3. If the further end-value b is variable, and if [int] [a to x]
[f](x)_dx_ = F(x), then if [f](x) is continuous at b, F(x) is
differentiable at b, and F'(b) = [f](b).
4. In case [f](x) is continuous throughout the interval F(x) is
continuous and differentiable throughout the interval, and F'(x) =
[f](x) throughout the interval.
5. In case [f]'(x) is continuous throughout the interval between a and
b,
_
/ b
| [f]'(x)dx = [f](b) - [f](a).
_/a
6. In case [f](x) is discontinuous at one or more points of the
interval between a and b, in which it is integrable,
_
/ x
| [f](x)dx
_/a
is a function of x, of which the four derivates at any point of the
interval are equal to the limits of indefiniteness of [f](x) at the
point.
7. It may be that there exist functions which are differentiable
throughout an interval in which their differential coefficients are
not integrable; if, however, F(x) is a function whose differential
coefficient, F'(x), is integrable in an interval, then
_
/ x
F(x) = | F'(x)dx + const.,
_/a
where a is a fixed point, and x a variable point, of the interval.
Similarly, if any one of the four derivates of a function is
integrable in an interval, all are integrable, and the integral of
either differs from the original function by a constant only.
The theorems (4), (6), (7) show that there is some discrepancy between
the indefinite integral considered as the function which has a given
function as its differential coefficient, and as a definite integral
with a variable end-value.
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."