numerical continuum of n dimensions, and its boundary consists
of all the points for which |x−a|＝12h. Now the points of
any domain, which does not extend to an infinite distance, may
be assigned to a finite number m of square domains of finite
breadths, so that every point of the domain is either within one
of these square domains or on its boundary, and so that no point
is within two of the square domains; also we may devise a rule
by which, as the number m increases indefinitely, the breadths
of all the square domains are diminished indefinitely. When
this process is applied to a homogeneous part, H, of the numerical
continuum Cn, then, at any stage of the process, there will be
some square domains of which all the points belong to H, and
there will generally be others of which some, but not all, of the
points belong to H. As the number m is increased indefinitely
the sums of the extents of both these categories of square
domains will tend to definite limits, which cannot be negative;
when the second of these limits is zero the domain H is said to
be “measurable,” and the first of these limits is its “extent”;
it is independent of the rule adopted for constructing the square
domains and contracting their breadths. The notion thus introduced
may be adapted by suitable modifications to continua of
lower dimensions in Cn.
The integral of a function ƒ(x) through a measurable domain H, which is a homogeneous part of the numerical continuum of n dimensions, is defined in just the same way as the integral through an interval, the extent of a square domain taking the place of the difference of the end-values of a partial interval; and the condition of integrability takes the same form as in the simple case. In particular, the condition is satisfied when the function is continuous throughout the domain. The definition of an integral through a domain may be adapted to any domain of measurable extent. The extensions to “improper” definite integrals may be made in the same way as for a function of one variable; in the particular case of a function which tends to become infinite at a point in the domain of integration, the point is enclosed in a partial domain which is omitted from the integration, and a limit is taken when the extent of the omitted partial domain is diminished indefinitely; a divergent integral may have different (principal) values for different modes of contracting the extent of the omitted partial domain. In applications to mathematical physics great importance attaches to convergent integrals and to principal values of divergent integrals. For example, any component of magnetic force at a point within a magnet, and the corresponding component of magnetic induction at the same point are expressed by different principal values of the same divergent integral. Delicate questions arise as to the possibility of representing the integral of a function of n variables through a domain Hn, as a repeated integral, of evaluating it by successive integrations with respect to the variables one at a time and of interchanging the order of such integrations. These questions have been discussed very completely by C. Jordan, and we may quote the result that all the transformations in question are valid when the function is continuous throughout the domain.
20. Representation of Functions in General.—We have seen that the notion of a function is wider than the notion of an analytical expression, and that the same function may be “represented” by one expression in one part of the domain of the argument and by some other expression in another part of the domain (§ 5). Thus there arises the general problem of the representation of functions. The function may be given by specifying the domain of the argument and the rule of calculation, or else the function may have to be determined in accordance with certain conditions; for example, it may have to satisfy in a prescribed domain an assigned differential equation. In either case the problem is to determine, when possible, a single analytical expression which shall have the same value as the function at all points in the domain of the argument. For the representation of most functions for which the problem can be solved recourse must be had to limiting processes. Thus we may utilize infinite series, or infinite products, or definite integrals; or again we may represent a function of one variable as the limit of an expression containing two variables in a domain in which one variable remains constant and another varies. An example of this process is afforded by the expression Lty＝∞xy / (x2y+1), which represents a function of x vanishing at x＝0 and at all other values of x having the value of 1/x. The method of series falls under this more general process (cf. § 6). When the terms u1, u2, . . . of a series are functions of a variable x, the sum sn of the first n terms of the series is a function of x and n; and, when the series is convergent, its sum, which is Ltn＝∞sn, can represent a function of x. In most cases the series converges for some values of x and not for others, and the values for which it converges form the “domain of convergence.” The sum of the series represents a function in this domain.
The apparently more general method of representation of a function of one variable as the limit of a function of two variables has been shown by R. Baire to be identical in scope with the method of series, and it has been developed by him so as to give a very complete account of the possibility of representing functions by analytical expressions. For example, he has shown that Riemann’s totally discontinuous function, which is equal to 1 when x is rational and to 0 when x is irrational, can be represented by an analytical expression. An infinite process of a different kind has been adapted to the problem of the representation of a continuous function by T. Brodén. He begins with a function having a graph in the form of a regular polygon, and interpolates additional angular points in an ordered sequence without limit. The representation of a function by means of an infinite product falls clearly under Baire’s method, while the representation by means of a definite integral is analogous to Brodén’s method. As an example of these two latter processes we may cite the Gamma function [Γ(x)] defined for positive values of x by the definite integral
e−tt x−1dt,
or by the infinite product
Ltn＝∞ nx/x(1 + x)(1 + 12x) . . . (1 +xn − 1)
The second of these expressions avails for the representation of the function at all points at which x is not a negative integer.
21. Power Series.—Taylor’s theorem leads in certain cases to a representation of a function by an infinite series. We have under certain conditions (§ 13)
ƒ(x)＝ƒ(a) + (x − a)rr !ƒ(r)(a)+Rn ;
and this becomes
ƒ(x)＝ƒ(a) + (x − a)rr !ƒ(r)(a),
provided that (α) a positive number k can be found so that at all points in the interval between a and a+k (except these points) ƒ(x) has continuous differential coefficients of all finite orders, and at a has progressive differential coefficients of all finite orders; (β) Cauchy’s form of the remainder Rn, viz. (x − a)n(n − 1)! (1 − θ)n−1 ƒn {a + θ(x − a)}, has the limit zero when n increases indefinitely, for all values of θ between 0 and 1, and for all values of x in the interval between a and a + k, except possibly a + k. When these conditions are satisfied, the series (1) represents the function at all points of the interval between a and a + k, except possibly a + k, and the function is “analytic” (§ 13) in this domain. Obvious modifications admit of extension to an interval between a and a − k, or between a − k and a + k. When a series of the form (1) represents a function it is called “the Taylor’s series for the function.”
Taylor’s series is a power series, i.e. a series of the form
an (x − a)n.
As regards power series we have the following theorems:
1. 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.
2. The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).
3. This function is analytic in the domain, and the power series representing it is the Taylor’s series for the function.
The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables.
22. Uniform Convergence.—We shall suppose that the domain of convergence of an infinite series of functions is an interval with the possible exception of isolated points. Let ƒ(x) be the sum of the series at any point x of the domain, and ƒn(x) the sum of the first n + 1 terms. The condition of convergence at a point a is that, after any positive number ε, however small, has been specified, it must be possible to find a number n so that |ƒm(a)−ƒp(a)|<ε for all values of m and p which exceed n. The sum, ƒ(a), is the limit of the sequence of numbers ƒn(a) at
