assigned by the rule of calculation real. In the most important | which exceeds all the others, or (B) there is a number S which cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.

diminishes when 1/e>x>0 and x diminishes towards zero, and it never becomes negative. It therefore has a limit on the right at x = 0. This limit is zero. The function represented by x sin (1/x) does not continually diminish towards zero as x diminishes towards zero, but is sometimes greater than zero and sometimes less than zero in any neighbourhood of x=0, however small. Nevertheless, the function has the limit zero at x=0.

9. Continuity of Functions.-A function f(x) of one variable x is said to be continuous at a point a if (1) f(x) is defined in an interval containing a; (2) f(x) has a limit at a; (3) f(a) is equal to this limit. The limit in question must be a limit for continuous variation, not for a restricted domain. If f(x) has a limit on the left at and f(a) is equal to this limit, the function may be said to be" continuous to the left" at a; similarly the function may be "continuous to the right" at a.

A function is said to be "continuous throughout an interval" when it is continuous at every point of the interval. This implies continuity to the right at the smaller end-value and continuity to the left at the greater end-value. When these conditions at the ends are not satisfied the function is said to be continuous Iwithin" the interval. By a "continuous function" of one variable we always mean a function which is continuous through

out an interval.

The principal properties of a continuous function are: 1. The function is practically constant throughout sufficiently small intervals. This means that, after any point a of the interval has been chosen, and any positive number, however small, has been specified, it is possible to find a number h, so that the difference between any two values of the function in the interval between a-h and a+h is less than e. There is an obvious modification if a is an end-point of the interval.

2. The continuity of the function is "uniform." This means that the number h which corresponds to any e as in (1) may be the same at all points of the interval, or, in other words, that the numbers h which correspond to e for different values of a have a positive inferior limit.

3. The function has a greatest value and a least value in the interval, and these are superior and inferior limits which are attained. 4. There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval. 5. If the interval is unlimited towards the right (or towards the left), the function has a limit at ∞ (or at ∞).

10. Discontinuity of Functions.-The discontinuities of a function of one variable, defined in an interval with the possible exception of isolated points, may be classified as follows:

(1) The function may become infinite, or tend to become infinite, at a point.

(2) The function may be undefined at a point.

(3) The function may have a limit on the left and a limit on the right at the same point; these may be different from each other, and at least one of them must be different from the value of the function at the point.

(4) The function may have no limit at a point, or no limit on the left, or no limit on the right, at a point.

11. Oscillation of Functions.-The difference between the greatest and least of the numbers f(a), f(a+0), f(a+0), f(a−0), f(a-o), when they are all finite, is called the "oscillation" or is the limit for h=o of the difference between the superior and "fluctuation " of the function f(x) at the point a. This difference inferior limits of the values of the function at points in the interval between a-h and a+h. The corresponding difference function in the interval." When any of the four limits of for points in a finite interval is called the "oscillation of the indefiniteness is infinite the oscillation is infinite in the sense explained in § 7.

of

the argument into partial intervals by means of points between

For the further classification of functions we divide the domain

the end-points. Suppose that the domain is the interval between a and b. Let intermediate points x1, x2... -, be taken so that b> xn-1>Xn-2 ..>>a. We may devise a rule by which, as n tend to zero as a limit. The interval is then said to be divided increases indefinitely, all the differences b-x-1, Xn-1-Xn-2,... x - a into indefinitely small partial intervals."

A function defined in an interval with the possible exception of isolated points may be such that the interval can be divided into a set of finite partial intervals within each of which the function is monotonous (§ 8). When this is the case the sum of the oscillations of the function in those partial intervals is finite, provided the function does not tend to become infinite. Further, in such a case the sum of the oscillations will remain below a fixed number for any mode of dividing the interval into indefinitely small partial intervals. A class of functions may be defined by the condition that the sum of the oscillations has this property, and such functions are said to have "restricted oscillation.' Sometimes the phrase "limited fluctuation" is used. It can be proved that any function with restricted oscillation is capable of being expressed as the sum of two monotonous functions, of which one never increases and the other never diminishes throughout the interval. Such a function has a limit on the right and a limit on the left at every point of the interval. This class of functions includes all those which have a finite number of maxima and minima in a finite-interval, and some which have an infinite number. It is to be noted that the class does not include all continuous functions.

12. Differentiable Function.-The idea of the differentiation of a continuous function is that of a process for measuring the rate of growth; the increment of the function is compared with the increment of the variable. If f(x) is defined in an interval containing the point a, and a-k and a+k are points of the interval, the expression f(a+b)-f(a)

h

(1)

represents a function of h, which we may call o(h), defined at all points of an interval for h between -k and k except the point o. Thus the four limits (+0), (+0), ☀(−0), $(−0) exist, and two or more of them may be equal. When the first two are equal either of them is the "progressive differential coefficient" of f(x) at the point a; when the last two are equal either of them is the "regressive differential coefficient " of f(x) at a; when all four are equal the function is said to be "differentiable" at a, and either of them is the "differential coefficient " of f(x) at a, or the "first derived function" of f(x) at a. It is denoted by

dx

or by f'(x). In this case (h) has a definite limit at h=0, or is determinately infinite at h=o (§ 7). The four limits here in question are called, after Dini, the "four derivates" of f(x) at a. In accordance with the notation for derived functions they may be denoted by

In case a function f(x), defined as above, has no limit at a point a, there are four limiting values which come into consideration. Whatever positive number h we take, the values of the function at points between a and a+h (a excluded) have a superior limit (or a greatest | df(x) value), and an inferior limit (or a least value); further, as I decreases, the former never increases and the latter never decreases; accordingly each of them tends to a limit. We have in this way two limits on the right-the inferior limit of the superior limits in diminishing neighbourhoods, and the superior limit of the inferior limits in diminishing neighbourhoods. These are. denoted by f(a+0) and f(a+o), and they are called the "limits of indefiniteness" on the right. Similar limits on the left are denoted by f(a−0) and ƒ(a−0). Unless f(x) becomes, or tends to become, infinite at a, all these must exist, any two of them may be equal, and at least one of them must be different from f(a), if f(a) exists. If the first two are equal there is a limit on the right denoted by f(a+o); if the second two are equal, there is a limit on the left denoted by f(a-o). In case the function becomes, or tends to become, infinite at a, one or more of these limits is infinite in the sense explained in §7; and now it is to be noted that, e.g. the superior limit of the inferior limits in diminishing neighbourhoods on the right of a may be negatively infinite; this happens if, after any number N, however great, has been specified, it is possible to find a positive number h, so that all the values of the function in the interval between a and ath (a excluded) are less than -N; in such a case f(x) tends to become negatively infinite when x decreases towards a; other modes of tending to infinite limits may be described in similar terms.

ƒ',(a), f'.(a), ƒ-(a), ƒ'-(a).

A function which has a finite differential coefficient at all points of an interval is continuous throughout the interval, but if the differential coefficient becomes infinite at a point of the interval the function may or may not be continuous throughout the interval; on the other hand a function may be continuous without being differentiable. This result, comparable in importance, from the point of view of the general theory of functions, with the discovery of Fourier's theorem, is due to G. F. B. Riemann; but the failure of an attempt made by Ampère to prove that every continuous function must be differentiable may be regarded as the first step in the theory. Examples of analytical expressions which represent continuous functions that are not differentiable have been given by Riemann, Weierstrass, Darboux and Dini (see § 24). The most important theorem in regard to differentiable functions is the theorem of intermediate value." (See INFINITESIMAI. Calculus:)

13. Analytic Function.—If f(x) and its first n differential | in partial intervals the sum of whose breadths can be diminished indefinitely. coefficients, denoted by f'(x), f'(x), . . . f(")-(x), are continuous in the interval between a and o+h, then

where R, may have various forms, some of which are given in the article INFINITESIMAL CALCULUS. This result is known as "Taylor's theorem."

When Talyor's theorem leads to a representation of the function by means of an infinite series, the function is said to be 'analytic" (cf. § 21)

14. Ordinary Function.-The idea of a curve representing a continuous function in an interval is that of a line which has the following properties: (1) the co-ordinates of a point of the curve of are a value x of the argument and the corresponding value y the function; (2) at every point the curve has a definite tangent; (3) the interval can be divided into a finite number of partial intervals within each of which the function is monotonous; (4) the property of monotony within partial intervals is retained after interchange of the axes of co-ordinates x and y. According to condition (2) y is a continuous and differentiable function of x, but this condition does not include conditions (3) and (4): there are continuous partially monotonous functions which are not differentiable, there are continuous differentiable functions which are not monotonous in any interval however small; and there are continuous, differentiable and monotonous functions which do not satisfy condition (4) (cf. § 24). A function which can be represented by a curve, in the sense explained above, is said to be" ordinary," and the curve is the graph of the function (§2). All analytic functions are ordinary, but not all ordinary functions are analytic.

15. Integrable Function.-The idea of integration is twofold. We may seek the function which has a given function as its differential coefficient, or we may generalize the question of finding the area of a curve. The first inquiry leads directly to the indefinite integral, the second directly to the definite integral. Following the second method we define "the definite integral of the function f(x) through the interval between a and b " to be the limit of the sum

Σƒ(x',) (x, —Xr-1)

Here x'; denotes

These partial intervals must be a set chosen out of some complete set obtained by the process used in the definition of integration. 4. The sum or product of two integrable functions is integrable. As regards integrable functions we have the following theorems: 1. If S and I are the superior and inferior limits (or greatest and

least values) of f(x) in the interval between a and b, f f(x)dx is intermediate between S(b−a) and I(b−a).

2. The integral is a continuous function of each of the end-values.

3. If the further end-value b is variable, and if (*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,

S.ƒ'(x)dx=f(b) —ƒ(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,

S. f(x)dx

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

F(x)=SF'(x)dx+const.,

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 a given function as its differential coefficient, and as a definite between the indefinite integral considered as the function which has integral with a variable end-value.

We have also two theorems concerning the integral of the product of two integrable functions f(x) and (x); these are known as "the The first theorem of the mean is that, if (x) is one-signed throughout the interval between first and second theorems of the mean." 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

MS +(x)dx = f(x)6(x)dx.

S® f(x)o(x)dx=f(a) ƒ*6(x)dx+f(b) fq$(x)dx.

(See FOURIER'S SERIES.)

The second theorem of the mean is that, if f(x) is monotonous when the interval is divided into ultimately indefinitely small throughout the interval, there is a number between a and b which has the property expressed by the equation partial intervals by points x1, x2, ... Xn-1. any point in the rth partial interval, xo is put for a, and x, for b. It can be shown that the limit in question is finite and independent of the mode of division into partial intervals, and of the choice of the points such as x',, provided (1) the function is defined for all points of the interval, and does not tend to become infinite at any of them; (2) for any one mode of division of the interval into ultimately indefinitely small partial intervals, the sum of the products of the oscillation of the function in each partial interval and the difference of the end-values of that partial interval has limit zero when ʼn is increased indefinitely. When these conditions are satisfied the function is said to be

"integrable" in the interval. The numbers a and b which limit the interval are usually called the "lower and upper limits." We shall call them the "nearer and further end-values." The above definition of integration was introduced by Riemann in his memoir on trigonometric series (1854). A still more general definition has been given by Lebesgue. As the more general definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of aggregates, we shall, in what follows, adhere to Riemann's

definition.

We have the following theorems:

1. Any continuous function is integrable.

2. Any function with restricted oscillation is integrable,

3. 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 a, however small, can be enclosed

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', of the definition: none of them is to be the point c. This comes to the same thing as definings, f(x)dx to be

Li S. f(x)dx+L!S,,f(x)dx,

(1)

where, to fix ideas, b is taken>a, and e and é 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

Í Hx)dx = Lt ff(x)dx,

2a

in which the positive number e is first diminished indefinitely, and the positive number h is afterwards increased indefinitely. The "theorems of the mean" (§ 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 f(x)dx, which has no value, is called a" divergent integral," and it may happen that there is a definite

value for

any

such

Li { S ̃ ̄f(x)dx+S+ef(x)dx }

་་

་་

boundary" of H. (4) When any two points a, b within H, are taken, it is possible to find a number e and a corresponding number m, and to choose points x', x", .x(m), so that the neighbourhood of a for e contains x', and consists exclusively of points within H., and similarly for x' and x", x" and x",...x) 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, 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. of C, 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.

"limit at

We take a positive fraction e and consider the neighbourhood of a for h, and from this neighbourhood we exclude the point a, and we Then we take x and x' to be any two of the retained points in the also exclude any point which is not in the domain of the argument. neighbourhood. The function f has a limit at a if for any positive & however small, there is corresponding h which has the property that f(x)-f(x) | <e, 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 argu. ment 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 an infinite distance" if, after any number e, however small, has been specified, a number N can be found which is such that f(x')−f(x)| <e, for all points x and x' (of the domain) of which one or more coordinates exceed N in absolute value. In the case of functions of domain. The definition of such a limit is verbally the same as the several variables great importance attaches to limits for a restricted corresponding definition in the case of functions of one variable (§ 6). For example, a function of x and x may have a limit at (x=0, x2=0) if we first diminish x without limit, keeping x constant, and afterwards diminish x 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 verbally with the corresponding definitions and explanations in the infinite, or tends to become infinite, at a point, are almost identical case of a function of one variable (§7). The definition of a continuous portant 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 xo it has the value of sin 14 tan(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).

provided that e and é 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 =, | 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 " dimensions (C) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x1, x2, . . . Xn, these values being any real numbers. The elements of such an aggregate are called "points," and the numbers x1, x2, . . . Xn the "co-ordinates " of a point. Denoting in general the points (X1, X2, ... Xn) and (x', x's... x'n) by x and x', the sum of the differences x1− x's |+|x2 − x′2 | + . . . + | x2 = x2. 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 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 is the aggregate constituted of all the points x, which are such that the "difference" denoted by [x-al<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, them-function (§ 9) admits of immediate extension; but it is very imselves 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 e, however small, a number m and points x', x', . . . x) of the aggregate can be found so that all the differences denoted by [x'-a], ]x" - x',...b-x(m) | are less than e. A perfect connected aggregate is a continuum. This is G. Cantor's definition. The definition of a continuum in C, 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 of C. Such a part would correspond to an interval in C, or to an arca bounded by a simple closed contour in C; and, besides being perfect and connected, it would have the following properties: (1) There are points of C, which are not points of H: these form a complementary aggregate H'. (2) There are points" within" H.; this means that for any such point there is a neighbourhood consisting exclusively of points of H. (3) The points of H, which do not lie "within" H are limiting points of H',; they are not points of H', but the neighbourhood of any such point for any number h, however small, contains points within H, and points of H': the aggregate of these points is called the

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|<h, where h is a chosen positive number; the extent of this domain is h",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 # dimensions, and its boundary consists
of all the points for which [x-a] = {h. 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, 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 intro-
duced may be adapted by suitable modifications to continua of
lower dimensions in Cn.

x, the sum s, 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
Li S, 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 o when x is irrational, can be represented by an analytical
to the problem of the representation of a continuous function by
expression. An infinite process of a different kind has been adapted
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 [F(x)] defined for positive values
of x by the definite integral

or by the infinite product

Lin.... n2/x(1+x)(1+}x) . (1 + 7 = 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)

f(x) = f(a) += '(x − a)' fir)(a) +Rai

The integral of a function f(x) through a measurable domain H, which is a homogeneous part of the numerical continuum of 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 and this becomes 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, 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 L!, = mxy/(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 1, ur, . . . of a series are functions of a variable

701

f(x) = f(a) + = (x − a)"ƒ«r3⁄4(a),

provided that (a) 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; (8) Cauchy's form of the remainder R, viz.
(x-a)",
{"? (x −0) n = 1 f(n) {a+0(x-a)}, has the limit zero when # in-
creases indefinitely, for all values of between o 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) repre-
sents 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
Σ an(x-a)".

م.

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). representing it is the Taylor's series for the function. 3. This function is analytic in the domain, and the power series

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 f(x) be the sum of the series at any point x of the domain, and f,(x) the sum of the first n+1 terms. The condition of convergence at a point a is that, after any positive number e, however small, has been specified, it must be possible to find a number so that [ƒ„(a)—ƒ,(a)|<e for all values of m and which exceed n. The sum, f(a), is the limit of the sequence of numbers f„(a) at