Part 22

We have also two theorems concerning the integral of the product of two integrable functions [f](x) and [phi](x); these are known as "the first and second theorems of the mean." The first theorem of the mean is that, if [phi](x) is one-signed throughout the interval between a and b, there is a number M intermediate between the superior and inferior limits, or greatest and least values, of [f](x) in the interval, which has the property expressed by the equation _ _ / b / b M | [phi](x)dx = | [f](x)[phi](x)dx. _/a _/a

The second theorem of the mean is that, if [f](x) is monotonous throughout the interval, there is a number [xi] between a and b which has the property expressed by the equation _ _ _ / b /[xi] / b | [f](x)[phi](x)dx = [f](a) | [phi](x)dx + [f](b) | [phi](x)dx. _/a _/a _/[xi]

(_See_ FOURIER'S SERIES.)

16. _Improper Definite Integrals._--We may extend the idea of integration to cases of functions which are not defined at some point, or which tend to become infinite in the neighbourhood of some point, and to cases where the domain of the argument extends to infinite values. If c is a point in the interval between a and b at which [f](x) is not defined, we impose a restriction on the points x'_r of the definition: none of them is to be the point c. This comes to the same thing as defining [int] [a to b] [f](x)dx to be _ _ / c-[epsilon] / b Lt | [f](x)dx + Lt | [f](x)dx, (1) _/a _/c+[epsilon]' [epsilon]=0 [epsilon]'=0

where, to fix ideas, b is taken > a, and [epsilon] and [epsilon]' are positive. The same definition applies to the case where [f](x) becomes infinite, or tends to become infinite, at c, provided both the limits exist. This definition may be otherwise expressed by saying that a

## partial interval containing the point c is omitted from the interval of

integration, and a limit taken by diminishing the breadth of this

## partial interval indefinitely; in this form it applies to the cases

where c is a or b.

Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definition _ _ / [oo] / h | [f](x)dx = Lt | [f](x)dx, _/a _/a h=[oo]

provided this limit exists. Similar definitions apply to _ _ /-[oo] / [oo] | [f](x)dx and to | [f](x)dx. _/a _/-[oo]

All such definite integrals as the above are said to be "improper." For example, [int] {0 to [oo]} (sin x / x)dx is improper in two ways. It means _ / h sin x Lt Lt | ----- dx, h=[oo] [epsilon]=0 _/[epsilon] x

in which the positive number [epsilon] is first diminished indefinitely, and the positive number h is afterwards increased indefinitely.

The "theorems of the mean" (S 15) require modification when the integrals are improper (see FOURIER'S SERIES).

When the improper definite integral of a function which becomes, or tends to become, infinite, exists, the integral is said to be "convergent." If [f](x) tends to become infinite at a point c in the interval between a and b, and the expression (1) does not exist, then the expression [int] [a to b][f](x)_dx_, which has no value, is called a "divergent integral, "and it may happen that there is a definite value for _ _ _ _ | / c-[epsilon] / b | Lt | | [f](x) dx + | [f](x) dx | |_ _/a _/c+[epsilon]' _|

provided that [epsilon] and [epsilon]' are connected by some definite relation, and both, remaining positive, tend to limit zero. The value of the above limit is then called a "principal value" of the divergent integral. Cauchy's principal value is obtained by making [epsilon]' = [epsilon], i.e. by taking the omitted interval so that the infinity is at its middle point. A divergent integral which has one or more principal values is sometimes described as "semi-convergent."

17. _Domain of a Set of Variables._--The numerical continuum of n dimensions (C_n) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x1, x2, ... x_n, these values being any real numbers. The elements of such an aggregate are called "points," and the numbers x1, x2 ... x_n the "co-ordinates" of a point. Denoting in general the points (x1, x2, ... x_n) and (x'1, x'2 ... x'_n) by x and x', the sum of the differences |x1 - x'1| + |x2 - x'2| + ... + |x_n - x'_n| may be denoted by |x - x'| and called the "difference of the two points." We can in various ways choose out of the continuum an aggregate of points, which may be an infinite aggregate, and any such aggregate can be the "domain" of a "variable point." The domain is said to "extend to an infinite distance" if, after any number N, however great, has been specified, it is possible to find in the domain points of which one or more co-ordinates exceed N in absolute value. The "neighbourhood" of a point a for a (positive) number h is the aggregate constituted of all the points x, which are such that the "difference" denoted by |x - a| < h. If an infinite aggregate of points does not extend to an infinite distance, there must be at least one point a, which has the property that the points of the aggregate which are in the neighbourhood of a for any number h, however small, themselves constitute an infinite aggregate, and then the point a is called a "limiting point" of the aggregate; it may or may not be a point of the aggregate. An aggregate of points is "perfect" when all its points are limiting points of it, and all its limiting points are points of it; it is "connected" when, after taking any two points a, b of it, and choosing any positive number [epsilon], however small, a number m and points x', x", ... x^(m) of the aggregate can be found so that all the differences denoted by |x' - a|, |x" - x'|, ... |b - x^(m)| are less than [epsilon]. A perfect connected aggregate is a _continuum_. This is G. Cantor's definition.

The definition of a continuum in C_n 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" H_n of C_n. 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 C_n, which are not points of H_n; these form a complementary aggregate H'_n. (2) There are points "within" H_n; this means that for any such point there is a neighbourhood consisting exclusively of points of H_n. (3) The points of H_n which do not lie "within" H_n are limiting points of H'_n; they are not points of H'_n, but the neighbourhood of any such point for any number h, however small, contains points within H_n and points of H'_n: the aggregate of these points is called the "boundary" of H_n. (4) When any two points a, b within H_n are taken, it is possible to find a number [epsilon] and a corresponding number m, and to choose points x', x", ... x^m, so that the neighbourhood of a for [epsilon] contains x', and consists exclusively of points within H_n, and similarly for x' and x", x" and x"', ... x^m and b. Condition (3) would exclude such an aggregate as that of the points within and upon two circles external to each other and a line joining a point on one to a point on the other, and condition (4) would exclude such an aggregate as that of the points within and upon two circles which touch externally.

18. Functions of Several Variables.--A function of several variables differs from a function of one variable in that the argument of the function consists of a set of variables, or is a variable point in a C_n when there are n variables. The function is definable by means of the domain of the argument and the rule of calculation. In the most important cases the domain of the argument is a homogeneous part H_n of C_n with the possible exception of isolated points, and the rule of calculation is that the value of the function in any assigned part of the domain of the argument is that value which is assumed at the point by an assigned analytical expression. The limit of a function at a point a is defined in the same way as in the case of a function of one variable.

We take a positive fraction [epsilon] and consider the neighbourhood of a for h, and from this neighbourhood we exclude the point a, and we also exclude any point which is not in the domain of the argument. Then we take x and x' to be any two of the retained points in the neighbourhood. The function [f] has a limit at a if for any positive [epsilon], however small, there is a corresponding h which has the property that |[f](x') - [f](x)| < [epsilon], whatever points x, x' in the neighbourhood of a for h we take (a excluded). For example, when there are two variables x1, x2, and both are unrestricted, the domain of the argument is represented by a plane, and the values of the function are correlated with the points of the plane. The function has a limit at a point a, if we can mark out on the plane a region containing the point a within it, and such that the difference of the values of the function which correspond to any two points of the region (neither of the points being a) can be made as small as we please in absolute value by contracting all the linear dimensions of the region sufficiently. When the domain of the argument of a function of n variables extends to an infinite distance, there is a "limit at an infinite distance" if, after any number [epsilon], however small, has been specified, a number N can be found which is such that |[f](x') - [f](x)| < [epsilon], for all points x and x' (of the domain) of which one or more co-ordinates exceed N in absolute value. In the case of functions of several variables great importance attaches to limits for a restricted domain. The definition of such a limit is verbally the same as the corresponding definition in the case of functions of one variable (S 6). For example, a function of x1 and x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1 without limit, keeping x2 constant, and afterwards diminish x2 without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x2. The definitions of superior and inferior limits, and of maxima and minima, and the explanations of what is meant by saying that a function of several variables becomes infinite, or tends to become infinite, at a point, are almost identical verbally with the corresponding definitions and explanations in the case of a function of one variable (S 7). The definition of a continuous function (S 9) admits of immediate extension; but it is very important to observe that a function of two or more variables may be a continuous function of each of the variables, when the rest are kept constant, without being a continuous function of its argument. For example, a function of x and y may be defined by the conditions that when x = 0 it is zero whatever value y may have, and when x [/=] 0 it has the value of sin {4tan^(-1)(y/x)}. When y has any particular value this function is a continuous function of x, and, when x has any particular value this function is a continuous function of y; but the function of x and y is discontinuous at (x = 0, y = 0).

19. _Differentiation and Integration._--The definition of partial differentiation of a function of several variables presents no difficulty. The most important theorems concerning differentiable functions are the "theorem of the total differential," the theorem of the interchangeability of the order of partial differentiations, and the extension of Taylor's theorem (see INFINITESIMAL CALCULUS).

With a view to the establishment of the notion of integration through a domain, we must define the "extent" of the domain. Take first a domain consisting of the point a and all the points x for which |x - a| < 1/2h, where h is a chosen positive number; the extent of this domain is h^n, n being the number of variables; such a domain may be described as "square," and the number h may be called its "breadth"; it is a homogeneous part of the numerical continuum of n dimensions, and its boundary consists of all the points for which |x - a| = 1/2h. 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 C_n, 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 C_n.

The integral of a function f(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 H_n, 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 (S 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 Lt_y = [oo]xy/(x^2y + 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. S 6). When the terms u1, u2, ... of a series are functions of a variable x, the sum s_n 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 Lt_n = [oo]s_n, 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. Broden. 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 Broden's method. As an example of these two latter processes we may cite the Gamma function [[Gamma](x)] defined for positive values of x by the definite integral _ / [oo] | e^(-t)t^(x - 1)dt, _/0

or by the infinite product

/ x \ L t_(n = [oo]) n^x/x (1 + x)(1 + 1/2x) ... ( 1 + ----- ). \ n - 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 (S 13)

_n-1 \ (x - a)^r [f](x) = [f](a) + /_ --------- [f]^(r) (a) + R_n; r=1 r!

and this becomes

_[oo] \ (x - a)^r [f](x) = [f](a) + /_ --------- [f]^(r) (a), r=1 r!

provided that ([alpha]) a positive number k can be found so that at all points in the interval between a and a + k (except these points) [f](x) has continuous differential coefficients of all finite orders, and at a has progressive differential coefficients of all finite orders; ([beta]) Cauchy's form of the remainder R_n, viz. [(x - a)^n / (n - 1)!] (1 - [theta])^(n - 1)[f]^n {a + [theta](x - a)}, has the limit zero when n increases indefinitely, for all values of [theta] 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" (S 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

_[oo] \ a_n (x - a)^n. /_ n=0

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.