Page images
PDF
EPUB
[ocr errors]

a finite value at a.

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

exceeds every value of the function but is such that, however interval, with the possible exception of isolated points. 6. Limits.-Let f(x) be a function of a variable number *; which exceed S - €. In the case (a) the function has a greatest

small a positive numbere we take, there are values of the function and let a be a point such that there are points of the domain value; in case (B) the function has a “superior limit ” S, and of the argument x in the neighbourhood of a for any number then there must be a point a which has the property that there h, however small. If there is a number 1 which has the property are points of the domain of the argument, in the neighbourhood that, after any positive number e, however small, has been of e for any h, at which the values of the function differ from specified, it is possible to find a positive number h, so that S by less thane. Thus S is the limit of the function at a, either il-f(x) <e for all points x of the domain (other than a) for for the domain of the argument or for some more restricted which lx-el<h, then L is the " limit of f(x) at the point a.” domain. If a is in the domain of the argument, and is, after The condition for the existence of L is that, after the positive omission of a, there is a superior limit S which is in this way the number e has been specified, it must be possible to find a positive limit of the function at a, il further f(a) = S, then S is the greatest number h, so that i f(x')-f(x)}<e for all points x and zof value of the function; in this case the greatest value is a limit the domain (other than a) for which (x-01<k and ix'-ak<h.

(at any rate for a restricted domain) which is attained; it may It is a fundamental theorem that, when this condition is be called a "superior limit which is attained." In like manner satisfied, there exists a perfectly definite number L which is the limit of f(x) at the point o as defined above. The limit of (x) smallest value may be an “inferior limit which is attained.”

we may have a “smallest value” or an “inferior limit," and a at the point e is denoted by Llo-aj(), or by limg=.|(x).

All that has been said here may be adapted to the description of If f(x) is a function of one variable x in a domain which extends greatest values, superior limits, &c., of a function in a restricted to infinite values, and if, alter e has been specified, it is possible to domain contained in the domain of the argument. In particular, find a number N, so that 1/(x')-f(x)\<e for all values of x and x' the domain of the argument may contain an interval; and therein which are in the domain and exceed N, then there is a number L the function may have a superior limit, or an inferior limit, which which has the property that ||(x)-L1 <e for all such values of x.

is attained. Such a limit is a maximum value or a minimum value In this case f(x) has a limit i at r=co. In like manner f(x) may of the function. have a limit at r=-0. This statement includes the case where Again, if, after any number N. however great, has been specified, the domain of the argument consists exclusively of positive integers. it is possible to find points of the domain of the argument at which The values of the function then form a sequence,

the value of the function exceeds N, the values of the function are Hm..., and this sequence can have a limit at n=o.

said to have an "infinite superior limit," and then there must be The principle common to the above definitions and theorems is a point a which has the property that there are points of the domain, called, after P. du Bois Reymond, "the general principle of con- in the neighbourhood of a sor any h, at which the value of the function vergence to a limit."

exceeds N. If the point a is in the domain of the argument the It must be understood that the phrase " * = " does not mean function is said to "tend to become infinite " at a; it has of course that x takes some particular value which is infinite. There is no

If the point a is not in the domain of the argu. such value. The phrase always refers to a limiting process in which, ment the function is said to " become infinite" at o; it has of as the process is carried out, the variable number 3 increases without course no value at a. In like manner we may have a (negatively) limit; it may, as in the above example of a sequence, increase by infinite inferior limit. Again, after any number N, however great, taking successively the values of all the integral numbers; in other has been specified and a number h found, so that all the values of cases it may increase by taking the values thai belong to any domain the function, at points in the neighbourhood of a for h, exceed N in which "extends to infinite values."

absolute value, all these values may have the same sign; the function A very important type of limits is furnished by infinile series. is then said to become, or to tend to become, determinately When a sequence of numbers , ua,... Un... is given, we may (positively or negatively) infinite "; otherwise it is said to become form a new sequence si. 2... .. Sa,... from it by the rules si = us. or to tend to become, “ indeterminately infinite.". Se... Sn= uit 149+ +um, or by the equivalent rules All the infinities that occur in the theory of functions are of the si = u, SA-SA-1 = un(n=2, 3,...). If the new sequence has a limit nature of variable finite numbers, with the single exception of the at »=s, this limit is called the “ sum of the infinite series infinity of an infinite aggregate. The latter is described as an uituzt.... and the series is said to be

convergent (see " actual infinity," the former as “improper infinities." There is no SERIES).

actual infinitely small corresponding to the actual infinity. A function which has not a limit at a point a may be such that, | The only " infinitely small". is zero. All "infinite values il a certain aggregate of points is chosen out of the domain of the the nature of superior and inferior limits which are not attained. argument, and the points x in the neighbourhood of a are restricted 8. Incrcasing and Decreasing Functions.--A function f(x) of one to belong to this aggregate. then the function has a limit at a.

For variable x, defined in the interval between a and b, is “ increasing example, sin (1/*) has limit zero at o il * is restricted to the throughout the interval ” if, whenever x and x' are two numbers aggregate 1/*, 1/21, 2/57, ,:. n!(n2+1)., ..., but if x takes all values in the neighbour in the interval and x'>x, then f(x')>}(tr); the 'unction “never hood of o, sin (1/x) has not a limit at o. Again, there may be a limit decreases throughout the interval ” if, x' and x being as before, at o if the points x in the neighbourhood of a are restricted by the 1(x')>f(x). Similarly for decreasing functions, and for functions condition that x-a is positive; then we have a " limit. on the which never increase throughout an interval. A function which right at a; similarly we may have a " limit on the left" at a point. Any such limit is described as a "limit for a restricted either never increases or never diminishes throughout an interval

"The limits on the left and on the right are de noted by is said to be “ monotonous throughout "the interval. If we take fla-o) ana f(a+o).

The limit L of /(*) at a stands in no necessary relation to the value in the above definition b>a, the definition may apply to a function of f(x) at a. Is the point a is in the domain of the argument, the under the restriction that x' is not b and r is not a; such a value of f(x) at a is assigned by the rule of calculation, and may be function is “monotonous within " the interval. In this case we difierent from L. In case f(a) = L the limit is said to be "attained." have the theorem that the function (if it never decreases) has If the point o is not in the domain of the argument, there is no value a limit on ihe left at b and a limit on the right at a, and these are for f(x) at d. In the case where f(x) is defined for all points in an the superior and inferior limits of its values at all points within interval containing a, except the point a, and has a limit L at a, we may arbitrarily annex the point a to the domain of the argument the interval (the ends excluded); the like holds mulatis mutandis and assign to f(a) the value L; the function may then be said to if the function never increases. If the function is monotonous be " extrinsically defined." The so-called " indeterminate forms

throughout the interval, f(b) is the greatest (or least) value (see INFINITESIMAL CALCULUS) are examples.

of /(x) in the interval; and is s(6) is the limit of S(x) on the left 7. Superior and Inferior Limits; Infinilies.—The value of a al b, such a greatest (or least) value is an example of a superior function at every point in the domain of its argument is finite, (or inserior) limit which is attained. In these cases the function since, by definition, the value can be assigned, but this does not

tends continually to its limit. necessarily imply that there is a number N which exceeds all

These theorems and definitions can be extended, with obvious the values (or is less than all the values). It may happen that, modifications, to the cases of a domain which is not an interval, or however great a number N we take, there are among the values extends to infinite values. By means of them we arrive at sufficient, of the function numbers which exceed N (or are less than-N). but not necessary, criteria for the existence of a limit; and these

are frequently easier to apply than the general principle of conver. If a number can be found which is greater than every value gence to a limit ($ 6), of which principle they are particular cases: of the function, then either (a) there is one value of the function For example, the function represented by a log (1 x) continually

[ocr errors]

are of

[ocr errors]

domain."

diminishes when ile>x>0 and x diminishes towards zero, and it

11. Oscillation of Functions.—The difference between the never becomes negative. It therefore has a limit on the right at =0.

This limit is zero. The function represented by * sin (1x) greatest and least of the numbers f(a), f(a+o)/(a+o), sa-o), does not continually diminish towards zero as x diminishes towards (a-o), when they are all finite, is called the “ oscillation" or zero, but is sometimes greater than zero and sometimes less than zero in any neighbourhood of x=0, however small. Nevertheless, is the limit for h=o of the difference between the superior and

“fluctuation "of the functions(x) at the point a. This difference the function has the limit zero at x=0. 9. Continuity of Functions.-A function f(x) of one variable a interval between ach and a+h.

inferior limits of the values of the function at points in the is said to be continuous at a point a if (1)/(x) is defined in an

The corresponding difference interval containing a; (2) f(x) has a limit at a; (3) 1(a) is for points in a finite interval is called the " oscillation of the equal to this limit. The limit in question must be a limit for function in the interval.” When any of the four limits of continuous variation, not for a restricted domain. If y(x) has indefiniteness is infinite the oscillation is infinite in the sense a limit on the left at a and f(a) is equal to this limit, the function explained in $ 7. may be said to be “ continuous to the left” at a; similarly the of the argument into partial intervals by means of points between

For the further classification of functions we divide the domain function may be “continuous to the right” at a.

the end-points. Suppose that the domain is the interval between a A function is said to be “continuous throughout an interval' and b. Let intermediate points x1, *?,: X-1, be taken so that when it is continuous at every point of the interval. This implies increases indefinitely, all the differences b-x-1, *-*

>x> a. We may devise a rule by which, as n continuity to the right at the smaller end-value and continuity tend to zero as a limit. The interval is then said to be divided to the left at the greater end-value. When these conditions at the into indefinitely small partial intervals." ends are not satisfied the function is said to be continuous A function defined in an interval with the possible exception of “ within " the interval. By a “continuous function ” of one isolated points may be such that the interval can be divided into a variable we always mean a function which is continuous through- set of finite partial intervals within each of which the function is

monotonous (88). When this is the case the sum of the oscillations out an interval.

of the function in those partial intervals is finite, provided the The principal properties of a continuous function are:

function does not tend to become infinite. Further, in such a case 1. The lunction is practically constant throughout sufficiently the sum of the oscillations will remain below a fixed number for any small intervals. This means that, after any point a of the interval mode of dividing the interval into indefinitely small partial intervals. has been chosen, and any positive number 4, however small, has A class of functions may be defined by the condition that the sum been specified, it is possible to find a number h, so that the difference of the oscillations has this property, and such functions are said between any two values of the function in the interval between to have a restricted oscillation. Sometimes the phrase " limited e-hand ath is less than 6. There is an obvious modification is a fluctuation" is used. It can be proved that any function with is an end-point of the interval.

restricted oscillation is capable of being expressed as the sum of 2. The continuity of the function is “uniform." This means two monotonous functions, of which one never increases and the other that the number k which corresponds to any e as in (1) may be the never diminishes throughout the interval. Such a function has a same at all points of the interval, or, in other words, that the numbers limit on the right and a limit on the left at every point of the interval. h which correspond to e for different values of a have a positive This class of functions includes all those which have a finite number inferior limit.

of maxima and minima in a finite-interval, and some which have an 3. The function has a greatest value and a least value in the infinite number. It is to be noted that the class does not include all interval, and these are superior and inferior limits which are attained. continuous functions.

4. There is at least one point of the interval at which the function 12. Differentiable Function.—The idea of the differentiation takes any value between its greatest and least values in the interval. 5. If the interval is unlimited towards the right (or towards the

of a continuous function is that of a process for measuring the lelt), the function has a limit at oo (or at --).

rate of growth; the increment of the function is compared with 10. Discontinuity of Functions. The discontinuities of a

the increment of the variable. If /(x) is defined in an interval function of one variable , defined in an interval with the possible containing the point e, and -k and a+k are points of the

interval, the expression exception of isolated points, may be classified as follows:

f(a+h)-f(a)

(1) (1) The function may become infinite, or tend to become

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

represents a function of h, which we may call $(h), defined at all

points of an interval for k between -k and k except the point o. (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 | Thus the four limits $(+0), 4(+o), f(-0), 0(-o) exist, and two other, and at least one of them must be different from the value or more of them may be equal. When the first two are equal of the function at the point.

either of them is the "progressive differential coeficient" of (4) The function may have no limit at a point, or no limit on !(x) at the point a; when the last two are equal either of them the left, or no limit on the right, at a point.

is the “ regressive differential coefficient " of f(x) at a; when all

four are equal the function is said to be a differentiable" at a, 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. "What

and either of them is the “differential coefficient ” of f(x) at a, between a and a +h (a excluded) have a superior dimrit for a greatest of(x) or by J'(x). In this case (h) has a definite limit at k=0, ever positive number k we take, the values of the function at points or the “first derived function " of f(x) at a. It is denoted by 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

or is determinately infinite at h=0 (87). The four limits here in the right--the inferior limit of the superior limits in diminishing question are called, after Dini, the “ four derivates " of /(x) at a. neighbourhoods, and the superior limit of the inferior limits in in accordance with the notation for derived functions they may diminishing neighbourhoods. These are. denoted by Ma+o) and be denoted by /(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 f(a -o).

51(a), [_(a), f-(a), L-(c). Unless f(x) becomes, or tends to become, infinite at a, all these must A function which has a finite differential coefficient at all points exist, any two of them may be equal, and at least one of them must. of an interval is continuous throughout the interval, but if the be different from f(a), ir [(a) exists. If the first two are equal there differential coefficient becomes infinite at a point of the interval is a limit on the right denoted by f(a+o); if the second two are the function may or may not be continuous throughout the interval; equal, there is a limit on the left denoted by f(a-o). In case the on the other hand a function may be continuous without being function becomes, or tends to become, infinite at a, one or more of differentiable. This result, comparable in importance, from the these limits is infinite in the sense explained in $.7; and now is point of view of the general theory of functions, with the discovery to be noted that, e.g. the superior limit of the inferior limits in of Fourier's theorcm, is due to G. F. B. Riemann; but the failure diminishing neighbourhoods on the right of a may be negatively of an attempt made by Ampère to prove that every continuous infinite; this happens is, after any number N, however great, has function must be differentiable may be regarded as the first step in been specihed, it is possible to find a positive number h, so that all the theory. Examples of analytical expressions which represent the values of the function in the interval between a and oth (a continuous functions that are not differentiable have been given by excluded) are less than - N; in such a case }(x) tends to become Ricmann, Weierstrass, Darboux and Dini (sce 24). The most negatively infinite when x decreases towards a: other modes of important theorem in regard to differentiable functions is the tending to infinite limits may be described in similar terms. "theorem of intermediate value." (See INFINITESIMAL. CALCULUS:)

)

is

13. Analytic Function.-11 S(x) and its first * differential in partial intervals the sum of whose breadths can be diminished coefficients, denoted by l'(x), f'(x), .../(*)-(x), are continuous

indefinitely.

These partial intervals must be a set chosen out of some complete in the interval between a and atk, then

set obtained by the process used in the definition of integration.. f(a+h) =f(e) +Hf"(a) + F"(a) +...

4. The sum or product of two integrable functions is integrable. As regards integrable sunctions we have the following theorems :

1. I1S and I are the superior and inferior limits (or greatest and +7)e

f(0-1)(c) +R where R, may have various forms, some of which are given in intermediate between S(b-a) and I(6-a).

least values) of f(x) in the interval between a and b. Sys(x)dx i the article INFINITESIMAL CALCULUS. This result is known as

2. The integral is a continuous function of each of the end-values. “ Taylor's theorem.”

3. If the further end-value B is variable, and if f* (x)dx=F(x). When Talyor's theorem leads to a representation of the function by means of an infinite series, the function is said to be then if f(x) is continuous at b, F(x) is differentiable at 6, and

F'() = f(b). "analytic" (cf. § 21)

4. In case f(x) is continuous throughout the interval F(x) is con14. Ordinary Function.—The idea of a curvé representing a tinuous and differentiable throughout the interval, and F'(x)=f(x) continuous function in an interval is that of a line which has the throughout the interval. following properties: (1) the co-ordinates of a point of the curve

5. In case f'(x) is continuous throughout the interval between a

and b, are a value x of the argument and the corresponding value y of the function; (2) at every point the curve has a definite tangent;

Sc="(x)dx=f(b) –s(a). (3) the interval can be divided into a finite number of partial 6. In case f(x) is discontinuous at one or more points of the interval intervals within each of which the function is monotonous; between a and b, in which it is integrable, (4) the property of monotony within partial intervals is retained after interchange of the axes of co-ordinates x and y: According is a function of x, of which the four derivates at any point of the

Sos(x)de to condition (2) y is a continuous and differentiable function interval arc equal to the limits of indefiniteness of f(x) at the point. of x, but this condition does not include conditions (3) and (4):

7. It may be that there exist functions which are differentiable there are continuous partially monotonous functions which are throughout an interval in which their differential coefficients are not differentiable, there are continuous differentiable functions not integrable; is, however, F(x) is a function whose differential which are not monotonous in any interval however small; and coefficient, F"(x), is integrable in an interval, then there are continuous, differentiable and monotonous functions

F(x)=S; F"(x)dx+const. which do not satisfy condition (4) (cf. § 24). A function which can be represented by a curve, in the sense explained above, is Similarly, if any one of the four derivates of a function is integrable

where a is a fixed point, and x a variable point, of the interval. said to be “ordinary," and the curve is the graph of the function in an inierval, all are integrable, and the integral of either differs from (82). All analytic functions are ordinary, but not all ordinary the original function by a constant only. functions are analytic.

The theorems (4). (6), (7) show that there is some discrepancy 15. Integrable Function. The idea of integration is twofold. between the indefinite integral considered as the function which has

a given function as its differential coefficient, and as a definite We may seek the function which has a given function as its integral with a variable end-value. differential coefficient, or we may generalize the question of We have also two theorems concerning the integral of the product finding the area of a curve. The first inquiry leads directly to the of two integrable functions f(x) and (x): these are known as "the

The first theorem of the indefinite integral, the second directly to the definite integral. first and second theorems of the mean."

mean is that, if $(x) is one-signed throughout the interval between Following the second method we define “the definite integrala and b, there is a number 1 intermediate between the superior of the function f(x) through the interval between a and b" to be and inferior limits, or greatest and least values, of f(x) in the interval, the limit of the sum

which has the property expressed by the equation

Mf*(r)dx= $*f(x)$(x)dx.
Ef(x)(x,-4-1)

The second theorem of the mean is that, if f(x) is monotonous when the interval is divided into ultimately indefinitely small has the property expressed by the equation

throughout the interval, there is a number & between a and b which partial intervals by points x1, x2, . . . Xn-l. Here x'; denotes any point in the rth partial interval, xo is put for a, and x, for b. $1«)(x)dx=f(0)S**(x)dx +F(0) Sf=(x)dx. It can be shown that the limit in question is finite and inde

(See Fourier's Series.) pendent of the mode of division into partial intervals, and of the

16. Improper Definite Integrals.-We may extend the idea of choice of the points such as x',, provided (1) the function is integration to cases of functions which are not defined at some defined for all points of the interval, and does not tend to become point, or which tend to become infinite in the neighbourhood of infinite at any of them; (2) for any one mode of division of the

some point, and to cases where the domain of the argument interval into ultimately indefinitely small partial intervals, the extends to infinite values. If c is a point in the interval between sum of the products of the oscillation of the function in each

Q and 6 at which /(x) is not defined, we impose a restriction on partial interval and the difference of the end-values of that the points x', of the definition: none of them is to be the point c. When these conditions are satisfied the function is said to be This comes to the same thing as defining S1(x)dx 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 where, to fix ideas, b is taken >a, and e and é are positive. The his memoir on trigonometric series (1854). A still more general same definition applies to the case where f(x) becomes infinite, or definition has been given by Lebesgue. As the more general tends to become infinite, at c, provided both the limits exist.

This definition may be otherwise expressed by saying that a definition cannot be made intelligible without the introduction of some rather recondite notions belonging to the theory of partial interval containing the point c is omitted from the aggregates, we shall, in what follows, adhere to Riemann's interval of integration, and a limit taken by diminishing the

breadth of this partial interval indefinitely; in this form it definition.

applies to the cases where c is a or b. We have the following theorems:

Again, when the interval of integration is unlimited to the 1. Any continuous function is integrable.

right, or extends to positively infinite values, we have as a 2. Any function with restricted oscillation is integrable,

definition 3. A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the

Xx)dx=L! S/(x)dx, function exceeds a given number o, however small, can be enclosed

[ocr errors]

XI. 6

2a

sinx

provided this limit exists. Similar definitions apply to " 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 $** $(x)dx, and to So S(r)dx.

m, and to choose points x', *", ...x("), so that the neighbourhood

of a for e contains x', and consists exclusively of points within H., All such definite integrals as the above are said to be “ improper." and similarly for x' and x", x' and x",...xmi and b. Condition For example. S.** dx is improper in two ways. It means

(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 Lt Li

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. in which the positive number e is first diminished indefinitely,

18. Funclions of Several Variables.-A function of several and the positive number h is afterwards increased indefinitely

variables difiers from a function of one variable in that the The theorems of the mean” ($ 15) require modification when argument of the function consists of a set of variables, or is a the integrals are improper (see FOURIER'S SERIES).

variable point in a Cn when there are n variables. The function When the improper definite integral of a function which is definable by means of the domain of the argument and the becomes, or tends to become, infinite, exists, the integral is said rule of calculation. In the most important cases the domain of to be “convergent.” If f(x) tends to become infinite at a point the argument is a homogeneous part H. of Cn with the possible c in the interval between a and b, and the expression (1) does not exception of isolated points, and the rule of calculation is that exist, then the expression S/(r)dt, which has no value, is called the value of the function in any assigned part of the domain a“ divergent integral," and it may happen that there is a definite of the argument is that value which is assumed at the point by value for

an assigned analytical expression. The limit of a function at a Le{ $*$(x)dx+ fitxf(x)dx}

point a is defined in the same way as in the case of a function of

one variable. provided that € and é are connected by some definite relation, We take a positive fraction e and consider the neighbourhood of a and both, remaining positive, tend to limit zero. The value of

for h, and from this neighbourhood we exclude the point a, and we the above limit then called a “principal value "of the divergent 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. integral. Cauchy's principal value is obtained by making é =€, neighbourhood. The function has a limit at a if for any positive e i.e. by taking the omitted interval so that the infinity is at however small, there is a corresponding h which has the property its middle point. A divergent integral which has one or more

that | S(x)-f(x) | <€, whatever points x, x' in the neighbourhood

of a for h' we take (a excluded). For example, when there are two principal values is sometimes described as “semi-convergent.”

variables x1, 73, and both are unrestricted, the domain of the argu. 17. Domain of a Set of Variables.-The numerical continuum ment is represented by a plane, and the values of the function are of n dimensions (Cr) is the aggregate that is arrived at by attribut-correlated with the points of the plane. The function has a limit ing simultaneous values to each of n variables xi, X3, . . . Xn,

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 these values being any real numbers. The elements of such an

the function which correspond to any two points of the region aggregate are called “points," and the numbers x1, x2, . . . Xn (neither of the points being c) can be made as small as we please the “co-ordinates" of a point. Denoting in general the points in absolute value by contracting all the linear dimensions of the (#1, #2, - .. Xn) and (x'ı, x': ... o'n) by x and x', the sum of region sufficiently. When the domain of the argument of a function the differences 1x1 - x'1 + 1 xz-a'z1+

+ 1 Xn-a'ml may

of n variables extends to an infinite distance, there is a "limit at

an infinite distance" is, alter any number e, however small, has been be denoted by Ix-x' and called the “ difference of the two specified, a number N can be found which is such that if(x')-f(x) <¢, points." We can in various ways choose out of the continuum for all points x and x' (of the domain) of which one or more coan aggregate of points, which may be an infinite aggregate, and ordinates exceed N in absolute value. In the case of functions of any such aggregate can be the “ domain " of a“ variable point.” | domain. The deinition of such a limit is verbally the same as the

several variables great importance attaches to limits for a restricted The domain is said to "extend to an infinite distance " if, after corresponding definition in the case of functions of one variable any number N, however great, has been specified, it is possible (8 6). For example, a function of xı and x3 may have a limit at to find in the domain points of which one or more co-ordinates (x1=0, x2 = 0) if we first diminish xi without limit, keeping x, conexceed N in absolute value. The " neighbourhood” of a point stant, and afterwards diminish x2 without limit. Expressed in a for a (positive) number he is the aggregate constituted of all the origin along the axis of xz.

geometrical language, this process amounts to approaching the

The definitions of superior and inferior points x, which are such that the “difference” denoted by limits, and of maxima and minima, and the explanations of what Ir-al<h. If an infinite aggregate of points does not extend is meant by saying that a function of several variables becomes to an infinite distance, there must be at least one point a, which infinite, or tends to become infinite, at a point, are almost identical

verbally with the corresponding definitions and explanations in the has the property that the points of the aggregate which are in case of a function of one variable (87). The definition of a continuous the neighbourhood of a sor any number h, however small, them- function (89) admits of immediate extension; but it is very imselves constitute an infinite aggregate, and then the point e is portant to observe that a function of two or more variables may be called a “limiting point” of the aggregate; it may or may not

a continuous function of each of the variables, when the rest are kept

constant, without being a continuous function of its argument. be a point of the aggregate. An aggregate of points is a perfect

For example, a sunction of x and y may be defined by the conditions when all its points are limiting points of it, and all its limiting that when x=0 it is zero whatever value y may have, and when points are points of it; it is connected” when, after taking * *o it has the value of sin (4 tan- \\/x)). When y has any particular any two points a, b of it, and choosing any positive number e,

value this function is a continuous function of x, and, when x has however small, a number m and points x', *", ...zm) of the but the function of x and y is discontinuous at (x=0, y=0).

any particular value this function is a continuous function of y; aggregate can be found so that all the differences denoted by

19. Differentiation and Integration.—The definition of partial lx'-a1,1x" —x'),...16-(m)| are less than e. A perfect connected aggregate is a continuum. This is G. Cantor's definition difficulty. The most important theorems concerning differ:

differentiation of a function of several variables presents no The definition of a continuum in C. leaves open the question of the number of dimensions of the continuum, and a further explana- entiable functions are the " theorem of the total difierential,” tion is necessary in order to define arithmetically what is meant by a the theorem of the interchangeability of the order of partial " homogeneous part " H. of Cn. Such a part would correspond to differentiations, and the extension of Taylor's theorem (see an interval in C. or to an arca bounded by a simple closed contour INFINITESIMAL CALCULUS). in Cz; and, besides being perfect and connected, it would have the following properties: (1) There are points of Ca, which are not points

With a view to the establishment of the notion of integration of Har these form a complementary aggregate H'm; (2) There are through a domain, we must define the “ extent " of the domain. points within " Hai this means that lor any such point there is Take first a domain consisting of the point a and all the points a a neighbourhood consisting exclusively of points of 'I!. (3) The for which lx-e/<th, where h is a chosen positive number; points of H, which do not lie" within HR are limiting points of

the extent of this domain is h“,n being the number of variables; H',; they are not points of H'n, but the neighbourhood of any such point for any number h, however small, contains points within H. such a domain may be described as “square," and the number k and points of H'n: the aggregate of these points is called the may be called its “breadth "; it is a homogeneous part of the numerical continuum of n dimensions, and its boundary consists | x, the sum sm of the first n terms of the series is a function of x of all the points for which \x-01 = sh. Now the points of and n; and, when the series is convergent, its sum, which is any domain, which does not extend to an infinite distance, may Lln - Sm, can represent a function of x. In most cases the series be assigned to a finite number m of square domains of finite converges for some values of x and not for others, and the values breadths, so that every point of the domain is either within one for which it converges form the " domain of convergence." of these square domains or on its boundary, and so that no point | The sum of the series represents a function in this domain. is within two of the square domains; also we may devise a rule The apparently more general method of representation of a by which, as the number m increases indefinitely, the breadths function of one variable as the limit of a function of two variables of all the square domains are diminished indefinitely. When has been shown by R. Baire to be identical in scope with the method this process is applied to a homogeneous part, H, of the numerical of series, and it has been developed by him so as to give a very continuum Cn, then, at any stage of the process, there will be analytical expressions. For example, he has shown that Riemann's some square domains of which all the points belong to H, and totally discontinuous sunction, which is equal to I when x is rational there will generally be others of which some, but not all, of the and 100 when x is irrational, can be represented by an analytical points belong to H. As the number m is increased indefinitely to the problem of the representation of a continuous function by

expression. . An infinite process of a different kind has been adapted the sums of the extents of both these categories of square T. Broden. He begins with a function having a graph in the form domains will tend to definite limits, which cannot be negative; of a regular polygon, and interpolates additional angular points in when the second of these limits is zero the domain H is said to

an ordered sequence without limit, The representation of a function be" measurable," and the first of these limits is iis " extent "; while the representation by means of a definite integral is analogous

by means of an infinite product falls clearly under Baire's method, it is independent of the rule adopted for constructing the square to Brodén's method. As an example of these two latter processes domains and contracting their breadths. The notion thus intro- we may çite the Gamma function 11(x)] defined for positive values duced may be adapted by suitable modifications to continua of of x by the definite integral lower dimensions in Cn.

Serra, The integral of a function f(x) through a measurable domain Hi or by the infinite product which is a homogeneous part of the numerical continuum of » dimensions, is defined in just the same way as the integral through

Lla-. -*/x(1+x)(1+3«). (1+1). an interval, the extent of a square domain taking the place of the

The second of these expressions avails for the representation of the difference of the end-values of a partial interval; and the condition function at all points at which x is not a negative integer. of integrability takes the same form as in the simple case. In par. ticular, the condition is satisfied when the function is continuous

21. Power Series.- Taylor's theorem leads in certain cases throughout the domain. The definition of an integral through a

to a representation of a function by an infinite series. We have domain may be adapted to any domain of measurable extent. The under certain conditions (13) extensions to "improper " definite integrals may be made in the same way as for a function of one variable; in ihe particular case

f(x) =f(a) +"(r a)"p"}(e) +Rai 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

$(x) f(a)+ (r-alpina), integral may have different (principal) values for dificrent modes of contracting the extent of the omitted partial domain. In applica- provided that (a) a positive number k can be found so that at vergent integrals and io principal values of divergent integrals. all points in the interval between a and a+k (except these points) For example, any component of magnetic force at a point within a /(x) has continuous differential coefficients of all finite orders, magnet, and the corresponding component of magnetic induction and at e has progressive differential coefficients of all finite at the same point are expressed by different principal values of the orders; (B) Cauchy's form of the remainder Ra, viz. same divergent integral. "Delicate questions arise as to the possibility -a)" of representing the integral of a function of n variables through a Fi)!

37(1-0)--ifinifa+0(x-a)}, has the limit zero when u indomain , as a repeated integral, of evaluating it by successive integrations with respect to the variables one at a time and of inter

creases indefinitely, for all values of 0 between o and 1, and for changing the order of such integrations. These questions have been all values of x in the interval between a and a+k, except possibly discussed very completely by C. Jordan, and we may, quote the a+k. When these conditions are satisfied, the series (1) repreresult that a! the transformations in question are valid when the sents the function at all points of the interval between a and atk, function is continuous throughout the domain.

except possibly a+k, and the function is "analytic ” ($ 13) in 20. Representation of Functions in General. — We have seen this domain. "Obvious modifications admit of extension to an that the notion of a function is wider than the notion of an

interval between a and e-k, or between a-k and a+k. When analytical expression, and that the same function may be

a series of the form (1) represents a function it is called “ lhe "represented " by one expression in one part of the domain of Taylor's series for the function.” the argument and by some other expression in another part of Taylor's series is a power series, i.e. a series of the form the domain (8 5). Thus there arises the general problem of the

3 An(x-a)". representation of functions. The function may be given by specifying the domain of the argument and the rule of calcula- As regards power series we have the following theorems: tion, or else the function may have to be determined in accord- 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 ance with certain conditions; for example, it may have to

in the interval between a-k and a+k, with the possible exception satisfy in a prescribed domain an assigned differential equation of one or both end-points. In either case the problem is to determine, when possible, a 2. The power series represents a continuous function in its domain single analytical expression which shall have the same value as of convergence (the end-points may have to be excluded). the function at all points in the domain of the argument. For representing it is the Taylor's series for the function,

3. This function is analytic in the domain, and the power series the representation of most functions for which the problem can The theory of power series has been developed chiefly from the be solved recourse must be had to limiting processes. Thus we point of view of the theory of functions of complex variables. may utilize infinite series, or infinite products, or definite in- 22. Uniform Convergence. We shall suppose that the domain tegrals; or again we may represent a function of one variable of convergence of an infinite series of functions is an interval with as the limit of an expression containing two variables in a domain the possible exception of isolated points. Let f(x) be the sum in which one variable remains constant and another varies of the series at any point x of the domain, and fu(x) the sum of An example of this process is afforded by the expression the first nti terms. The condition of convergence at a point L!, --Xy/(.roy+1), which represents a sunction of r vanishing at o is that, after any positive number e, however small, has been x=0 and at all other values of x having the value of 1/x. The specified, it must be possible to find a number n so that method of series falls under this more general process (cf. § 6). Um(a)-fo(a)l<e for all values of m and p which exceed n. When the terms *, uz, ...of a series are functions of a variable | The sum, lla), is the limit of the sequence of numbers sala) at

« PreviousContinue »