GEOGRAPHICAL NAMES

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 notationf (x) for a function of a variablex 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 asax e bx -Fc, or rational algebraic functions which are not integral, such asalxn+a2xn-T+... +an, b i x m +b 2 x m_T +... +b,,,, or irrational algebraic functions, such asx, 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 Italianfunzione, 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.

wheref„, (x,y),.. . mean homogeneous expressions inx andy 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 sinx and tanx, or inverse trigonometrical functions, such as sin l x, or exponential functions, such ase x and a x, or logarithmic functions, such as logx 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 expressionxy/(x 2 + y 2), in which bothx andy 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.

In the case of a function of one variablex, any value ofx and the corresponding valuey of the function can be the co-ordinates of a point in a plane. To any value ofx there corresponds a pointN on the axis ofx, in accordance with the rule thatx is the abscissa ofN. The corresponding value ofy determines a pointP in accordance with the rule thatx is the abscissa andy the ordinate ofP. The ordinatey gives the value of the function which corresponds to that value of the variablex which is specified byN; and it may be described as " the value of the function atN." Since there is a one-to-one correspondence of the pointsN and the numbersx, we may also describe the ordinate as " the value of the function atx." In simple cases the aggregate of the pointsP 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.

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 ofordinal 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 acontinuum. In fact the notion of a onedimensional continuum first becomes precise in virtue of the establishment of the system of real numbers.

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, oraggregate, 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 numberN, however great, is specified, it is possible to find in the aggregate numbers which exceedN in absolute value. Such an aggregate is always infinite. The " neighbourhood of a number (or point) a for a positive numberh " is the aggregate of all numbers (or points)x for which the absolute value ofx - a denoted by Ix - a I, does not exceedh. 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 toy whenx is any point in the domain; theny is said to be a " function " of the variablex, andx 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 expressioni/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.

Let f(x) be a function of a variable numberx; and let a be a point such that there are points of the domain of the argumentx in the neighbourhood of a for any numberh, however small. If there is a numberL which has the property that, after any positive number however small, has been specified, it is possible to find a positive numberh, so thatIL - f(x)(x) I < e for all points x of the domain (other than a) for which lx - al<h, thenL is the " limit off(x) at the pointa." The condition for the existence ofL is that, after the positive number has been specified, it must be possible to find a positive numberh, so that If(x') - f(x)l< e for all pointsx andx 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 numberL which is the limit off(x) at the point a as defined above. The limit off(x) at the point a is denoted byLtx=a f(x), or by lim a=a f(x).

Iff(x) is a function of one variablex in a domain which extends to infinite values, and if, after has been specified, it is possible to find a numberN, so that If(x') - f(x) I < e for all values of x and x' which are in the domain and exceedN, then there is a numberL which has the property that If(x) - L for all such values ofx. In this casef(x) has a limitL at x = co. In like mannerf(x) may have a limit atx= - 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,u2, .. un, ..., and this sequence can have a limit atn= 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 byinfinite 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 ruless 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 +--u2 + ..., 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 pointsx 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 thatx - a is positive; then we have a " limit on the right " ata; 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 byf(a - o) andf(a +o). The limitL off(x) at a stands in no necessary relation to the value off(x) ata. If the point a is in the domain of the argument, the value off(x) at a is assigned by the rule of calculation, and may be different fromL. In casef(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 forf(x) ata. In the case wheref(x) is defined for all points in an interval containinga, except the pointa, and has a limitL ata, we may arbitrarily annex the point a to the domain of the argument and assign tof(a) the valueL; the function may then be said to be " extrinsically defined." The so-called " indeterminate forms " (see are examples.

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 numberN which exceeds all the values (or is less than all the values). It may happen that, however great a numberN we take, there are among the values of the function numbers which exceedN (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 numberS 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 exceedS - 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 anyh, at which the values of the function differ fromS by less than E. ThusS is the limit of the function ata, 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 ofa, there is a superior limitS which is in this way the limit of the function ata, if furtherf (a) = S, thenS 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 amaximum value or aminimum value of the function.

Again, if, after any numberN, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceedsN, 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 anyh, at which the value of the function exceedsN. If the point a is in the domain of the argument the function is said to " tend to become infinite " ata; it has of course a finite value ata. If the point a is not in the domain of the argument the function is said to " become infinite " ata; it has of course no value ata. In like manner we may have a (negatively) infinite inferior limit. Again, after any numberN, however great, has been specified and a numberh found, so that all the values of the function, at points in the neighbourhood of a forh, exceedN 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.

A functionf(x) of one variablex, defined in the interval between a andb, is " increasing throughout the interval " if, wheneverx andx' are two numbers in the interval andx'> x, then f(x')> f (x); the function " never decreases throughout the interval " if,x andx 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 definitionb> a, the definition may apply to a function under the restriction thatx is notb andx is nota; 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 atb and a limit on the right ata, and these are the superior and inferior limits of its values at all points within the interval (the ends excluded); the like holdsmutatis mutandis if the function never increases. If the function is monotonous throughout the interval,f(b) is the greatest (or least) value off (x) in the interval; and iff (b) is the limit off (x) on the left atb, 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 andx diminishes towards zero, and it never becomes negative. It therefore has a limit on the right atx=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 atx =o.

9.Continuity of Functions. - A functionf(x) of one variablex is said to be continuous at a point a if (i)f(x) is defined in an interval containinga; (2)f(x) has a limit ata; (3) f(a) is equal to this limit. The limit in question must be a limit for continuous variation, not for a restricted domain. Iff(x) has a limit on the left at a andf(a) is equal to this limit, the function may be said to be " continuous to the left " ata; similarly the function may be " continuous to the right " ata. 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 numberh, so that the difference between any two values of the function in the interval betweena-h anda+h is less thane. 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 numberh which corresponds to any a as in (I) may be the same at all points of the interval, or, in other words, that the numbersh 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 pointa, there are four limiting values which come into consideration. Whatever positive numberh we take, the values of the function at points between a anda+h (a excluded) have a superior limit (or a greatest value), and an inferior limit (or a least value); further, ash 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) andf(a+o), and they are called the " limits of indefiniteness " on the right. Similar limits on the left are denoted byf(a- o) andf(a- o). Unlessf(x) becomes, or tends to become, infinite ata, all these must exist, any two of them may be equal, and at least one of them must be different from f(a), iff(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 ata, 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 numberN, however great, has been specified, it is possible to find a positive numberh, so that all the values of the function in the interval between a anda+h (a excluded) are less than-N; in such a casef(x) tends to become negatively infinite when x decreases towards a; other modes of tending to infinite limits may be described in similar terms.

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 functionf(x) at the pointa. 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 betweena-h anda+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 andb. Let intermediate points x 1, x,.xn_l, be taken so thatb> n _ 1 > x n _ 2.. > i > a. We may devise a rule by which, as n increases indefinitely, all the differencesb -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. Iff(x) is defined in an interval containing the pointa, anda - k and a--1--k are points of the interval, the expressionf(a+h) -f(a) (I) h represents a function ofh, which we may call (/)(h), defined at all points of an interval forh between-k andk 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 " off(x) at the pointa; when the last two are equal either of them is the " regressive differential coefficient " off(x) ata; when all four are equal the function is said to be " differentiable " ata, and either of them is the " differential coefficient " off(x) ata, or the " first derived function " off(x) ata. It is denoted byd dx) or byf' (x) . In this case 4)(h) has a definite limit at h = o, or is determinately infinite ath=o (§ 7). The four limits here in question are called, after Dini, the " four derivates " off(x) ata. In accordance with the notation for derived functions they may be denoted byf+(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 andad-h, then2 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 valuex of the argument and the corresponding valuey 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-ordinatesx andy. According to condition (2)y is a continuous and differentiable function ofx, 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 functionf(x) through the interval between a andb " to be the limit of the sumf (x 'T) -i) when the interval is divided into ultimately indefinitely small partial intervals by points x 1, x 2,. .. x,,_ l. Herex', denotes any point in the rth partial interval,x 0 is put fora, andx„ forb. 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 whenn 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 numbera, 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. IfS and I are the superior and inferior limits (or greatest and least values) off(x) in the interval between a andb, f f(x)dx is intermediate betweenS(b - a) and I(b - a).

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

3. If the further end-valueb is variable, and if ff(x)dx = F(x), then iff(x) is continuous atb, F(x) is differentiable atb, and F'(b)=f(b). 4. In casef(x) is continuous throughout the intervalF(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 andb, f'(x)dx =f(b) - f(a). 6. In casef(x) is discontinuous at one or more points of the interval between a andb, in which it is integrable, xf(x)dx is a function ofx, of which the four derivates at any point of the interval are equal to the limits of indefiniteness off(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 xF(x) = a F'(x)dx+const., where a is a fixed point, andx 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 functionsf(x) andcp(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 andb, there is a numberM intermediate between the superior and inferior limits, or greatest and least values, off(x) in the interval, which has the property expressed by the equationMcS f (x) dx = f f (x)q The second theorem of the mean is that, iff(x) is monotonous throughout the interval, there is a number E between a andb which has the proper d by the equationb =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. Ifc is a point in the interval between a andb at whichf(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 ff(x)dx to beLt 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 wheref(x) becomes infinite, or tends to become infinite, atc, provided both the limits exist.. This definition may be otherwise expressed by saying that a partial interval containing the pointc 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 wherec is a orb. Again, when the interval of integration is unlimited to the right, or extends to positively infinite values, we have as a definitionax f(x)dx = Lt f h f(x)dx provided this limit exists. Similar definitions apply toa f(x)dx, and toff(x)dx. All such definite integrals as the above are said to be " improper." x For example, fis improper in two ways. It meansLi ( sinh=œ =0.e x in which the positive number e is first diminished indefinitely, and the positive numberh 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." Iff(x) tends to become infinite at a pointc in the interval between a andb, and the expression (I) does not exist, then the expressionf(x)dx, which has no value, is called d a " divergent integral," and it may happen that there is a definite value forLt 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 ofn dimensions (C,) is the aggregate that is arrived at by attributing simultaneous values to each ofn variablesx 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 andx', the sum of the differences Ix i -x', I + I x 2 -x' 2 I + ... -iIxn-x', I may be denoted byI 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 numberN, however great, has been specified, it is possible to find in the domain points of which one or more co-ordinates exceedN in absolute value. The " neighbourhood " of a point a for a (positive) numberh is the aggregate constituted of all the pointsx, 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 pointa, which has the property that the points of the aggregate which are in the neighbourhood of a for any numberh, 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 pointsa, b of it, and choosing any positive number E, however small, a numberm 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 acontinuum. 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 aggregateH' 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 ofH,, 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 numberh, however small, contains points withinHn and points of H'„: the aggregate of these points is called the boundar y " ofH n . (4) When any two pointsa, b withinH, are taken, it is possible to find a number and a corresponding numberm, and to choose points x', x",...x(m), so that the neighbourhood of a for containsx', and consists exclusively of points within H,,, and similarly for x' and x", x" and x"',. x(m) andb. 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 aC 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 partH,, 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 forh, and from this neighbourhood we exclude the pointa, and we also exclude any point which is not in the domain of the argument. Then we takex and x' to be any two of the retained points in the neighbourhood. The functionf has a limit at a if for any positive however small, there is a correspondingh which has the property that (f(x')-f(x) whatever points in the neighbourhood of a forh we take (a excluded). For example, when there are two variablesx1, 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 pointa, 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 ofn 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 diminishx 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 ofx andy may be defined by the conditions that when x = 0 it is zero whatever valuey may have, and whenx o it has the value of sin {4 tan 1 (y/x) {. Wheny has any particular value this function is a continuous function ofx, and, whenx has any particular value this function is a continuous function ofy; but the function ofx andy 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 pointsx 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 numberh may be called its " breadth "; it is a homogeneous part of theLt numerical continuum ofn dimensions, and its boundary consists of all the points for whichIx - 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 toH, and there will generally be others of which some, but not all, of the points belong toH. 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 domainH 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 functionf(x) through a measurable domainII, 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 domainn 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 expressionxy / (x 2 y+i), which represents a function ofx vanishing at x = o and at all other values ofx having the value of i/x. The method of series falls under this more general process (cf. § 6). When the termsu 1, u 2, ... of a series are functions of a variablex, the sums n of the firstn terms of the series is a function ofx andn; and, when the series is convergent, its sum, which isLt,, = o,s,,, can represent a function ofx. 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 whenx 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 ofx by the definite integral e ttx-1dt, or by the infinite productLtn =? n z '+Zx) ' The second of these expressions avails for the representation of the f unction at all points at whichx 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 becomesf (x) =f(a) a)r fir)(a),r provided that (a) a positive numberk can be found so that at all points in the interval between a anda+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=ifcn>}a+B(x - a)}, has the limit zero whenn in-1). creases indefinitely, for all values of0 between o and i, and for all values ofx in the interval between a anda+k, except possiblya+k. When these conditions are satisfied, the series (I) represents the function at all points of the interval between a anda+k, except possiblya+k, and the function is " analytic " (§ 13) in this domain. Obvious modifications admit of extension to an interval between a anda - k, or betweena - k anda+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 numberk which has the property that the series converges absolutely in the interval betweena - k anda+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. Letf(x) be the sum of the series at any pointx of the domain, andfn(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 numbern so that if n (a) - f p (a)I <e for all values ofm andp which exceedn. The sum, f(a), is the limit of the sequence of numbersfn(a) at n=. The convergence is said to be " uniform " in an interval if, after specification of e, the same numbern suffices at all points of the interval to make If(x)-f m (x) I < e for all values of m which exceedn. 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 numberN we take, e can be taken so small that, at some point in the neighbourhood ofa, n must be taken> to make I f (x) - fn(x) I < e 'thenm>n; then the series does not converge uniformly in the neighbourhood ofa. The distinction may be otherwise expressed thus: Choose a first and e afterwards, then the numbern is finite; choose e first and allow a to vary, then the number n becomes a function ofa, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however smalle may be, the convergence is uniform.

For example, the series sin x - z sin 2x++ sinis convergent for all real values ofx, and, when R->x> - 7r its sum is a x; but, whenx is but a little less than7, 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 ofx = 7r. Another example is afforded by the seriesnx (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 putx= 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 2fr(x) =f(x) converges uniformly in the interval r=0 between a andr = 0 ,l afr(x)dx, .or a series which converges unformly may be integrated term by term.

(6) Iff' 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 have4;/(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 functionsfr(x) have limits ata, thenf(x) has a limit ata, which is ELt 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 coefficientsa n, b n of such a series, in order that it may represent a given functionf(x) in the interval between -c andc, was given by Fourier, viz, we haveao = 1 '_' an = f f(x) cosdx,b n =f f(x) sinedx. The interval between -c andc 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=nLt 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 of (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, iff(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 pointx 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 whichf(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 functionf(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 pointsa, (2) the improper definite integral off(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 forf(x) formed by Fourier's rule converges at the point a of the periodic interval, and iff(x) is continuous ata, the sum of the series isf(a); but it has been proved by P. du Bois Reymond that the function may be continuous ata, and yet the series formed by Fourier's rule may be divergent ata. 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) Iff(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 off(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) Iff(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 inx of the nth degree, so that the difference between the value off(x) and the value of the polynomial shall be less than e at all points of the interval.

Again it can be proved that, iff(x) is continuous throughout a given interval, polynomials inx of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal tof(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 seriesa n cos (b n x7r), where a is positive and less n=0 than unity, andb 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 functionf(x) represented by it has the property that there is, in the neighbourhood of any point xo, an infinite aggregate of pointsx', 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 whenx' - xo approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a functionf(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for whichF'(x)=f(x); and, iff(x) is one-signed throughout any intervalF(x) is monotonous throughout that interval, but yetF(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 formZa n cpn(x), where Za n is an absolutely convergent series of numbers, and4n(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, letfo(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 numbern can afterwards be found so that, at a point a of the domain, the valuef(a) of a certain functionf(x) is connected with the sum of the first n+ terms of the series by the relation If (a) - Zf r (a) It mustr=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 seriesfo(x) +f 1 (x) + ... is then an r=0 asymptotic expansion for the functionf(x) in the domain.

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

n! = y (27r) Inn+ie-n+e where0 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 ofa(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 pointa, and that each of'the functionsf r (x) has a limit ata. If we first putx= a, and then sum the series, we have the valuef (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 off(x) at a; if we first replace each functionf r (x) by its limit ata, and then sum the series, we may arrive at a value different from either of the foregoing. If the functionf(x) is continuous at a, the first and second results are equal; if the functionsfr(x) are all continuous ata, 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.

- 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), andSeries divergentes (Paris, 1901), alsoFonctions 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), andSerie 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), andDuff.- 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 " inActa Math. tt. ii., vii., andMath. 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), andStetigkeit u. irrationale Zahlen (Brunswick, 1872); Dirichlet, " Convergence des series trigonometriques,"Crelle, Bd. iv. P. Du Bois Reymond,Allgemeine Functionentheorie (Tubingen, 1882), and many memoirs inCrelle and inMath. 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 theEncyclopcidie 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 withElliptic 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 ofarea 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 formx+iy, wherex, y are ordinary real numbers, andi 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 symboli 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 numbersx, 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 variablez=x+iy, the distance OP be calledr, and the positive angle less than 2rr between Ox and OP be called0, we may write z = r(cos0+i sin0); thenr is called the modulus or absolute value of z and often denoted by I z I and0 is called the phase or amplitude of z, and often denoted by ph (z); strictly the phase is ambiguous by additive multiples of 27r. Ifz' = 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 cos0+i sin0; 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 particularz = i corresponds to = o.

Moreover, i being a rational function of z, both E and i are continuous differentiable functions ofx andy, save when is infinite; writing = f (x, y) = f (z -iy, y), the fact that this is really independent ofy leads at once toof/ax+iaflay =o, and hence toa = 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 toNlax, 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 coefficientsaE/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 ofx, 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 top, and therefore within a circle in the I'-plane whose centre is the origin, of radiusp, there are no points 1representing 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 numberm exists such that for every and every positivep, 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 calleduniform 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, whereina 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 whichr'< 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, ora 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 andt, which, since I zoI +t < R, is absolutely convergent; it may then be arranged according to powers oft. Thus we may writef(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 A1n=1 fort =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 findf (zo +t) = Etn f(n) (zo) /n!, where f (n) (zo) is the nth differential coefficient. Repeating for this power series, int, 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!ise 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 whenx, y are real, and z = x+iy, exp (z) =exp (x) exp (iy). Now the functions Uo=siny, Vo= i - cosy, =y -siny, V i =1y 2 - +cos = 6y 3 -y+siny, V= 4 -2y 2 +I-cos y,... all vanish fory =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, fory positive, that each of these functions is positive; proceeding to a limit we hence infer that COSy= -2y2+y4-..., sin y=ysy3+izay5-..., for positive, and hence, for all values ofy. We thus have exp (iy) = cosy+i siny, and exp (z) =exp (x). (cosy+i siny). In other words, the modulus of exp (z) is exp (x) and the phase isy. 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 period2kiri, wherek 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 linesy =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 Iis real this becomes the logarithm of 3 .; in general X(3-) =log 13-I+i ph (3-)+2k7ri, wherek is an integer; and when 1describes a closed circuit surrounding the origin the phase of I increases by 27r, ork increases by unity. Differentiating the series for I we have4/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 forft- - 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, whereq is a positive integer, there areq values possible for-llq, of the form exp [-X(d exp (2 q'-), withk = o, i, ... q - i, all other values ofk leading to one of these; the qth power of any one of these values is 3-; whena = plq, wherep, 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 numberi 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, cosi = 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 sidea, to whose perimeter is attached a definite direction of description, which we take to be counterclockwise; another square, also of sidea, 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 aregion 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 apath 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 radiusro, 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 orproper 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 callextensive; 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 pointz 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) forsome point zo within or upon the boundary of the sub-region, (2) forevery 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 calledsuitable: 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 whichf(x,y) > H, but points (x,y) exist for whichf(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 off(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 off(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 functionf(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 radiusr - I zo -, which is greater than jzo - i I, and increases tor as Izoi- 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 fromr 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 numberr 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 functionf(x,y) by f(z), and letf 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; whenf(z) = z, by takingf 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 numbersf(zo), F(zo), such that, whatever real positive quantity i may be, a real positive number e exists for which the conditionI' 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 realvariables 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 integralff (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 asf(z) =.f (zo)-{-(z - F(zo)-I-ne(z - zo), where I0 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 hencea(a+b+c) <2a(b+c) <47r0/a; the integralff(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 integralf 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&, andf(z) differs arbitrarily little from f(zo) ifr is sufficiently small; the value of the integral round this circle is therefore, ultimately, whenr vanishes, equal to 21rif (zo). Hence f(zo) = rifttf (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 fdtf (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 wholeinterior of the original region.

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 =fZ uo(z)dz+f u1(z)dz-{-...+r u1(z)dz - fz R(z)dz, z0 o wherein, in absolute value, f <eL,if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.

(2) Iff(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the regionf f (x, y)dz = o, then (z) = f af(x, y)dz, for interior points zo, z, is a differentiable function of z, having for its differential coefficient the functionf (x,y), which is therefore also a differentiable function of z at interior points.

(3) Hence if the series uo(z)+ui(z)d-... to ao be uniformly convergent over a region, its terms being differentiable functions of z, then its sum S(z) is a differentiable function of z, whose differential coefficient, given_byI (S(t)dt 2, i s obtainable by differentiating the 2?ra (t - z) series. This theorem, unlike (I), does not hold for functions of a real variable.

(4) If the region of definition of a differentiable function f(z) include the region bounded by two concentric circles of radiir, R, with centre at the origin, and zo be an interior point of this region, f(zo) =2? i f x ()d- Z,I rif t (t zpt, whe r e the integrals are both counterclockwise round the two circumferences respectively; putting in the first (t - zo)-1 = zo"/tn+1, and in the second (t - zo)-1= - to/zo"+1, n=0 n = o we find f(zo) _ Anzo n, wherein An = a?if t (dt, taken round any circle, centre the origin, of radius intermediate betweenr and R. Particular cases are: (a) when the region of definition of the function includes the whole interior of the outer circle; then we may taker =o, the coefficients A n for which n <o all vanish, and the function f(zo) is expressed for the whole interior Izo f <R by a power series A n zo n. In other words,about every interior point cof the region of definition a differentiable function of zis expressible by a power series in z - c; a very important result.

(13) If the region of definition, though not including the origin, extends to within arbitrary nearness of this on all sides, and at the same time the product z m f(z) has a finite limit when I z I diminishes to zero, all the coefficients A n for whichn< - m vanish, and we have.f (zo) = A-mzo m +A-m+izom +i + ... -I-A_ i zo 1 -I-Ao +Aizo ... to 00. Such a case occurs, for instance, when f(z) = cosec z, the number m being unity.

§ 6.Singular Points. - The region of existence of a differentiable function of z is an unclosed aggregate of points, each of which is an interior point of a neighbourhood consisting wholly of points of the aggregate, at every point of which the function is definite and finite and possesses a unique finite differential coefficient. Every point of the plane, not belonging to the aggregate, which is a limiting point of points of the aggregate, such, that is, that points of the aggregate lie in every neighbourhood of this, is called asingular point of the function.

About every interior point zo of the region of existence the function may be represented by a power series in z - zo, and the series converges and represents the function over any circle centre at zo which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting z= i/?', that if the region of existence of the function contains all points of the plane for which I z I > R, then the function is representable for all such points by a power series in z1 - or; in such case we say that the region of existence of the function contains the point z = co. A series in z 1 has a finite limit when I z I = 00; a series in z cannot remain finite for all points z for which I z I > R; for if, for I z I = R, the sum of a power series Za n e in z is in absolute value less than M, we have Ian I <Mr - n, and therefore, if M remains finite for all values ofr however great, an = o. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = 00; such is, for instance, the case of the function exp (z) =Ezn/n!. This may be regarded as a particular case of a well-known result (§ 7), that the circumference of convergence of any power series representing the function contains at least one singular point. As an extreme case functions exist whose region of existence is circular, there being a singular point in every arc of the circumference, however small; for instance, this is the case for the functions represented for I z I < i by the series E z m, where m = n 2, the series /zm wherem=n!, and the series I ?zm/(m+i)(m+2) where m=an, n=1 a being a positive integer, although in the last case the series actually converges for every point of the circle of convergence I z =I. If z be a point interior to the circle of convergence of a series representing the function, the series may be rearranged in powers of z - zo; as zo approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in z - zo diminish to zero; when, however, a circle can be put about zo, not containing any singular point of the function, but containing points outside the circle of convergence of the original series, then the series in z - zo gives the value of the function for these external points. If the function be supposed to be given only for the interior of the original circle, by the original power series, the series in z - zo converging beyond the original circle gives what is known as ananalytical continuation of the function. It appears from what has been proved that the value of the function at all points of its region of existence can be obtained from its value, supposed given by a series in one original circle, by a succession of such processes of analytical continuation.

This suggests an entirely different way of formulating the fundamental parts of the theory of functions of a complex variable, which appears to be preferable to that so far followed here.

Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z-zo, about any point zo interior to its circle of convergence, and the new series converges certainly for I z-zo I <rIzol, ifr be the original radius of convergence. If for every position of zo this is the greatest radius of convergence of the derived series, then the original series represents a function existing only within its circle of convergence. If for some position of zo the derived series converges for Iz - zoI <r-1zoI+D, then it can be shown that for points z, interior to the original circle, lying in the annulusr- IzoI <Iz-zoI <r-Izol+D, the value represented by the derived series agrees with that represented by the original series. If for another point z 1 interior to the original circle the derived series converges for Iz-z i l<r-Iz 1 I+E, and the two circles Iz-zol = r - I zp l -i-D, =r - IziI+E have interior points common, lying beyond I z I = r, then it can be shown that the values represented by these series at these common points agree. Either series then can be used to furnish an analytical continuation of the function as originally defined. Continuing this process of continuation as far as possible, we arrive at the conception of the function as defined by an aggregate of power series of which every one has points of convergence common with some one or more others; the whole aggregate of points of the plane which can be so reached constitutes the region of existence of the function; the limiting points of this region are the points in whose neighbourhood the derived series have radii of convergence diminishing indefinitely to zero; these are the singular points. The circle of convergence of any of the series has at least one such singular point upon its circumference. So regarded the function is called amonogenic function, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called amonogenic analytical function, or simply ananalytical function; all that is necessary to define it is the value of the function and of all its differential coefficients, at some one point of the plane; in the method previously followed here it was necessary to suppose the function differentiable at every point of its region of existence. The theory of the integration of a monogenic function, and Cauchy's theorem, thatff(z)dz =o over a closed path, are at once deducible from the corresponding results applied to a single power series for the interior of its circle of convergence. There is another advantage belonging to the theory of monogenic functions: the theory as originally given here applies in the first instance only to single valued functions; a monogenic function is by no means necessarily single valued - it may quite well happen that starting from a particular power series, converging over a certain circle, and applying the process of analytical continuation over a closed path back to an interior point of this circle, the value obtained does not agree with the initial value. The notion of basing the theory of functions on the theory of power series is, after Newton, largely due to Lagrange, who has some interesting remarks in this regard at the beginning of hisTheorie des fonctions analytiques. He applies the idea, however, primarily to functions of a real variable for which the expression by power series is only of very limited validity; for functions of a complex variable probably the systematization of the theory owes most to Weierstrass, whose use of the word monogenic is that adopted above. In what follows we generally suppose this point of view to be regarded as fundamental.

Apole is a singular point of the function f(z) which is not a singularity of the functioni /f (z); this latter function is therefore, by the definition, capable of representation about this point, zo, by a series [f (z)] -1 = /a n (z -zo) ". If herein ao is not zero we can hence derive a representation forf(z) as a power series about zo, contrary to the hypothesis that zo is a singular point for this function. Hence ao = o; suppose also ai =o,a 2 = o, ...

buta m o. Then [ f (z)]- 1 = (z-z0)m[am+am+1(z-zo)'+' ...], and hence (z-zo) m f(z) =a;1+Ebn(z-zo)", namely, the expression of f(z) about z=z 0 contains a finite number of negative powers of z-zo and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.

The integralff(z)dz (z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 27riA 1, where A 1 is the coefficient of (z -zo)-' in the expansion of f(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called theresidue of f(z) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points,the integral 2_ri f f (z)dzround the boundary of this region is equal to the sum of the residues at the included poles, a very important result. Any singular point of a function which is not a pole is called anessential singularity; if it be isolated the function is capable, in the neighbourhood of this point, of approaching arbitrarily near to any assigned value. For, the point being isolated, the function can be represented, in its neighbourhood, as we have proved, by a series an(z - zo) n;it thus cannot remain finite in the immediate neighbour hood of the point. The point is necessarily an isolated essential singularity also of the function If(z) -A}- 1, for if this were expressible by a power series about the point, so would also the functionf(z) be; as {f(z)-A}- 1 approaches infinity, so does f(z) approach the arbitrary value A. Similar remarks apply to the point z = oo, the function being regarded as a function of =z 1 In the neighbourhood of an essential singularity, which is a limiting point also of poles, the function clearly becomes infinite. For an essential singularity which is not isolated the same result does not necessarily hold.

A single valued function is said to be anintegral function when it has no singular points except z= co. Such is, for instance, an integral polynomial, which has z= oo for a pole, and the functions exp (z) which has z= ao as an essential singularity. A function which has no singular points for finite values of z other than poles is called ameromorphic function. If it also have a pole at z= GO it is arational function; for then, ifa l,. .. a s be its finite poles, of orders m l,m 2j. ..m s , the product (z-a i) m i. .. (z-a s) m s f(z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z 1)-m ar(z 1)r; thus (z-a l) m l. .. (z-as)m,f(z) / is capable of a form /b r z r , butz--- m Eb r z r remains finite forr -0 r=0 z = oo. Thereforeb r+1 = b r+2 =. .. = o, and f(z) is a rational function.

If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to beal, a2, a 31.. . with Ja2!- Ia31. and limit Ia n ] =cc. Aboutas the function is expressible as An(z-as)n; let fs(z) = E A n (z-as) n be the sum of the negative powers in this expansion. Assuming z=o not to be a singular point, let fs(z) be expanded in powers of z, in the form / Cnzn, and As be chosen so n0 that F,(z) = fs(z) - E Cnz n = E Cnz n is, for IzI <r s< last, less in absolute 1µs value than the general term ss of a fore-agreed convergent series of real positive terms. Then the series 4(z) = F s (z) converges uni formly in any finite region of the plane, other than at the pointsas, and is expressible about any point by a power series, and neara s , 41(z)-f s (z) is expressible by a power series in z-as. Thus F(z) -0(z) is an integral function. In particular when all the finite singularities of F(z) are poles, F(z) is hereby expressed as the sum of an integral function and a series of rational functions. The condition<os is imposed only to render the series /F8(z) uniformly convergent; this condition may in particular cases be -1 satisfied by a series ZG s (z) where G s (z) = f s (z) - E Cnz n and <µ3.

An example of the theorem is the function it cot 7rz-z 1 for which, taking at first only half the poles, f s (z) = 1/(z-s); in this case the series EF s (z) where F s (z) = (z-s)--'+s--1 is uniformly convergent; thus it cot 2rz-z 1 -[(z-s)-1-1-s-9, wheres=o is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.

Considering an integral function f(z), if there be no finite positions of z for which this function vanishes, the function X[f(z)] is at once seen to be an integral function, 41(z), or f(z) =exp [0(z)]; if however great R may be there be only a finite number of values of z for whichf(z) vanishes, say z = a l, ... a,,, then it is at once seen that f(z) = exp [41(z)]. (z-ai)hl... (z-a, n) h m, where 41(z) is an integral function, and h1,...h, n are positive integers. If, however,f(z) vanish for z = a1,a 2 ,. .. where la i la2le ... and limit lad= oo , and if for simplicity we assume that z-o is not a zero and all the zerosa 1, a 2 ,... are of the first order, we find, by applying the preceding theorem to the functi on idf?z), thatf(z) =exp [0(z)] II {(1-z/an) exp 4n(z)},f (Z) dz l where 41(z) is an integral function, and 41n(z) is an integral polynomial of the form 41n(z) = a n+ 2 a2+... +s -o. The numbers may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to takes=n. In particular for the function we have '7rx sinT x oo ((I n) exp / 5 ' where n=o is excluded from the product. (Or again we have rIx) =xecxn 1 (I+n) exp n), where C is a constant, and r(x) is a function expressible whenx is real and positive by the integrale ttx-idt. There exist interesting investigations as to the connexion of the value ofs above, the law of increase of the modulus of the integral function f(z), and the law of increase of the coefficients in the series f(z) =lc/ 7 ,z" as n increases (see the bibliography below underIntegral Functions). It can be shown, moreover, that an integral function actually assumes every finite complex value, save, in exceptional cases, one value at most. For instance, the function exp (z) assumes every finite value except zero (see below under § 21,Modular Functions). The two theorems given above, the one, known as MittagLeffler's theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point z = oo as their only limiting point, the other, Weierstrass's factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows: I. Ifa l, a 2, a 3, ... be an infinite series of isolated points having the points of the aggregate (c) as their limiting points, so that in any neighbourhood of a point of (c) there exists an infinite number of the pointsa l, a 2, ..., and with every pointa i there be associated a polynomial in (z-a t)- 1, say g i; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point a i a pole whereat the expansion consists of the terms gi, together with a power series in z-a i; the function is expressible as an infinite series of termsgi-yi, where 'yi is also a rational function.

II. With a similar aggregate (a), with limiting points (c), suppose with every pointa i there is associated a positive integerr t . Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to orderr i at the point ai, but not elsewhere, expressible in the form II cn)rn' exp (g.), 'n=1 - cn where with every point an is associated a proper pointc n of (c), and?i (an-c, Sgn=rn 1s z- A . being a properly chosen positive integer.

If it should happen that the points (c) determine a path dividing the plane into separated regions, as, for instance, if an= exp (iir-N/ 2.n), when(c) consists of the points of the circle I z I = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.

§ 9.Construction of a Monogenic Function with a given Region of Existence. - A series of isolated points interior to a given region can be constructed in infinitely many ways whose limiting points are the boundary points of the region, or are boundary points of the region of such denseness that one of them is found in the neighbourhood of every point of the boundary, however small. Then the application of the last enunciated theorem gives rise to a function having no singularities in the interior of the region, but having a singularity in a boundary point in every small neighbourhood of every boundary point; this function has the given region as region of existence.

§ I o.Expression of a Monogenic Function by of Rational Functions in a given Region. - Suppose that we have a region R 0 of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called Co. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of Co; leta, b, ... be finite points upon C or outside R. Then whenb is near enough toa, the fraction (a - b)/(z - b) is arbitrarily small for all positions of z; say z-b I<?' for Ia-b the rational function of the complex variable t, In, a tal_ t - in whichn is a positive integer, is not infinite att= a, but has a pole att= b. By takingn large enough, the value of this function, for all positions z oft belonging to Ro, differs as little as may be desired from (t - a) - 1 . By taking a sum of terms such as F =EA t ci [I - ()fl] P we can thus build a rational function differing, in value, in Ro, as little as may be desired from a given rational function f=?A,,(t-a)-p, and differing, outside R or upon the boundary of R, fromf, in the fact that whilef is infinite att = a, F is infinite only atb. By a succession of steps of this kind we thus have the theorem that, given a rational function oft whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in Ro arbitrarily little from the former, whose poles are all at the pointc. Now any monogenic functionf(t) whose region of definition includes C and the interior of R can be represented at all points z in R 0 byf (z) = Iff(t)dt 27rtt- z ' where the path of integration is C. This integral is the limit of a sum/ s_ I z'f(ti)(ti+1- ti) 27riti - z ' where the pointst i are upon C; and the proof we have given of the existence of the limit shows that the sum S converges to f(z) uniformly in regard to z, when z is in R 0, so that we can suppose, when the subdivision of C into intervals t i+i-t i has been carried sufficiently far, that- f(z) e, for all points z of Ro, where e is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to Ro. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from f(z), for all points z of Ro, with poles at arbitrary positions outside Ro which can be reached by finite continuous curves lying outside R from the points of C.

In particular, to take the simplest case, if Co, C be simple closed polygons, and F be a path to which C approximates by taking the number of sides of C continually greater, we can find a rational function differing arbitrarily little from f(z) for all points of Ro whose poles are at one finite pointc external to r. By a transformation of the formt-c = r-', with the appropriate change in the rational function, we can suppose this pointc to be at infinity, in which case the rational function becomes a polynomial. Suppose E2, to be an indefinitely continued sequence of real positive numbers, converging to zero, and P r to be the polynomial such that, within 3 ,.--f(z) then the infinite series of polynomials P i (z) P2 (z) - (z) --1-1 13 3(z) - P2(z)}+...

whose sum to n terms is Pn(z), converges for all finite values of z and represents f(z) within Co.

When C consists of a series of disconnected polygons, some of which may include others, and, by increasing indefinitely the number of sides of the polygons C, the points C become the boundary points r of a region, we can suppose the poles of the rational function, constructed to approximate to f(z) within R 0, to be at points of r. A series of rational functions of the form (z)-H i 3 (z)-H 2 then, as before, represents f(z) within R 0. And Ro may be taken to coincide as nearly as desired with the interior of the region bounded by F.

§ I I.Expression of (I-z)- 1by means of Polynomials. Applications. - We pursue the ideas just cursorily explained in some further detail.

Letc be an arbitrary real positive quantity; putting the complex variable = i--ir k , enclose the points = I, = I -}-c by means of (i.) the straight lines i = =a, from E =I to = 1+c, (ii.) a semicircle convex to = o of equation Q-1) 2 +1 7 2= a 2 , (iii.) a semicircle concave to = o of equation (E-I-c)2-}-v2 = a 2. The quantitiesc and a are to remain fixed. Take a positive integer r so thatY (a) is less than unity, and puta = r () . Now takeci=l+c/r, c 2 =I+2c/r, ... cr=l+c; ifn 11 n 2, ...n r be positive integers, the rationalfunction I I _ c:;.) nl 1 is finite at =I, and has a pole of order n 1 at =c1; the rational function I I r { I - 1 c i nl) I - (c2 Z_ 1) "2 "1.is thus finite except for=c 2 , where it has a pole of order n1n2; finally, writing ns xs c 8 f the rational function U =. .. (I-x) n i n 2

nr-1 has a pole only at = I+c, of ordern 1 n 2 ...

The difference is of the form (I-0 - 1 P, where P, of the form I-(I-pi) (I -P2). .. (I-Pk), in which there are equalities among p i, p 2, ...p k , is of the form therefore, if I r i I = I p i I, we have I PI <Er 1 +Er l r 2 +Er i r 2 r 3 + ... < (I +ri) (I +r2) ... (I +r k)-I; now, so long as i is without the closed curve above described round = I, = I +c, we have I I ?I<a -Cm <lr<a, and tJ hence 7 J (I - S)-1 - V <a-1{ (i +o"i) (I -Ia n2) n 1(I +an3)"1"2.. .

(I +anr)n1n2.

• nr-1 - I}

Take an arbitrary real positivee, and a positive number, so that 0µ-I<ea, then a value ofn, such that a n d <4(I and therefore (7 ni /(I - a ni) < 1 c, and values for n 2, 113, ... such that a n t < I a2n1, n1 a n 3 < I a 3n la n r < n 1 n 2

nr-1 .

niI a n r n l; then, as I +x <e x, we have I(-?')-1-UI<a 1 {exp (a nl +n l a n 2+n i n 2 a n 3+... +...nr_ianr)-I}, and therefore less than a-'t exp (ani+a2n1+. .. +anrnl) - I which is less than 11 1 d Lexp I I a n1) I ] and therefore less than 0.

The rational function U, with a pole at ?'= I+c, differs therefore from (I-f)- 1, for all points outside the closed region put about =I, ?-= i+c, by a quantity numerically less than E. So long as a remains the same, r and a will remain the same, and a less value of e will require at most an increase of the numbers n 1, n2, ... nr; but if a be taken smaller it may be necessary to increaser, and with this the complexity of the function U.

c (c+I)z. Now put z-c+ I-?' ? =c+z ' thereby the points i-=0, I, I+c become the points z =o, I, Po, the function (I-z)- 1 being given by (I -z)' =c(c+I)-1(I _ i-)-1+ (c+ I) -1; the function U becomes a rational function of z with a pole only at z = c, that is, it becomes a polynomial in z, say- z i H - c, where H is also a polynomial in z, and I I z -H - c+I LI I ?' U ] ' the lines n = ± a become the two circles expressed, ifz = x+iy, by (x+c)2+y2= t c (c a + i) y, the points (n =o, E= I-a), (n =o, E = I +c+a) become respectively the points (y=o,c(i-a)/(c+a),(y =o, x=-c(i+c+a)/a), whose limiting positions for a =o are respectively (y=o, x = I), (y =o, x= - oo). The circle (x+c) 2 +y 2 =c(c+i)y/a can be writteny = (x+-)2+ (x+-) 4 1+1 / [u2-(x+c)2]}-2, 2122µ where i = Zc(c+I)/a; its ordinatey, for a given value of x, can therefore be supposed arbitrarily small by taking a sufficiently small.

We have thus proved the following result; taking in the plane of z any finite region of which every interior and boundary point is at a finite distance, however short, from the points of the real axis for which i x co, we can take a quantitya, and hence, with an arbitraryc, determine a numberr; then corresponding to an arbitrarye s we can determine a polynomial Ps, such that, for all points interior to the region, we have (i-z1) -Ps1«8; thus the series of polynomials P l +(P2-Pi)+(P3-P2)+

constructed with an arbitrary aggregate of real positive numbers, i e 2, ea, ... with zero as their limit, converges uniformly and represents (I-z)- 1 for the whole region considered.

§ 12.Expansion of a Monogenic Function in Polynomials, over a Star Region. - Now consider any monogenic functionf(z) of which the origin is not a singular point; joining the origin to any singular point by a straight line, let the part of this straight line, produced beyond the singular point, lying between the singular point and z =, be regarded as a barrier in the plane, the portion of this straight line from the origin to the singular point being erased. Consider next any finite region of the plane, whose boundary points constitute a path of integration, in a sense previously explained, of which every point is at a finite distance greater than zero from each of the barriers before explained; we suppose this region to be such that any line joining the origin to a boundary point, when produced, does not meet the boundary again. For every pointx in this region R we can then writedt 2?tr2f(x) = tI-xt-1' wheref(x) represents a monogenic branch of the function, in case it be not everywhere single valued, andt is on the boundary of the region. Describe now another region Ro lying entirely within R, and let x be restricted to be within R, or upon its boundary; then for any pointt on the boundary of R, the points z of the plane for whichzt1 is real and positive and equal to or greater than I, being points for which I z 1=1 t I I z I> I t I, are without the region Ro, and not infinitely near to its boundary points. Taking then an arbitrary real positive e we can determine a polynomial inxt - 1 , sayP(xt-1), such that for all pointsx in Ro we have (I-xt-1)-1-P(xt-1) I <e; the form of this polynomial may be taken the same for all pointst on the boundary of R, and hence, if E be a proper variable quantity of modulus not greater thane, 12if) - f dt f(t)P(xt - 1)I = If f (t) E IceLM, where L is the length of the path of integration, gration, the boundary of R, and M is a real positive quantity such that upon this boundary I t1 f (t)I <M. If now P(xt1) =co+cixl-1+...+cment-m, and 2711t - r -1f(t)dt =µr, 2z this gives c01-10+ clµ,x+

+cm,umx m } I eLM/27r, where the quantities Po, 1 c i, 112, ... are the coefficients in the expansion off(x) about the origin.

If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points off(x) to x = oo, it is possible, corresponding to an arbitrary real positive number a, to determine a number m, and a polynomial Q(x), of orderm, such that for all interior points of this region If(x)-Q(x) I <aHence as before, within this regionf(x) can be represented by a series of polynomials, converging uniformly; whenf(x) is not a single valued function the series represents one branch of the function.

The same result can be obtained without the use of Cauchy's integral. We explain briefly the character of the proof. If a monogenic function oft, 0(t) be capable of expression as a power series int-x about a pointx, for It p, and for all points of this circle I4(t) we know that I (p(")(x) I<gp n (n!). Hence, taking I z I' we have (µ+n)<(2)", we have 4)(µ+n) jx) un I < ?(µ+n)(x) (p+n)µ I zl n <g n (3) n (P) n C gn. (Ii+n)! Pµ+ 2 3P42 and therefore 0cµ>(x = E?(?ni (x) z n +?µ, n=o where I c p. g 2 I n < p2 m.

Pµn =m+1 Now draw barriers as before, directed from the origin, joining the singular point of 4)(z) to z = cc, take a finite region excluding all these barriers, letp be a quantity less than the radii of convergence of all the power series developments of (I)(z) about interior points of this region, so chosen moreover that no circle of radiusp with centre at an interior point of the region includes any singular point ofcp(z), letg be such that) 4 (z) I<g for all circles of radiusp whose centres are interior points of the region, and, x being any interior point of the region, choose the positive integer n so thatI x I < 3p; then take the points a i = x/n,a 2 = 2x/n, a 3 = 3x/n, ...a n = x; it is supposed that the region is so taken that, whateverx may be, all these are interior points of the region. Then by what has been said, replacingx, z respectively by o and x/n, we have where I A3µ I < 2g/p/ A 2 m 2, the numbers ml, m 2 being respectively n21 and n2"-2.

Applying then the original inequality toct.(µ)(a 3) = w)(a2+x/n), and then using the series j ust obtained, we find a series for (1-)(a3). This process being continued, we finally obtain m1 m2 mn (0) (xl (h) cp(x) = E E ...E--R -` n)+E' Al = 0 A2 =0 An =0 whereh =Al+A2+...+An, K=AI! A 2 !... An!, m1 = n 2n, m2=n2n-2,..., mn = n 2, I <2g/27%.

By this formula c i 5(x) is represented, with any required degree of accuracy, by a polynomial, within the region in question; and thence can be expressed as before by a series of polynomials converging uniformly (and absolutely) within this region.

u=J (x,v)y x ' conversely if =4(uo), =- 4'(uo) and, n be any values satisfying n2= 4.4 which are sufficiently near respectively to xo, whilev is defined byn) v - = - then, n are respectively42(v) and -4'(v); for this equation leads to an expansion for - xo in terms ofv = uo, and only one such expansion, and this is obtained by the same work as would be necessary to expand 4(v) whenv is near to uo; the function 0(u) can therefore be continued by the help of this equation, from v=uo, provided the lower limit of i-xoI necessary for the expansions is not zero in the neighbourhood of any value (xo,yo). In fact the function 4(u) can have only a finite number of poles in any finite part of the plane of u; each of these can be surrounded by a small circle, and in the portion of the finite part of the plane ofu which is outside these circles, the lower limit of the radii of convergence of the expansions of 4(u) is greater than zero; the same will therefore be the case for the lower limit of the radii necessary for the continuations spoken of above provided that the values of (, n) considered do not lead to infinitely increasing values ofv; there does not exist, however, any definite point in the neighbourhood of which the integralf E,n xo v o) - 77 increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if Um) be any point, where n 2 =4 3 -g2-g3, andv be defined by v = ((°°) (f,',)y' then =4(v) and n= -4'(v). Thus this equation determines (, n) without ambiguity. In particular the additive indeterminatenesses of the integral obtained byclosed circuits of the point of integration are periods of the function by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be f`° dx _ w=2J '=2 Je 3 these quantities cannot therefore have a real ratio, for else, being periods of a monogenic function, they would, as we have previously seen, be each integral multiples of another period; there would then be a closed path for (x,y), starting from an arbitrary point (xo,yo), other than one enclosing two of the points (e i ,o), (e2,o), (e 3 ,o), (oo, oo), which leads back to the initial point (xo,yo), which is impossible. On the whole, therefore, it appears that the function 4(u) agrees with the functionl3(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14,Doubly Periodic Functions. Modular Functions. - One result of the previous theory is the remarkable fact that if_' r °° dx w= co 2 ,J -; y e3 y wherey 2 = 4(x - e i)(x -e 2)(x - e 3), then we have e l = (Zw)- 2 +E'{[(m+ 1)w+m'w'] -2 -[mw and a similar equation for ea, where the summation refers to all integer values of m andm other than the one pair m = o, m'= o. This, with similar results, has led to the consideration of functions of the complex ratioco /w.