the sum of the residues of φ (z) is zero, that for the function ƒ (z) , the sum of the orders of its vanishing at points belonging to one parallelogram, Σm, is equal to the sum of the orders of its poles, Σn; which is briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function ƒ (z) − A, where A is an arbitrary constant, we have the result, that the function ƒ (z) assumes the value A in one of the parallelograms as many times as it becomes infinite. ”

“ A domain which consists of all the real numbers which exceed some fixed number may be described as an “interval unlimited towards the right”; similarly we may have an interval” unlimited towards the left.”
In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of a variable. The methods of definition of particular types of aggregates, and the theorems relating to them, form a branch of analysis called the “theory of aggregates” (Mengenlehre, Théorie des ensembles, Theory of sets of points) . ”

“ Indeterminateness of Algebraic Integrals.—The theorem that the integral ∫ xa ƒ (z) dz is independent of the path from a to z, holds only on the hypothesis that any two such paths are equivalent, that is, taken together from the complete boundary of a region of the plane within which ƒ (z) is finite and single valued, besides being differentiable. Suppose that these conditions fail only at a finite number of isolated points in the finite part of the plane. ”

“ These theorems and definitions can be extended, with obvious modifications, to the cases of a domain which is not an interval, or extends to infinite values. By means of them we arrive at sufficient, but not necessary, criteria for the existence of a limit; and these are frequently easier to apply than the general principle of convergence to a limit (§ 6) , of which principle they are particular cases. For example, the function represented by x log (1/x) continually diminishes when 1/e > x > 0 and x diminishes towards zero, and it never becomes negative. ”

“ If the power series converges at any point except a there is a number k which has the property that the series converges absolutely in the interval between a − k and a + k, with the possible exception of one or both end-points. ”

“ There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued—it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. ”

“ In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Σanφn (x) , where Σan is an absolutely convergent series of numbers, and φn (x) is an analytic function whose absolute value never exceeds unity. ”

“ There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ (z) , and the law of increase of the coefficients in the series ƒ (z) = Σanzn as n increases (see the bibliography below under Integral Functions) . It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21, Modular Functions) . ”

“ It thus appears that if a single valued monogenic function of z be periodic, either all its periods are real multiples of one of them, and then all are of the form MΩ, where Ω is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods, and N, N′ are integers. ”

“ By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c. ”

“ A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the function exceeds a given number σ, however small, can be enclosed in partial intervals the sum of whose breadths can be diminished indefinitely. ”

“ 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. ”

“ A perfect connected aggregate is a continuum. This is G. Cantor’s definition.
The definition of a continuum in Cn leaves open the question of the number of dimensions of the continuum, and a further explanation is necessary in order to define arithmetically what is meant by a “homogeneous part” Hn of Cn. Such a part would correspond to an interval in C1, or to an area bounded by a simple closed contour in C2; and, besides being perfect and connected, it would have the following properties: (1) There are points of Cn, which are not points of Hn; these form a complementary aggregate H′n. ”

“ 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. ”

“ Similarly if z0 be any finite value of z which is not a root of the polynomial ƒ (z) to which s2 is equal, and s0 denotes a particular one of the determinations of s for z=z0, the integral
wherein ƒ′ (z) = dƒ (z) /dz, becomes infinite for z = z0, s = s0, but not for z = z0, s = −s0. its infinite term in the former case being the negative of 2s0 (z − z0) . For no other finite or infinite value of z is the integral infinite. ”