Function

FUNCTION, 1 in mathematics, a variable number the value of which depends upon the values of one or more other variable numbers. The theory of functions is conveniently divided into (I.) Functions of Real Variables, wherein real, and only real, numbers are involved, and (II.) Functions of Complex Variables, wherein complex or imaginary numbers are involved.

I. Functions Of Real Variables I. Historical. - The word function, defined in the above sense, was introduced by Leibnitz in a short note of date 1694 concerning the construction of what we now call an " envelope " (Leibnizens mathematische Schriften, edited by C. I. Gerhardt, Bd. v. p. 306), and was there used to denote a variable length related in a defined way to a variable point of a curve. In 1698 James Bernoulli used the word in a special sense in connexion with some isoperimetric problems (Joh. Bernoulli, Opera, t. i. p. 255). He said that when it is a question of selecting from an infinite set of like curves that one which best fulfils some function, then of two curves whose intersection determines the thing sought one is always the " line of the function " (Linea functionis) . In 1718 John Bernoulli (Opera, t. ii. p. 241) defined a " function of a variable magnitude " as a quantity made up in any way of this variable magnitude and constants; and in 1730 (Opera, t. iii. p. 174) he noted a distinction between " algebraic " and " transcendental " functions. By the latter he meant integrals of algebraic functions. The notation f (x) for a function of a variable x was introduced by Leonhard Euler in 1734 (Comm. Acad. Petropol. t. vii. p. 186), in connexion with the theorem of the interchange of the order of differentiations. The notion of functionality or functional relation of two magnitudes was thus of geometrical origin; but a function soon came to be regarded as an analytical expression, not necessarily an algebraic expression, containing the variable or variables. Thus we may have rational integral algebraic functions such as ax e bx -Fc, or rational algebraic functions which are not integral, such as alxn+a2xn-T+... +an, b i x m +b 2 x m_T +... +b,,,, or irrational algebraic functions, such as x, or, more generally the algebraic functions that are determined implicitly by an algebraic e quation, as, for instance, fn (x, y) + fn-T (x, y) + ... Xf 0=0 1 The word " function " (from Lat. fungi, to perform) has many uses, with the fundamental sense of an activity special or proper to an office, business or profession, or to an organ of an animal or plant, the definite work for which the organ is an apparatus. From the use of the word, as in the Italian funzione, for a ceremony of the Roman Church, " function " is often employed for a public ceremony of any kind, and loosely of a social entertainment or gathering.

where f„, (x,y),.. . mean homogeneous expressions in x and y having constant coefficients, and having the degrees indicated by the suffixes, and fo is a constant. Or again we may have trigonometrical functions, such as sin x and tan x, or inverse trigonometrical functions, such as sin l x, or exponential functions, such as e x and a x, or logarithmic functions, such as log x and log (I +x). We may have these functional symbols combined in various ways, and thus there arises a great number of functions. Further we may have functions of more than one variable, as, for instance, the expression xy/(x 2 + y 2), in which both x and y are regarded as variable. Such functions were introduced into analysis somewhat unsystematically as the need for them arose, and the later developments of analysis led to the introduction of other classes of functions.

2. Graphic Representation

In the case of a function of one variable x, any value of x and the corresponding value y of the function can be the co-ordinates of a point in a plane. To any value of x there corresponds a point N on the axis of x, in accordance with the rule that x is the abscissa of N. The corresponding value of y determines a point P in accordance with the rule that x is the abscissa and y the ordinate of P. The ordinate y gives the value of the function which corresponds to that value of the variable x which is specified by N; and it may be described as " the value of the function at N." Since there is a one-to-one correspondence of the points N and the numbers x, we may also describe the ordinate as " the value of the function at x." In simple cases the aggregate of the points P which are determined by any particular function (of one variable) is a curve, called the "graph of the function" (see § 14). In like manner a function of two variables defines a surface.

3. The Variable

Graphic methods of representation, such as those just described, enabled mathematicians to deal with irrational values of functions and variables at the time when there was no theory of irrational numbers other than Euclid's theory of incommensurables. In that theory an irrational number was the ratio of two incommensurable geometric magnitudes. In the modern theory of number irrational numbers are defined in a purely arithmetical manner, independent of the measurement of any quantities or magnitudes, whether geometric or of any other kind. The definition is effected by means of the system of ordinal numbers (see Number). When this formal system is established, the theory of measurement may be founded upon it; and, in particular, the co-ordinates of a point are defined as numbers (not lengths), which are assigned in accordance with a rule. This rule involves the measurement of lengths. The theory of functions can be developed without any reference to graphs, or co-ordinates or lengths. The process by which analysis has been freed from any consideration of measurable quantities has been called the " arithmetization of analysis." In the theory so developed, the variable upon which a function depends is always to be regarded as a number, and the corresponding value of the function is also a number. Any reference to points or coordinates is to be regarded as a picturesque mode of expression, pointing to a possible application of the theory to geometry. The development of " arithmetized analysis " in the 19th century is associated with the name of Karl Weierstrass.

All possible values of a variable are numbers. In what follows we shall confine our attention to the case where the numbers are real. When complex numbers are introduced, instead of real ones, the theory of functions receives a wide extension, which is accompanied by appropriate limitations (see below, II. Functions of Complex Variables). The set of all real numbers forms a continuum. In fact the notion of a onedimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

4. Domain of a Variable

Theory of Aggregates. - The notion of a " variable " is that of a number to which we may assign at pleasure any one of the values that belong to some chosen set, or aggregate, of numbers; and this set, or aggregate, is called the " domain of the variable." This domain may be an " interval," that is to say it may consist of two terminal numbers, all the numbers between them and no others. When this is the case the number is said to be " continuously variable." When the domain consists of all real numbers, the variable is said to be " unrestricted." A domain which consists of all the real numbers which exceed some fixed number may be described as an " interval unlimited towards the right "; similarly we may have an interval " unlimited towards the left." In more complicated cases we must have some rule or process for assigning the aggregate of numbers which constitute the domain of a variable. The methods of definition of particular types of aggregates, and the theorems relating to them, form a branch of analysis called the"theory of aggregates" (Mengenlehre, Theorie des ensembles, Theory of sets of points). The notion of an " aggregate " in general underlies the system of ordinal numbers. An aggregate is said to be " infinite " when it is possible to effect a one-to-one correspondence of all its elements to some of its elements. For example, we may make all the integers correspond to the even integers, by making I correspond to 2, 2 to 4, and generally n to 2n. The aggregate of positive integers is an infinite aggregate. The aggregates of all rational numbers and of all real numbers and of points on a line are other examples of infinite aggregates. An aggregate whose elements are real numbers is said to " extend to infinite values " if, after any number N, however great, is specified, it is possible to find in the aggregate numbers which exceed N in absolute value. Such an aggregate is always infinite. The " neighbourhood of a number (or point) a for a positive number h " is the aggregate of all numbers (or points) x for which the absolute value of x - a denoted by I x - a I, does not exceed h. 5. General Notion of Functionality. - A function of one variable was for a long time commonly regarded as the ordinate of a curve; and the two notions (1) that which is determined by a curve supposed drawn, and (2) that which is determined by an analytical expression supposed written down, were not for a long time clearly distinguished. It was for this reason that Fourier's discovery that a single analytical expression is capable of representing (in different parts of an interval) what would in his time have been called different functions so profoundly struck mathematicians (§ 23). The analysts who, in the middle of the 19th century, occupied themselves with the theory of the convergence of Fourier's series were led to impose a restriction on the character of a function in order that it should admit of such representation, and thus the door was opened for the introduction of the general notion of functional dependence. This notion may be expressed as follows: We have a variable number, y, and another variable number, x, a domain of the variable x, and a rule for assigning one or more definite values to y when x is any point in the domain; then y is said to be a " function " of the variable x, and x is called the " argument " of the function. According to this notion a function is, as it were, an indefinitely extended table, like a table of logarithms; to each point in the domain of the argument there correspond values for the function, but it remains arbitrary what values the function is to have at any such point.

For the specification of any particular function two things are requisite: (r) a statement of the values of the variable, or of the aggregate of points, to which values of the function are to be made to correspond, i.e. of the " domain of the argument "; (2) a rule for assigning the value or values of the function that correspond to any point in this domain. We may refer to the second of these two essentials as " the rule of calculation." The relation of functions to analytical expressions may then be stated in the form that the rule of calculation is: " Give the function the value of the expression at any point at which the expression has a determinate value," or again more generally, " Give the function the value of the expression at all points of a definite aggregate included in the domain of the argument." The former of these is the rule of those among the earlier analysts who regarded an analytical expression and a function as the same thing, and their usage may be retained without causing confusion and with the advantage of brevity, the analytical expression serving to specify the domain of the argument as well as the rule of calculation, e.g. we may speak of " the function i/x." This function is defined by the analytical expression i/x at all points except the point x = o. But in complicated cases separate statements of the domain of the argument and the rule of calculation cannot be dispensed with. In general, when the rule of calculation is determined as above by an analytical expression at any aggregate of points, the function is said to be " represented " by the expression at those points.

When the rule of calculation assigns a single definite value for a function at each point in the domain of the argument the function is " uniform " or " one-valued." In what follows it is to be understood that all the functions considered are one-valued, and the values assigned by the rule of calculation real. In the most important cases the domain of the argument of a function of one variable is an interval, with the possible exception of isolated points.

6. Limits

Let f(x) be a function of a variable number x; and let a be a point such that there are points of the domain of the argument x in the neighbourhood of a for any number h, however small. If there is a number L which has the property that, after any positive number however small, has been specified, it is possible to find a positive number h, so that IL - f(x)(x) I < e for all points x of the domain (other than a) for which lx - al <h, then L is the " limit of f(x) at the point a." The condition for the existence of L is that, after the positive number has been specified, it must be possible to find a positive number h, so that I f(x') - f(x)l< e for all points x and x of the domain (other than a) for which Ix - al <h and Ix' - aI<h. It is a fundamental theorem that, when this condition is satisfied, there exists a perfectly definite number L which is the limit of f(x) at the point a as defined above. The limit of f(x) at the point a is denoted by Ltx=a f(x), or by lim a=a f(x).

If f(x) is a function of one variable x in a domain which extends to infinite values, and if, after has been specified, it is possible to find a number N, so that I f(x') - f(x) I < e for all values of x and x' which are in the domain and exceed N, then there is a number L which has the property that I f(x) - L for all such values of x. In this case f(x) has a limit L at x = co. In like manner f(x) may have a limit at x= - Go . This statement includes the case where the domain of the argument consists exclusively of positive integers. The values of the function then form a " sequence," u1, u 2, .. u n, ..., and this sequence can have a limit at n= co .

The principle common to the above definitions and theorems is called, after P. du Bois Reymond, " the general principle of convergence to a limit." It must be understood that the phrase " x = oo " does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit; it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which " extends to infinite values." A very important type of limits is furnished by infinite series. When a sequence of numbers u1, u 2 ,... u, ... is given, we may form a new sequence sl, s2, ... s,,, ...from it by the rules s 1 = u,, s2=ui+u2,

. s,,=u i +u. +u,,, or by the equivalent rules s 1 = u, s n - s,,,_ 1 = u n (n = 2, 3, ...). If the new sequence has a limit at n = co, this limit is called the " sum of the infinite series " u1 +--u 2 + ..., and the series is said to be " convergent " (see A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin (1 /x) has limit zero at o if x is restricted to the aggregate I/ir, I/27r, ... I/n7r, .. or to the aggregate I/27r, 2//51r, ... n /(n 2 +I)7r,.. ., but if x takes all values in the neighbourhood of 0, sin (I /x) has not a limit at o. Again, there may be a limit at a if the points x in the neighbourhood of a are restricted by the condition that x - a is positive; then we have a " limit on the 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 domain." The limits on the left and on the right are denoted by f(a - o) and f(a +o). The limit L of f(x) at a stands in no necessary relation to the value of f(x) at a. If the point a is in the domain of the argument, the value of f(x) at a is assigned by the rule of calculation, and may be different from L. In case f(a) =L the limit is said to be " attained." If the point a is not in the domain of the argument, there is no value for f(x) at a. In the case where f(x) is defined for all points in an 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 and assign to f(a) the value L; the function may then be said to be " extrinsically defined." The so-called " indeterminate forms " (see are examples.

7. Superior and Inferior Limits; Infinities

The value of a function at every point in the domain of its argument is finite, since, by definition, the value can be assigned, but this does not necessarily imply that there is a number N which exceeds all the values (or is less than all the values). It may happen that, however great a number N we take, there are among the values of the function numbers which exceed N (or are less than - N).

If a number can be found which is greater than every value of the function, then either (a) there is one value of the function which exceeds all the others, or (0) there is a number S which exceeds every value of the function but is such that, ho wever small a positive number we take, there are values of the function which exceed S - e. In the case (a) the function has a greatest value; in case (13) the function has a " superior limit " S, and then there must be a point a which has the property that there are points of the domain of the argument, in the neighbourhood of a for any h, at which the values of the function differ from S by less than E. Thus S is the limit of the function at a, either for the domain of the argument or for some more restricted domain. If a is in the domain of the argument, and if, after omission of a, there is a superior limit S which is in this way the limit of the function at a, if further f (a) = S, then S is the greatest value of the function; in this case the greatest value is a limit (at any rate for a restricted domain) which is attained; it may be called a " superior limit which is attained." In like manner we may have a " smallest value " or an " inferior limit," and a smallest value may be an "inferior limit which is attained." All that has been said here may be adapted to the description of greatest values, superior limits, &c., of a function in a restricted domain contained in the domain of the argument. In particular, the domain of the argument may contain an interval; and therein the function may have a superior limit, or an inferior limit, which is attained. Such a limit is a maximum value or a minimum value of the function.

Again, if, after any number N, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceeds N, the values of the function are said to have an " infinite superior limit," and then there must be a point a which has the property that there are points of the domain, in the neighbourhood of a for any h, at which the value of the function exceeds N. If the point a is in the domain of the argument the function is said to " tend to become infinite " at a; it has of course a finite value at a. If the point a is not in the domain of the argument the function is said to " become infinite " at a; it has of course no value at a. In like manner we may have a (negatively) infinite inferior limit. Again, after any number N, however great, has been specified and a number h found, so that all the values of the function, at points in the neighbourhood of a for h, exceed N in absolute value, all these values may have the same sign; the function is then said to become, or to tend to become, " determinately (positively or negatively) infinite "; otherwise it is said to become or to tend to become, " indeterminately infinite." All the infinities that occur in the theory of functions are of the nature of variable finite numbers, with the single exception of the infinity of an infinite aggregate. The latter is described as an " actual infinity," the former as " improper infinities." There is no " actual infinitely small " corresponding to the actual infinity. The only " infinitely small " is zero. All " infinite values " are of the nature of superior and inferior limits which are not attained.

8. Increasing and Decreasing Functions

A function f(x) of one variable x, defined in the interval between a and b, is " increasing throughout the interval " if, whenever x and x' are two numbers in the interval and x'> x, then f(x') > f (x); the function " never decreases throughout the interval " if, x and x being as before, f (x') >f(x). Similarly for decreasing functions, and for functions which never increase throughout an interval. A function which either never increases or never diminishes throughout an interval is said to be " monotonous throughout " the interval. If we take in the above definition b> a, the definition may apply to a function under the restriction that x is not b and x is not a; such a function is " monotonous within " the interval. In this case we have the theorem that the function (if it never decreases) has a limit on the left at b and a limit on the right at a, and these are the superior and inferior limits of its values at all points within the interval (the ends excluded); the like holds mutatis mutandis if the function never increases. If the function is monotonous throughout the interval, f(b) is the greatest (or least) value of f (x) in the interval; and if f (b) is the limit of f (x) on the left at b, such a greatest (or least) value is an example of a superior (or inferior) limit which is attained. In these cases the function tends continually to its limit.

These theorems and definitions can be extended, with obvious modifications, to the cases of a domain which is not an interval, or extends to infinite values. By means of them we arrive at sufficient, but not necessary, criteria for the existence of a limit; and these are frequently easier to apply than the general principle of convergence to a limit (§ 6), of which principle they are particular cases. For example, the function represented by x log (I/x) continually diminishes when tie >x>o and x diminishes towards zero, and it never becomes negative. It therefore has a limit on the right at x=o. This limit is zero. The function represented by x sin (I/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=o, however small. Nevertheless, the function has the limit zero at x =o.

9. Continuity of Functions. - A function f(x) of one variable x is said to be continuous at a point a if (i) 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 a 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 " within " the interval. By a " continuous function " of one variable we always mean a function which is continuous throughout an interval.

The principal properties of a continuous function are: I. 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 e, 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 a as in (I) 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 co (or at -ce).

io. 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: (r) 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.

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 value), and an inferior limit (or a least value); further, as h 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+o) 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- o) and f(a- o). 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 +0); 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 a+h (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.

Oscillation of Functions

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

For the further classification of functions we divide the domain of the argument into partial intervals by means of points between the end-points. Suppose that the domain is the interval between a and b. Let intermediate points x 1, x,. xn_l, be taken so that b> n _ 1 > x n _ 2.. > i > a. We may devise a rule by which, as n increases indefinitely, all the differences b -x n _ 1, x n _ 1 -x n _ 2, ... x, -a tend to zero as a limit. The interval is then said to be divided 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.

i 2. 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--1--k are points of the interval, the expression f(a+h) -f(a) (I) h represents a function of h, which we may call (/)(h), defined at all points of an interval for h between -k and k except the point o. Thus the four limits ¢(-}-o), y6(+o), 4(-o), c)( -o) 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 d dx) or by f' (x) . In this case 4)(h) has a definite limit at h = o, 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 f+(a), f-(a), .f'-(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 Ampere 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 Infinitesimal Calculus.) 13. Analytic Function. - If f(x) and its first n differential coefficients, denoted by f'(x), f"(x), ... f(") (x), are continuous in the interval between a and ad-h, then 2 f(a + h) =f(a)+hf'(a)+2,if"(a)+... (n - I)! f (n1) (a)+--R„, 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: (I) the co-ordinates of a point of the curve 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; (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 f (x 'T) -i) when the interval is divided into ultimately indefinitely small partial intervals by points x 1, x 2,. .. x,,_ l. Here x', denotes any point in the rth partial interval, x 0 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 (I) 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 n 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 Piave the following theorems: I. 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 in partial intervals the sum of whose breadths can be diminished indefinitely.

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: I. 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 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, f'(x)dx =f(b) - f(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 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) = a F'(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 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 cp(x); these are known as " the first and second theorems of the mean." The first theorem of the mean is that, if 4(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 McS f (x) dx = f f (x)q The second theorem of the mean is that, if f(x) is monotonous throughout the interval, there is a number E between a and b which has the proper d by the equation b =f(a) f q5(x)dx-f f(b) f co(x)dx. (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',. of the definition: none of them is to be the point c. This comes to the same thing as defining f f(x)dx to be Lt f ° f (x)dx+Lt f b E, f(x)dx, (I) e=0 a F'=0 c+ where, to fix ideas, b is taken>a, and e and E 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 ax f(x)dx = Lt f h f(x)dx provided this limit exists. Similar definitions apply to a f(x)dx, and tof f(x)dx. All such definite integrals as the above are said to be " improper." x For example, f is improper in two ways. It means Li ( sin h=œ =0. e x 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 (I) does not exist, then the expression f(x)dx, which has no value, is called d a " divergent integral," and it may happen that there is a definite value for Lt f(x)dx+f.+E,f(x)dx provided that e and e 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 'e =E, ' 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,) is the aggregate that is arrived at by attributing simultaneous values to each of n variables x 1, x 2, ... xn, these values being any real numbers. The elements of such an aggregate are called " points," and the numbers x,, x 2,. .. xn the " co-ordinates " of a point. Denoting in general the points (x l , x2,. .. x,,) and (x' 1 , x' 2. .. x'n) by x and x', the sum of the differences I x i -x', I + I x 2 -x' 2 I + ... -iI xn-x', I may be denoted by I x-x' I 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 Ix-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, 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 E, however small, a number m and points x', x", ... x(m) of the aggregate can be found so that all the differences denoted by ix' - a I, I x" - x' I, ...lb- b 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 n of C. Such a part would correspond to an interval in Cl, or to an area bounded by a simple closed contour in and, besides being perfect and connected, it would have the following properties: (I) There are points of C., which are not points of H n; these form a complementary aggregate H' n . (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 Hn and points of H'„: the aggregate of these points is called the boundar y " of H n . (4) When any two points a, b within H, are taken, it is possible to find a number and a corresponding number m, and to choose points x', x",...x(m), so that the neighbourhood of a for contains x', and consists exclusively of points within H,,, 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.

IS. 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,, 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 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 however small, there is a corresponding h which has the property that (f(x')-f(x) whatever points 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 however small, has been specified, a number N can be found which is such that I f (x') -f (x) 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 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 (§ 6). For example, a function of x i and x 2 may have a limit at (x 1 =0, x 2 =o) if we first diminish x i without limit, keeping x 2 constant, and afterwards diminish x 2 without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x 2. 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 (§ 7). The definition of a continuous function (§ 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 o it has the value of sin {4 tan 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 = o, y= o).

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 -a '<2h, where a chosen positive number; the extent of this domain is 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 Lt numerical continuum of n dimensions, and its boundary consists of all the points for which Ix - al =P. 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 introduced may be adapted by suitable modifications to continua of lower dimensions in C,,.

The integral of a function f(x) through a measurable domain II, 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 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 (§ 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 xy / (x 2 y+i), which represents a function of x vanishing at x = o and at all other values of x having the value of i/x. The method of series falls under this more general process (cf. § 6). When the terms u 1, u 2, ... 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,, = o,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 I when x is rational and to o 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 [P(x)] defined for positive values of x by the definite integral e ttx-1dt, or by the infinite product Ltn =? n z '+Zx) ' The second of these expressions avails for the representation of the f unction 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)+Z 1 (x a) ro.)(a)+Rn; y and this becomes f (x) =f(a) a)r fir)(a), r 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; (/3) Cauchy's form of the remainder R,,, viz.

(

a) .a) i ('' (i - e)n=if cn>}a+B(x - a)}, has the limit zero when n in- 1). creases indefinitely, for all values of 0 between o and i, 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 (I) represents the function at all points of the interval between a and a+k, except possibly a+k, and the function is " analytic " (§ 13) in this domain. Obvious modifications admit of extension to an interval between a and a - k, or between a - k and a+k. When a series of the form (I) represents a function it is called " the Taylor's series for the function." Taylor's series is a power series, i.e. a series of the form an(x - a)..

n=0 As regards power series we have the following theorems: 1. If the power series converges at any point except a there is a number k which has the property that the series converges absolutely in the interval between a - k and a+k, with the possible exception of one or both end-points.

2. The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).

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

The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables.

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 fn(x) the sum of the first n+ i 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 n so that if n (a) - f p (a)I <e for all values of m and p which exceed n. The sum, f(a), is the limit of the sequence of numbers fn(a) at n=. The convergence is said to be " uniform " in an interval if, after specification of e, the same number n suffices at all points of the interval to make I f(x)-f m (x) I < e for all values of m which exceed n. The numbers n corresponding to any e, however small, are all finite, but, when e is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, e can be taken so small that, at some point in the neighbourhood of a, n must be taken > to make I f (x) - fn(x) I < e 'then m>n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and e afterwards, then the number n is finite; choose e first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small e may be, the convergence is uniform.

For example, the series sin x - z sin 2x++ sin is convergent for all real values of x, and, when R->x> - 7r its sum is a x; but, when x is but a little less than 7, the number of terms which must be taken in order to bring the sum at all near to the value of Ix is very large, and this number tends to increase indefinitely as x approaches x. This series does not converge uniformly in the neighbourhood of x = 7r. Another example is afforded by the series nx (n -1)x of which the remainder after n terms n =0n 2 x 2 +1 (n+1) 2 x 2 +I' - is nx/(n 2 x 2 +1). If we put x= I /n, for any value of n, however great, the remainder is -; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of when x is near to zero - it must, in fact, be large compared with I/x. The series does not converge uniformly in the neighbourhood of x = o.

As regards series whose terms represent continuous functions we have the following theorems: (1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.

(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval.

(3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.

(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.

(5) If 2 fr(x) =f(x) converges uniformly in the interval r=0 between a and r = 0 ,l a fr(x)dx, .or a series which converges unformly may be integrated term by term.

(6) If f' converges uniformly in an interval, then r=0 fr(x) converges in the interval, and represents a continuous r=0 differentiable function, 4)(x); in fact we have 4;/(x)= ? o r or a series can be differentiated term by term if the series of derived functions converges uniformly.

A series whose terms represent functions which are not continuous throughout an interval may converge uniformly in the interval. If / fr(x),=f(x), is such a series, and if all the r=0 functions fr(x) have limits at a, then f(x) has a limit at a, which is E Lt fr(x). A similar theorem holds for limits on the left r=0x=a or on the right.

23. Fourier's Series. - An extensive class of functions admit of being represented by series of the form a0+ n ?i (an cos n ? x -I-bn 'sinnLx' l and the rule for determining the coefficients a n, b n of such a series, in order that it may represent a given function f(x) in the interval between - c and c, was given by Fourier, viz, we have ao = 1 '_' an = f f(x) cosdx, b n =f f(x) sinedx. The interval between - c and c may be called the " periodic interval," and we may replace it by any other interval, e.g. that between o and 1, without any restriction of generality. When this is done the sum of the series takes the form 1 r=n Lt J / f(z)cos{2rir (z - x)} dz, n=? Or= -n and this is sin {(2n-I-I)(z - x)7r } dz.

n= Lt ? f o f (z) sin {(z - x)ir} Fourier's theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier's time a function of this character was regarded as completely arbitrary.

By a discussion of the integral (ii.) based on the Second Theorem of the Mean (§ 15) it can be shown that, if f(x) has restricted oscillation in the interval (§ I I), the sum of the series is equal to a { f(x+o) + f(x - o)} at any point x within the interval, and that it is equal tc { f (+o) +f (I - o) } at each end of the interval. (See the article Fourier'S Series.) It therefore represents the function at any point of the periodic interval at which the function is continuous (except possibly the end-points), and has a definite value at each point of discontinuity. The condition of restricted oscillation includes all the functions contemplated in the statement of the theorem and some others. Further, it can be shown that, in any partial interval throughout which f(x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function f(x) which tends to become infinite at a finite number of points a of the interval, provided (I) f(x) tends to become determinately infinite at each of the points a, (2) the improper definite integral of f(x) through the interval is convergent, (3) f(x) has not an infinite number of discontinuities or of maxima or minima in the interval.

24. Representation of Continuous Functions by Series. - If the series for f(x) formed by Fourier's rule converges at the point a of the periodic interval, and if f(x) is continuous at a, the sum of the series is f(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier's rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as " terminated Fourier's series " and " terminated power series," according to the two following theorems: (I) If f(x) is continuous throughout the interval between o and 27, and if any positive number e however small is specified, it is possible to find an integer n, so that the difference between the value of f(x) and the sum of the first n terms of the series for f(x), formed by Fourier's rule with periodic interval from o to air, shall be less than e at all points of the interval. This result can be extended to a function which is continuous in any given interval.

(2) If f(x) is continuous throughout an interval, and any positive number e however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of f(x) and the value of the polynomial shall be less than e at all points of the interval.

Again it can be proved that, if f(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to f(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised.

Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series a n cos (b n x7r), where a is positive and less n=0 than unity, and b is an odd integer exceeding (1+27r)/a. It can be shown that this series is uniformly convergent in every interval, (ii.) and that the continuous function f(x) represented by it has the property that there is, in the neighbourhood of any point xo, an infinite aggregate of points x', having xo as a limiting point, for which { f (x') - f (xo) }/(x' - x 0 ) tends to become infinite with one sign when x' - xo approaches zero through positive values, and infinite with the opposite sign when x' - xo approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function f(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F'(x)=f(x); and, if f(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Za n cpn(x), where Za n is an absolutely convergent series of numbers, and 4n(x) is an analytic function whose absolute value never exceeds unity.

25. Calculations with Divergent Series. - When the series described in (I) and of § 24 diverge, they may, nevertheless, be used for the approximate numerical calculation of the values of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the property of representing a function approximately when the expansion is not carried too far are called " asymptotic expansions." Sometimes they are called " semi-convergent series "; but this term is avoided in the best modern usage, because it is often used to describe series whose convergence depends upon the order of the terms, such as the series. .

In general, let fo(x)+f1(x)+... be a series of functions which does not converge in a certain domain. It may happen that, if any number however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value f(a) of a certain function f(x) is connected with the sum of the first n+ terms of the series by the relation I f (a) - Z f r (a) It must r=p also happen that, if any number N, however great, is specified, a number n'(>n) can be found so that, for all values of m which exceed n', r > N. The divergent series fo(x) +f 1 (x) + ... is then an r=0 asymptotic expansion for the function f(x) in the domain.

The best known example of an asymptotic expansion is Stirling's formula for n! when n is large, viz.

n! = y (27r) Inn+ie-n+e where 0 is some number lying between o and I. This formula is included in the asymptotic expansion for the Gamma function. We have in fact log {r(x)} =(x -2) log x - x+z log 27r+05(X), where a(x) is the function defined by the definite integral =J {(I - 1 _ t 1_ z}t -1e txdt. 0 The multiplier of e tx under the sign of integration can be expanded in the power series __41_12+.67/4_ where i are " Bernoulli's numbers " given by the formula B m = 2.2m! (27r)-2m (r 2m).

r=1 When the series is integrated term by term, the right-hand member of the equation for takes the form B 1 I 1.2 x 3.4x 3+ 5.6 55 - "' This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value of a(x) is less than the last of the retained terms. Stirling's formula is obtained by retaining the first term only. Other well-known examples of asymptotic expansions are afforded by the descending series for Bessel's functions. Methods of obtaining such expansions for the solutions of linear differential equations of the second order were investigated by G. G. Stokes (Math. and Phys. Papers, vol. ii. p. 329), and a general theory of asymptotic expansions has been developed by H. Poincare. A still more general theory of divergent series, and of the conditions in which they can be used, as above, for the purposes of approximate calculation has been worked out by E. Borel. The great merit of asymptotic expansions is that they admit of addition, subtraction, multiplication and division, term by term, in the same way as absolutely convergent series, and, they admit also of integration term by term; that is to say, the results of such operations are asymptotic expansions for the sum, difference, product, quotient, or integral, as the case may be.

26. Interchange of the Order of Limiting Operations. - When we require to perform any limiting operation upon a function which is itself represented by the result of a limiting process, the question of the possibility of interchanging the order of the two processes always arises. In the more elementary problems of analysis it generally happens that such an interchange is possible; but in general it is not possible. In other words, the performance of the two processes in different orders may lead to two different results; or the performance of them in one of the two orders may lead to no result. The fact that the interchange is possible under suitable restrictions for a particular class of operations is a theorem to be proved.

Among examples of such interchanges we have the differentiation and integration of an infinite series term by term (§ 22), and the differentiation and integration of a definite integral with respect to a parameter by performing the like processes upon the subject of integration (§ 19). As a last example we may take the limit of the sum of an infinite series of functions at a point in the domain of convergence. Suppose that the series 2 f,.(x) represents a function r=0 (fx) in an interval containing a point a, and that each of'the functions f r (x) has a limit at a. If we first put x= a, and then sum the series, we have the value f (a); if we first sum the series for any x, and afterwards take the limit of the sum at x=a, we have the limit of f(x) at a; if we first replace each function f r (x) by its limit at a, and then sum the series, we may arrive at a value different from either of the foregoing. If the function f(x) is continuous at a, the first and second results are equal; if the functions fr(x) are all continuous at a, the first and third results are equal; if the series is uniformly convergent, the second and third results are equal. This last case is an example of the interchange of the order of two limiting operations, and a sufficient, though not always a necessary, condition, for the validity of such an interchange will usually be found in some suitable extension of the notion of uniform convergence.

Authorities

- Among the more important treatises and memoirs connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905); O. Biermann, Analytische Functionen (Leipzig, 1887); E. Borel, Theorie des fonctions (Paris, 1898) (containing an introductory account of the Theory of Aggregates), and Series divergentes (Paris, 1901), also Fonctions de variables reelles (Paris, 1905); T. J. I'A. Bromwich, Introduction to the Theory of Infinite Series (London, 1908); H. S. Carslaw, Introduction to the Theory of Fourier's Series and Integrals (London, 1906); U. Dini, Functionen e. reellen Grosse (Leipzig, 1892), and Serie di Fourier (Pisa, 1880); A. Genocchi u. G. Peano, Duff.- u. Int.-Rechnung (Leipzig, 1899); J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898); A. Harnack, Duff. and Int. Calculus (London, 1891); E. W. Hobson, The Theory of Functions of a real Variable and the Theory of Fourier's Series (Cambridge, 1907); C. Jordan, Cours d'analyse (Paris, 1893-1896); L. Kronecker, Theorie d. einfachen u. vielfachen Integrale (Leipzig, 1894) H. Lebesgue, Lecons sur l'integration (Paris, 1904); M. Pasch, Di f.- u. Int.-Rechnung (Leipzig, 1882); E. Picard, Traite d'analyse (Paris, 1891); O. Stolz, Allgemeine Arithmetik (Leipzig, 1885), and Duff.- u. Int.- Rechnung (Leipzig, 1893-1899); J. Tannery, Theorie des fonctions (Paris, 1886); W. H. and G. C. Young, The Theory of Sets of Points (Cambridge, 1906); Broden," Stetige Functionen e. reellen Vera,nderlichen," Crelle, Bd. cxviii.; G. Cantor, A series of memoirs on the " Theory of Aggregates " and on " Trigonometric series " in Acta Math. tt. ii., vii., and Math. Ann. Bde. iv.-xxiii.; Darboux, " Fonctions discontinues," Ann. Sci. Ecole normale sup. (2), t. iv.; Dedekind, Was rind u. was sollen d. Zahlen? (Brunswick, 1887), and Stetigkeit u. irrationale Zahlen (Brunswick, 1872); Dirichlet, " Convergence des series trigonometriques," Crelle, Bd. iv. P. Du Bois Reymond, Allgemeine Functionentheorie (Tubingen, 1882), and many memoirs in Crelle and in Math. Ann.; Heine, " Functionenlehre," Crelle, Bd. lxxiv.; J. Pierpont, The Theory of Functions of a real Variable (Boston, 1905); F. Klein, " Allgemeine Functionsbegriff," Math. Ann. Bd. xxii.; W. F. Osgood, " On Uniform Convergence," Amer. J. of Math. vol. xix.; Pincherle, " Funzioni analitiche secondo Weierstrass," Giorn. di mat. t. xviii.; Pringsheim, " Bedingungen d. Taylorschen Lehrsatzes," Math. Ann. Bd. xliv.; Riemann, " Trigonometrische Reihe," Ges. Werke (Leipzig, 1876); Schoenflies, " Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten," Jahresber. d. deutschen Math.- Vereinigung, Bd. viii. Study, Memoir on " Functions with Restricted Oscillation," Math. Ann. Bd. xlvii.; Weierstrass, Memoir on " Continuous Functions that are not Differentiable," Ges. math. Werke, Bd. ii. p. 71 (Berlin, 1895), and on the " Representation of Arbitrary Functions," ibid. Bd. iii. p. 1; W. H. Young, " On Uniform and Non-uniform Convergence," Proc. London Math. Soc. (Ser. 2) t. 6. Further information and very full references will be found in the articles by Pringsheim, Schoenflies and Voss in the Encyclopcidie der math. Wissenschaften, Bde. i., ii. (Leipzig, 1898, 1899). (A. E. H. L.) II. - Functions Of Complex Variables In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (§ 1), Complex numbers, states what a complex number is; (§ 2), Plotting of simple expressions involving complex numbers, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; (§ 3), Limiting operations, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by X(z); (§ 4), Functions of a complex variable in general, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy's theorem relating thereto; (§ 5), Applications, considers the differentiation and integration of series of functions of a complex variable, proves Laurent's theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (§ 6), Singular points, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (§ 7), Monogenic Functions, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; (§ 8), Some elementary properties of single valued functions, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler's theorem, and Weierstrass's theorem for the primary factors of an integral function, stating generalized forms for these, leading to the theorem of (§ q), The construction of a monogenic function with a given region of existence, with which is connected (§ io), Expression of a monogenic function by rational functions in a given region, of which the method is applied in (§ II), Expression of (I - z) - ' by polynomials, to a definite example, used here to obtain (§ 12), An expansion of an arbitrary function by means of a series of polynomials, over a star region, also obtained in the original manner of MittagLeffler; (§ 13), Application of Cauchy's theorem to the determination of definite integrals, gives two examples of this method; (§ 14), Doubly Periodic Functions, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section (§ 20) below, dealing with Elliptic Integrals; (§ 15), Potential Functions, Conformal representation in general, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (§ 16), Multiple-valued Functions, Algebraic Functions, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (§ 17), Integrals of Algebraic Functions, enunciating Abel's theorem; (§ 18), Indeterminateness of Algebraic Integrals, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (§ ig), Reversion of an algebraic integral, mentions a problem considered below in detail for an elliptic integral; (§ 20), Elliptic Integrals, considers the algebraic reduction of any elliptic integral to one of three standard forms, and proves that the function obtained by reversion is single-valued; (§ 21), Modular Functions, gives a statement of some of the more elementary properties of some functions of great importance, with a definition of Automorphic Functions, and a hint of the connexion with the theory of linear differential equations; (§ 22), A property of integral functions, deduced from the theory of modular functions, proves that there cannot be more than one value not assumed by an integral function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the periods; (§ 23), Geometrical applications of Elliptic Functions, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function q3(u); (§ 24), Integrals of Algebraic Functions in connexion with the theory of plane curves, discusses the generalization to curves of any deficiency; (§ 25), Monogenic Functions of several independent variables, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (§ 26), Multiply-Periodic Functions and the Theory of Surfaces, attempts to show the nature of some problems now being actively pursued.

Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more elementary results are stated only, without proof, as, for instance: the monogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic; that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning of area than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

§ r. Complex Numbers. - Complex numbers are numbers of the form x+iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that 12= - I.

Taking in a plane two rectangular axes Ox, Oy, we assume that every point of the plane is definitely associated with two real numbers x, y (its co-ordinates) and conversely; thus any point of the plane is associated with a single complex number; in particular, for every point of the axis Ox, for which y=O, the associated number is an ordinary real number; the complex numbers thus include the real numbers. The axis Ox is often called the real axis, and the axis Oy the imaginary axis. If P be the point associated with the complex variable z=x+iy, the distance OP be called r, and the positive angle less than 2rr between Ox and OP be called 0, we may write z = r(cos 0+i sin 0); then r is called the modulus or absolute value of z and often denoted by I z I and 0 is called the phase or amplitude of z, and often denoted by ph (z); strictly the phase is ambiguous by additive multiples of 27r. If z' = x'+iy' be represented by P', the complex argument z'+z is represented by a point P" obtained by drawing from P' a line equal to and parallel to OP; the geometrical representation involves for its validity certain properties of the plane; as, for instance, the equation z'+z=z+z' involves the possibility of constructing a parallelogram (with OP"as diagonal). It is important constantly to bear in mind, what is capable of easy algebraic proof (and geometrically is Euclid's proposition III. 7), that the modulus of a sum or difference of two complex numbers is generally less than (and is never greater than) the sum of their moduli, and is greater than (or equal to) the difference of their moduli; the former statement thus holds for the sum of any number of complex numbers. We shall write E(10) for cos 0+i sin 0; it is at once verified that E(ia). E (il) =E[i(a+(3)], so that the phase of a product of complex quantities is obtained by addition of their respective phases.

§ 2. Plotting and Properties of Simple Expressions involving a Complex Number. - If we put = (z - i)/(z+i), and, putting = +-in, take a new plane upon which, n are rectangular co-ordinates, the equations %= (x2+y2 - I)/[x2+ I)2], ri = - 2xy/[x 2 +(y+I) 2 ] will determine, corresponding to any point of the first plane, a point of the second plane. There is the one exception of z = - i, that is, x = o, y= - I, of which the corresponding point is at infinity. It can now be easily proved that as z describes the real axis in its plane the point describes once a circle of radius unity, with centre at o, and that there is a definite correspondence of point to point between points in the z-plane which are above the real axis and points of the c-plane which are interior to this circle; in particular z = i corresponds to = o.

Moreover, i being a rational function of z, both E and i are continuous differentiable functions of x and y, save when is infinite; writing = f (x, y) = f (z -iy, y), the fact that this is really independent of y leads at once to of/ax+iaflay =o, and hence to a = an _ _an a 2 E a 2 _ ax ay' ay ax' axe+ay2 -O; so that is not any arbitrary function of x, y, and when is known n is determinate save for an additive constant. Also, in virtue of these equations, if ?', ?' be the values of corresponding to two near values of z, say z and z', the ratio (3-'-3')/(z'-z) has a definite limit when z' = z, independent of the ultimate phase of z' -z, this limit being therefore equal to Nlax, that is, a lax+ianlax. Geometrically this fact is interpreted by saying that if two curves in the z-plane intersect at a point P, at which both the differential coefficients aE/ax, an/ax are not zero, and P', P" be two points near to P on these curves respectively, and the corresponding points of the 3--plane be Q, Q', Q", then (I) the ratios PP"/PP', QQ"/QQ' are ultimately equal, (2) the angle P'PP" is equal to Q'QQ", (3) the rotation from PP' to PP" is in the same sense as from QQ' to QQ", it being understood that the axes of, n in the one plane are related as are the axes of x, y. Thus any diagram of the z-plane becomes a diagram of the c-plane with the same angles; the magnification, however, which is equal to [(u) 2+ ?y/ 2 ] varies from point to point. Conversely, it appears subsequently that the expression of any copy of a diagram (say, a map) which preserves angles requires the intervention of the complex variable.

As another illustration consider the case when 31s a polynomial in z, =pozn+pizn-l+.. .+pn; H being an arbitrary real positive number, it can be shown that a radius R can be found such for every IzI >R we have W > H; consider the lower limit of 131 for I z I <R; as, 2 -1-77 2 is a real continuous function of x, y for I z I < R, there is a point (x, y), say (xo, ye), at which III is least, say equal to p, and therefore within a circle in the I'-plane whose centre is the origin, of radius p, there are no points 1 representing values corresponding to I z I < R. But if 3"o be the value of 3' corresponding to (xo, ye), and the expression of -Io near zo= xo +iyo, in terms of z-zo, be A(z-zo)m+ B(z-zo)m+1+..., where A is not zero, to two points near to (xo, ye), say (x i, yi) or z i and z 2 = zo+ (z 1 - zo) (cosj+i sin, will corre spond two points near to 3'o, say I-i, and 21 o - ' 1 situated so that is between them. One of these must be within the circle (p). We infer then that p=o, and have proved that every polynomial in z vanishes for some value of z, and can therefore be written as a product of factors of the form z-a, where a denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

§ 3. Limiting Operations. - In order that a complex number = +-irl may have a limit it is necessary and sufficient that each of and has a limit. Thus an infinite series w o+wi+W2+

, whose terms are complex numbers, is convergent if the real series formed by taking the real parts of its terms and that formed by the imaginary terms are both convergent. The series is also convergent if the real series formed by the moduli of its terms is convergent; in that case the series is said to be absolutely convergent, and it can be shown that its sum is unaltered by taking the terms 'in any other order. Generally the necessary and sufficient condition of convergence is that, for a given real positive a number m exists such that for every and every positive p, the batch of terms wn-}-w.+1+ ... +w n+p is less than in absolute value. If the terms depend upon a complex variable z, the convergence is called uniform for a range of values of z, when the inequality holds, for the same and m, for all the points z of this range.

The infinite series of most importance are those of which the general term is a n z n, wherein a n is a constant, and z is regarded as variable, n =o, 3, ... Such a series is called a power series. If a real and positive number M exists such that for z=zo and every n, II anzo n I < M, a condition which is satisfied, for instance, if the series converges for z = zo, then it is at once proved that the series converges absolutely for every z for which IzI < I zo I, and converges uniformly over every range I z I < r' for which r'< I zo I To every power series there belongs then a circle of convergence within which it converges absolutely and uniformly; the function of z represented by it is thus continuous within the circle (this being the result of a general property of uniformly convergent series of continuous functions); the sum for an interior point z is, however, continuous with the sum for a point zo on the circumference, as z approaches to zo provided the series converges for z=zo, as can be shown without much difficulty. Within a common circle of convergence two power series Ea n z n, n can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r i be less than the radius of convergence of a series Ea n z n and for I z I =ri, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for IzI = r i every term is also less than M in absolute value, namely, I anI < Mrr'. If in every arbitrarily small neighbourhood of z=o there be a point for which two converging power series Zane, Ebnz n agree in value, then the series are identical, or a n = bn; thus also if Ea n z n vanish at z = o there is a circle of finite radius about z =o as centre within which no other points are found for which the sum of the series is zero. Considering a power series f(z) = Ea n z n of radius of convergence R, if Izol <R and we put z=zo+t with 111 <R - Izol, the resulting series Ean(zo+t) n may be regarded as a double series in zo and t, which, since I zo I +t < R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write f(z) = EAnt n; hence Ao = f (zo), and we have [ f (zo+t) -f (zo)]/t = A n t n - 1, wherein the continuous series on the right reduces to A1 n=1 for t =o; thus the ratio on the left has a definite limit when t=o, equal namely to A l or Enanzo n-1. In other words, the original series may legitimately be differentiated at any interior point zo of its circle of convergence. Repeating this process we find f (zo +t) = E tn f(n) (zo) /n!, where f (n) (zo) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z=o for Eanz n, we infer that for the series f(z) every point which reduces it to zero is an isolated point, and of such points only a finite number lie within a circle which is within the circle of convergence of f(z).

Perhaps the simplest possible power series!is e z =exp (z) = I +z 2 /2! + z3 /3 ! + ... of which the radius of convergence is infinite. By multiplication we have exp (z).exp (z 1) =exp (z+z i). In particular when x, y are real, and z = x+iy, exp (z) =exp (x) exp (iy). Now the functions Uo=sin y, Vo= i - cos y, =y -sin y, V i =1y 2 - +cos = 6y 3 -y+sin y, V = 4 -2y 2 +I-cos y,... all vanish for y =o, and the differential coefficientcient of any one after the first is the preceding one; as a function (of a real variable) is increasing when its differential coefficient is positive, we infer, for y positive, that each of these functions is positive; proceeding to a limit we hence infer that COS y= -2y2+y4-..., sin y =ysy3+izay5-..., for positive, and hence, for all values of y. We thus have exp (iy) = cos y+i sin y, and exp (z) =exp (x). (cos y+i sin y). In other words, the modulus of exp (z) is exp (x) and the phase is y. Hence also exp (z+2zri)=exp (x)[cos (y+27r) +i sin (y+27r)], which we express by saying that exp (z) has the period 2714, and hence also the period 2kiri, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x=o, we at once prove that exp (z) has no other periods.

Taking in the plane of z an infinite strip lying between the lines y =o, y =27 and plotting the function 3- = exp (z) upon a new plane, it follows at once from what has been said that every complex value of 1' arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation 3- = exp (z) thus defines z, regarded as depending upon 3', with only an additive ambiguity 2k7ri, where k is an integer. We write z = X(3'); when I is real this becomes the logarithm of 3 .; in general X(3-) =log 13-I +i ph (3-)+ 2k7ri, where k is an integer; and when 1 describes a closed circuit surrounding the origin the phase of I increases by 27r, or k increases by unity. Differentiating the series for I we have 4/dz = -, so that z, regarded as depending upon ?, is also differentiable, with dz/d1" = 31. On the other hand, consider the series I -I-z(1'-i)2+ a (? -1) 3 - ...; it converges when 3- =2 and hence converges for ft- - i I < 1; its differential coefficient is, however, 1- (3' - 0+ (3' - i) 2 - ..., that is, ('+-')'. Wherefore if OM denote this series, for 13-- I I < I, the difference a(1') -0q), regarded as a function of and n, has vanishing differential coefficients; if we take the value of X(3) which vanishes when 3'= i we infer thence that for Il'-il<I, (- n)n -i (? i) n. It is to be remarked that it is impossible for 3while subject to I - i I <1 to make a circuit about the origin. For values of 3for which 13-- i I. i, we can also calculate X(1') with the help of infinite series, utilizing the fact that X(I'3-') = X(3-) +X(r).

The function A(3-) is required to define 3-a when 3and a are complex numbers; this is defined as exp [aX(3-)], that is as E an[X(t')]"/n!.

n= When a is a real integer the ambiguity of X(3') is immaterial here, since exp [aX(3-)+2kaxi]=exp[aX(1')]; when a is of the form i/q, where q is a positive integer, there are q values possible for -llq, of the form exp [-X(d exp (2 q'-), with k = o, i, ... q - i, all other values of k leading to one of these; the qth power of any one of these values is 3-; when a = plq, where p, q are integers without common factor, q being positive, we have I' P14 = (3 -i le)P. The definition of the symbol 3-a is thus a generalization of the ordinary definition of a power, when the numbers are real. As an example, let it be required to find the meaning of i i; the number i is of modulus unity and phase fir; thus a(i)=i(27r+2k7r); thus i' =exp (-27r-2kir)=exp (-27r) exp (-2kir), is always real, but has an infinite number of values.

The function exp (z) is used also to define a generalized form of the cosine and sine functions when z is complex; we write, namely, cos z = 2 [exp (iz) -}- exp (- iv)] and sin z = - Zi[exp (iz) - exp (- iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = i + t /2 ! + 1 /4! -}-... is obviously greater than unity.

§4. Of Functions of a Complex Variable in General. - We have in what precedes shown how to generalize the ordinary rational, algebraic and logarithmic functions, and considered more general cases, of functions expressible by power series in z. With the suggestions furnished by these cases we can frame a general definition. So far our use of the plane upon which z is represented has been only illustrative, the results being capable of analytical statement. In what follows this representation is vital to the mode of expression we adopt; as then the properties of numbers cannot be ultimately based upon spatial intuitions, it is necessary to indicate what are the geometrical ideas requiring elucidation.

Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counterclockwise; another square, also of side a, may be added to this, so that there is a side common; this common side being erased we have a composite region with a definite direction of perimeter; to this a third square of the same size may be attached, so that there is a side common to it and one of the former squares, and this common side may be erased. If this process be continued any number of times we obtain a region of the plane bounded by one or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of description, which is such that the region is on the left of the describing point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the greatest side of a triangle, and a lower limit for the smallest angle. In the former method, each square may be divided into four others by lines through its centre parallel to its sides; in the latter method each triangle may be divided into four others by lines joining the middle points of its sides; this halves the sides and preserves the angles. When we speak of a region of the plane in general, unless the contrary is stated, we shall suppose it capable of being generated in this latter way by means of a finite number of triangles, there being an upper limit to the length of a side of the triangle and a lower limit to the size of an angle of the triangle. We shall also require to speak of a path in the plane; this is to be understood as capable of arising as a limit of a polygonal path of finite length, there being a definite direction or sense of description at every point of the path, which therefore never meets itself. From this the meaning of a closed path is clear. The boundary points of a region form one or more closed paths, but, in general, it is only in a limiting sense that the interior points of a closed path are a region.

There is a logical principle also which must be referred to. We frequently have cases where, about every, interior or boundary, point zo of a certain region a circle can be put, say of radius ro, such that for all points z of the region which are interior to this circle, for which, that is, j z - zo I<ro, a certain property holds. Assuming that to r 0 is given the value which is the upper limit for zo, of the possible values, we may call the points I z - zo I <ro, the neighbourhood belonging to or proper to zo, and may speak of the property as the property (z,z 0). The value of ro will in general vary with zo; what is in most cases of importance is the question whether the lower limit of ro for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z,zo) is of a kind which we may call extensive; such, namely, that if it holds, for some position of zo and all positions of z, within a certain region, then the property (z,z i) holds within a circle of radius R about any interior point z 1 of this region for all points z for which the circle I z - z i I = R is within the region. Also in this case ro varies continuously with zo. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (I) for some point zo within or upon the boundary of the sub-region, (2) for every point z within or upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z,z 0) holds for all points z and some point zoof the region, be called suitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say; after a certain stage all these will be interior to the proper region of 1; this, however, is contrary to the supposition that they are all unsuitable.

We now make some applications of this result (B). Suppose a definite finite real value attached to every interior or boundary point of the region, say f(x,y). It may have a finite upper limit H for the region, so that no point (x,y) exists for which f(x,y) > H, but points (x,y) exist for which f(x,y) > H - e, however small e may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of f(x,y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of f(x,y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function f(x,y) is the radius ro of the neighbourhood proper to any point zo, spoken of above. We can hence prove the statement (A) above.

Suppose the property (z,z 0) extensive, and, if possible, that the lower limit of ro is zero. Let then O be a point such that the lower limit of ro is zero for points zo within a circle about however small; let be the radius of the neighbourhood proper to; take zo so that the property (z,r0), being extensive, holds within a circle, centre zo, of radius r - I zo -, which is greater than jzo - i I, and increases to r as I zoi- I diminishes; this being true for all points zo near the lower limit of ro is not zero for the neighbourhood of 1, contrary to what was supposed. This proves (A). Also, as is here shown that ror - I zo - f I, may similarly be shown that - I zo -. Thus ro differs arbitrarily little from r when I zo - is sufficiently small; that is, ro varies continuously with zo. Next suppose the function f(x,y), which has a definite finite value at every point of the region considered, to be continuous but not necessarily real, so that about every point z0, within or upon the boundary of the region, n being an arbitrary real positive quantity assigned beforehand, a circle is possible, so that for all points z of the region interior to this circle, we have I <In, and therefore (x',y') being any other point interior to this circle, - f (x,y) I <,,. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point zo the circular area within which, for any two points (x,y), (x',y'), we have! f (x',y') - f (x,y) I .co. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x,y), (x',y') within a circle I z - zo I =r about any point zo, we have I f(x',y') - f(x,y) and, in particular, I - f (xo,yo) I < n, where i is an arbitrary real positive quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are z 0, z, and let z 1, ... z n _ 1 be intermediate points of the path, in order; denote the continuous function f(x,y) by f(z), and let f r denote any quantity such that ! f r - f(z r) I I f(Z r+1 ) - f(zr) j; consider the sum (Z i - z0) fo + (Z 2 - z l)fl +. .. -]- (z - zn -1)f n-1.

By the definition of a path we can suppose, n being large enough, that the intermediate points z 11 ... z„ ^1 are so taken that if zi, zi+i be any two points intermediate, in order, to z r and z r+ii we have zi+1 - z i I < I z r+i - z r I; we can thus suppose I z 1 - zo 1, I z2 - z1 I, ...

i z - z,_ 1 Iall to converge constantly to zero. This being so, we can show that the sum above has a definite limit. For this it is sufficient, as in the case of an integral of a function of one real variable, to prove this to be so when the convergence is obtained by taking new points of division intermediate to the former ones. If, however, zr, l, zr, 2 i ... zr, ,?_1 be intermediate in order to z r and z r+1r and I fr, i - f (zr,i) (z r,i+1) - f (zr,i) I, the difference between Z(z r+i - zr)fr and (Z,,i - Zr)fr,o+(zr,2 - zr,l)fr,l+

+(Zr+1 - zr,m-1)fr,m-11, which is equal to EZ(zr,i+i - Zr,i) (r is, when I z r+i - z r I is small enough, to ensure I f (z r+1) - f (z r) I <'i, less in absolute value than z - Z which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2nS, and is therefore arbitrarily small.

The limit in question is called ýf(v)dz. In particular when f(z) =I, it is obvious from the definition that its value is z - zo; when f(z) = z, by taking f r = z (z r+1 - zr), it is equally clear that its value is 1(z° - zo 2); these results will be applied immediately.

Suppose now that to every interior and boundary point zo of a certain region there belong two definite finite numbers f(zo), F(zo), such that, whatever real positive quantity i may be, a real positive number e exists for which the condition I' f(z) - (zo) - F(zo) I < z - zo which we describe as the condition (z,zo), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation lz - zol <e. Then f(zo) is called a differentiable function of the complex variable zo over this region, its differential coefficient being F(zo). The function f(zo) is thus a continuous function of the real variables xo,' yo, where - - over the region; it will appear that F(zo) is also continuous and in fact also a differentiable function of zo.

Supposing n to be retained the same for all points zo of the region, and ao to be the upper limit of the possible values of € for the point zo, it is to be presumed that ao will vary with zo, and it is not obvious as yet that the lower limit of the values of ao as zo varies over the region may not be zero. We can, however, show that the region can be divided into a finite number of sub-regions for each of which the condition (z, zo), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of zo, within or upon the boundary of this sub-region. This is proved above as result (B).

Hence it can be proved that, for a differentiable function f(z), the integral f z 1 f (z)dz has the same value by whatever path within the region we pass from z 1 to z. This we prove by showing that when taken round a closed path in the region the integral ff (z)dz vanishes. Consider first a triangle over which the condition (z, zo) holds, for some position of zo and every position of z, within or upon the boundary of the triangle. Then as f(z) =.f (zo)-{-(z - F(zo)-I-ne(z - zo), where I 0 I < we have ff (z) dz = [f(zo) - zoF (z o)] f dz - f - F (z o)fzdz -fin fe (z - zo)dz, which, as the path is closed, is nf0(z - zo)dz. Now, from the theorem that the absolute value of a sum is less than the sum of the absolute values of the terms, this last is less, in absolute value, than nap, where a is the greatest side of the triangle and p is its perimeter; if A be the area of the triangle, we have 0 = lab sin C> (a/7r)ba, where a is the least angle of the triangle, and hence a(a+b+c) <2a(b+c) <47r0/a; the integral ff(z)dz (z)dz round the perimeter of the triangle is thus <47rn0/a. Now consider any region made up of triangles, as before explained, in each of which the condition (z, zo) holds, as in the triangle just taken. The integral f f (z)dz round the boundary of the region is equal to the sum of the values of the integral round the component triangles, and thus less in absolute value than 41rnK/a, where K is the whole area of the region, and a is the smallest angle of the component triangles. However small n be taken, such a division of the region into a finite number of component triangles has been shown possible; the integral round the perimeter of the region is thus arbitrarily small. Thus it is actually zero, which it was desired to prove. Two remarks should be added: (I) The theorem is proved only on condition that the closed path of integration belongs to the region at every point of which the conditions are satisfied. (2) The theorem, though proved only when the region consists of triangles, holds also when the boundary points of the region consist of one or more closed paths, no two of which meet.

Hence we can deduce the remarkable result that the value of f(z) at any interior point of a region is expressible in terms of the value of f(z) at the boundary points. For consider in the original region the function f(z)/(z - zo), where zo is an interior point: this satisfies the same conditions as f(z) except in the immediate neighbourhood of zo. Taking out then from the original region a small regular polygonal region with zo as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral f dzf(z) round the boundary of z - zo the original region is equal to the same integral taken counterclockwise round a small circle having zo as centre; on this circle, however, if z - zo=rE(iO), dz/(z - zo)=id&, and f(z) differs arbitrarily little from f(zo) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to 21rif (zo). Hence f(zo) = rif ttf (z0 ' w here this integral is round the 27 boundary of the original region. From this it appears that F(zo) =11m f (z) - f (z o) _ i f dtf (t) z - zo 27ri (t - zo)2 also round the boundary of the original region. This form shows, however, that F(zo) is a continuous, finite, differentiable function of zo over the whole interior of the original region.

§ 5. Applications

The previous results have manifold applications.

(I) If an infinite series of differentiable functions of z be uniformly convergent along a certain path lying with the region of definition of the functions, so that S(z) =uo(z)+ui(z)+... un_1(z)-+Rn(z), where I Rn(z) I <e for all points of the path, we have f S(z)dz =f Z uo(z)dz+f u1(z)dz-{-...+ r u1(z)dz - f z R(z)