Function

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, Théorie 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 1 correspond to 2, 2 to 4, and generally n to 2n. The aggregate of positive integers is an infinite aggregate. The aggregates of all rational numbers and of all real numbers and of points on a line are other examples of infinite aggregates. An aggregate whose elements are real numbers is said to “extend to infinite values” if, after any number N, however great, is specified, it is possible to find in the aggregate numbers which exceed N in absolute value. Such an aggregate is always infinite. The “neighbourhood of a number (or point) a for a positive number h” is the aggregate of all numbers (or points) x for which the absolute value of x − a denoted by |x − a|, does not exceed h.

For the specification of any particular function two things are requisite: (1) 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 1/x.” This function is defined by the analytical expression 1/x at all points except the point x = 0. 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.

If ƒ(x) is a function of one variable x in a domain which extends to infinite values, and if, after ε has been specified, it is possible to find a number N, so that |ƒ(x′) − ƒ(x)| < ε for all values of x and x′ which are in the domain and exceed N, then there is a number L which has the property that |ƒ(x) − L| < ε for all such values of x. In this case ƒ(x) has a limit L at x = ∞. In like manner ƒ(x) may have a limit at x = −∞. 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 at n = ∞.

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 = ∞” does not mean that x takes some particular value which is infinite. There is no such value. The phrase always refers to a limiting process in which, as the process is carried out, the variable number x increases without limit: it may, as in the above example of a sequence, increase by taking successively the values of all the integral numbers; in other cases it may increase by taking the values that belong to any domain which “extends to infinite values.”

A very important type of limits is furnished by infinite series. When a sequence of numbers u1, u2, ... un, ... is given, we may form a new sequence s1, s2, ... sn, ... from it by the rules s1 = u1, s2 = u1 + u2, ... sn = u1 + u2 + ... + un or by the equivalent rules s1 = u, sn − sn−1 = un(n = 2, 3, ...). If the new sequence has a limit at n = ∞, this limit is called the “sum of the infinite series” u1 + u2 + ..., and the series is said to be “convergent” (see Series).

A function which has not a limit at a point a may be such that, if a certain aggregate of points is chosen out of the domain of the argument, and the points x in the neighbourhood of a are restricted to belong to this aggregate, then the function has a limit at a. For example, sin(1/x) has limit zero at 0 if x is restricted to the aggregate 1/π, 1/2π, ... 1/nπ, ... or to the aggregate 1/2π, 2/5π, ... n/(n2 + 1)π, ..., but if x takes all values in the neighbourhood of 0, sin (1/x) has not a limit at 0. Again, there may be a limit at a if the points x in the neighbourhood of a are restricted by the condition that x − a is positive; then we have a “limit on the right” at a; similarly we may have a “limit on the left” at a point. Any such limit is described as a “limit for a restricted domain.” The limits on the left and on the right are denoted by ƒ(a − 0) and ƒ(a + 0).

The limit L of ƒ(x) at a stands in no necessary relation to the value of ƒ(x) at a. If the point a is in the domain of the argument, the value of ƒ(x) at a is assigned by the rule of calculation, and may be different from L. In case ƒ(a) = L the limit is said to be “attained.” If the point a is not in the domain of the argument, there is no value for ƒ(x) at a. In the case where ƒ(x) is defined for all points in an interval containing a, except the point a, and has a limit L at a, we may arbitrarily annex the point a to the domain of the argument and assign to ƒ(a) the value L; the function may then be said to be “extrinsically defined.” The so-called “indeterminate forms” (see Infinitesimal Calculus) are examples.

All that has been said here may be adapted to the description of greatest values, superior limits, &c., of a function in a restricted domain contained in the domain of the argument. In particular, the domain of the argument may contain an interval; and therein the function may have a superior limit, or an inferior limit, which is attained. Such a limit is a maximum value or a minimum value of the function.

Again, if, after any number N, however great, has been specified, it is possible to find points of the domain of the argument at which the value of the function exceeds N, the values of the function are said to have an “infinite superior limit,” and then there must be a point a which has the property that there are points of the domain, in the neighbourhood of a for any h, at which the value of the function exceeds N. If the point a is in the domain of the argument the function is said to “tend to become infinite” at a; it has of course a finite value at a. If the point a is not in the domain of the argument the function is said to “become infinite” at a; it has of course no value at a. In like manner we may have a (negatively) infinite inferior limit. Again, after any number N, however great, has been specified and a number h found, so that all the values of the function, at points in the neighbourhood of a for h, exceed N in absolute value, all these values may have the same sign; the function is then said to become, or to tend to become, “determinately (positively or negatively) infinite”; otherwise it is said to become or to tend to become, “indeterminately infinite.”

All the infinities that occur in the theory of functions are of the nature of variable finite numbers, with the single exception of the infinity of an infinite aggregate. The latter is described as an “actual infinity,” the former as “improper infinities.” There is no “actual infinitely small” corresponding to the actual infinity. The only “infinitely small” is zero. All “infinite values” are of the nature of superior and inferior limits which are not attained.

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

The principal properties of a continuous function are:

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

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

3. The function has a greatest value and a least value in the interval, and these are superior and inferior limits which are attained.

4. There is at least one point of the interval at which the function takes any value between its greatest and least values in the interval.

5. If the interval is unlimited towards the right (or towards the left), the function has a limit at ∞ (or at −∞).

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

For the further classification of functions we divide the domain of the argument into partial intervals by means of points between the end-points. Suppose that the domain is the interval between a and b. Let intermediate points x1, x2 ... xn−1, be taken so that b > xn−1 > xn−2 ... > x1 > a. We may devise a rule by which, as n increases indefinitely, all the differences b − xn−1, xn−1 − xn−2, ... x1 − 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.

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

We have the following theorems:—

1. Any continuous function is integrable.

2. Any function with restricted oscillation is integrable.

3. A discontinuous function is integrable if it does not tend to become infinite, and if the points at which the oscillation of the function exceeds a given number σ, 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:

1. If S and I are the superior and inferior limits (or greatest and least values) of ƒ(x) in the interval between a and b, ∫ ba ƒ(x)dx is intermediate between S(b − a) and I(b − a).

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

3. If the further end-value b is variable, and if ∫ xa ƒ(x)dx = F(x), then if ƒ(x) is continuous at b, F(x) is differentiable at b, and F′(b) = ƒ(b).

4. In case ƒ(x) is continuous throughout the interval F(x) is continuous and differentiable throughout the interval, and F′(x) = ƒ(x) throughout the interval.

5. In case ƒ′(x) is continuous throughout the interval between a and b,

∫ ba ƒ′(x)dx = ƒ(b) − ƒ(a).

6. In case ƒ(x) is discontinuous at one or more points of the interval between a and b, in which it is integrable,

∫ xa ƒ(x)dx

is a function of x, of which the four derivates at any point of the interval are equal to the limits of indefiniteness of ƒ(x) at the point.

7. It may be that there exist functions which are differentiable throughout an interval in which their differential coefficients are not integrable; if, however, F(x) is a function whose differential coefficient, F′(x), is integrable in an interval, then

F(x) = ∫ xa F′(x)dx + const.,

where a is a fixed point, and x a variable point, of the interval. Similarly, if any one of the four derivates of a function is integrable in an interval, all are integrable, and the integral of either differs from the original function by a constant only.

The theorems (4), (6), (7) show that there is some discrepancy between the indefinite integral considered as the function which has a given function as its differential coefficient, and as a definite integral with a variable end-value.

We have also two theorems concerning the integral of the product of two integrable functions ƒ(x) and φ(x); these are known as “the first and second theorems of the mean.” The first theorem of the mean is that, if φ(x) is one-signed throughout the interval between a and b, there is a number M intermediate between the superior and inferior limits, or greatest and least values, of ƒ(x) in the interval, which has the property expressed by the equation

M ∫ ba φ(x)dx = ∫ ba ƒ(x)φ(x)dx

The second theorem of the mean is that, if ƒ(x) is monotonous throughout the interval, there is a number ξ between a and b which has the property expressed by the equation

∫ ba ƒ(x) φ(x)dx = ƒ(a) ∫ ξa φ(x)dx + ƒ(b) ∫ bξ φ(x)dx.

(See Fourier’s Series.)

The definition of a continuum in Cn 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” Hn of Cn. Such a part would correspond to an interval in C1, or to an area bounded by a simple closed contour in C2; and, besides being perfect and connected, it would have the following properties: (1) There are points of Cn, which are not points of Hn; these form a complementary aggregate H′n. (2) There are points “within” Hn; this means that for any such point there is a neighbourhood consisting exclusively of points of Hn. (3) The points of Hn which do not lie “within” Hn are limiting points of H′n; they are not points of H′n, but the neighbourhood of any such point for any number h, however small, contains points within Hn and points of H′n: the aggregate of these points is called the “boundary” of Hn. (4) When any two points a, b within Hn are taken, it is possible to find a number ε and a corresponding number m, and to choose points x′, x″, ... x(m), so that the neighbourhood of a for ε contains x′, and consists exclusively of points within Hn, and similarly for x′ and x″, x″ and x″′, ... x(m) and b. Condition (3) would exclude such an aggregate as that of the points within and upon two circles external to each other and a line joining a point on one to a point on the other, and condition (4) would exclude such an aggregate as that of the points within and upon two circles which touch externally.

We take a positive fraction ε and consider the neighbourhood of a for h, and from this neighbourhood we exclude the point a, and we also exclude any point which is not in the domain of the argument. Then we take x and x′ to be any two of the retained points in the neighbourhood. The function ƒ has a limit at a if for any positive ε, however small, there is a corresponding h which has the property that |ƒ(x′) − ƒ(x)| < ε, whatever points x, x′ in the neighbourhood of a for h we take (a excluded). For example, when there are two variables x1, x2, and both are unrestricted, the domain of the argument is represented by a plane, and the values of the function are correlated with the points of the plane. The function has a limit at a point a, if we can mark out on the plane a region containing the point a within it, and such that the difference of the values of the function which correspond to any two points of the region (neither of the points being a) can be made as small as we please in absolute value by contracting all the linear dimensions of the region sufficiently. When the domain of the argument of a function of n variables extends to an infinite distance, there is a “limit at an infinite distance” if, after any number ε, however small, has been specified, a number N can be found which is such that |ƒ(x′) − ƒ(x)| < ε, for all points x and x′ (of the domain) of which one or more co-ordinates exceed N in absolute value. In the case of functions of several variables great importance attaches to limits for a restricted domain. The definition of such a limit is verbally the same as the corresponding definition in the case of functions of one variable (§ 6). For example, a function of x1 and x2 may have a limit at (x1 = 0, x2 = 0) if we first diminish x1 without limit, keeping x2 constant, and afterwards diminish x2 without limit. Expressed in geometrical language, this process amounts to approaching the origin along the axis of x2. The definitions of superior and inferior limits, and of maxima and minima, and the explanations of what is meant by saying that a function of several variables becomes infinite, or tends to become infinite, at a point, are almost identical verbally with the corresponding definitions and explanations in the case of a function of one variable (§ 7). The definition of a continuous function (§ 9) admits of immediate extension; but it is very important to observe that a function of two or more variables may be a continuous function of each of the variables, when the rest are kept constant, without being a continuous function of its argument. For example, a function of x and y may be defined by the conditions that when x = 0 it is zero whatever value y may have, and when x ≠ 0 it has the value of sin {4 tan−1 (y/x)}. When y has any particular value this function is a continuous function of x, and, when x has any particular value this function is a continuous function of y; but the function of x and y is discontinuous at (x = 0, y = 0).

The integral of a function ƒ(x) through a measurable domain H, which is a homogeneous part of the numerical continuum of n dimensions, is defined in just the same way as the integral through an interval, the extent of a square domain taking the place of the difference of the end-values of a partial interval; and the condition of integrability takes the same form as in the simple case. In particular, the condition is satisfied when the function is continuous throughout the domain. The definition of an integral through a domain may be adapted to any domain of measurable extent. The extensions to “improper” definite integrals may be made in the same way as for a function of one variable; in the particular case of a function which tends to become infinite at a point in the domain of integration, the point is enclosed in a partial domain which is omitted from the integration, and a limit is taken when the extent of the omitted partial domain is diminished indefinitely; a divergent integral may have different (principal) values for different modes of contracting the extent of the omitted partial domain. In applications to mathematical physics great importance attaches to convergent integrals and to principal values of divergent integrals. For example, any component of magnetic force at a point within a magnet, and the corresponding component of magnetic induction at the same point are expressed by different principal values of the same divergent integral. Delicate questions arise as to the possibility of representing the integral of a function of n variables through a domain Hn, 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.

The apparently more general method of representation of a function of one variable as the limit of a function of two variables has been shown by R. Baire to be identical in scope with the method of series, and it has been developed by him so as to give a very complete account of the possibility of representing functions by analytical expressions. For example, he has shown that Riemann’s totally discontinuous function, which is equal to 1 when x is rational and to 0 when x is irrational, can be represented by an analytical expression. An infinite process of a different kind has been adapted to the problem of the representation of a continuous function by T. Brodén. He begins with a function having a graph in the form of a regular polygon, and interpolates additional angular points in an ordered sequence without limit. The representation of a function by means of an infinite product falls clearly under Baire’s method, while the representation by means of a definite integral is analogous to Brodén’s method. As an example of these two latter processes we may cite the Gamma function [Γ(x)] defined for positive values of x by the definite integral

∫ ∞0 e−t tx−1 dt,

or by the infinite product

The second of these expressions avails for the representation of the function at all points at which x is not a negative integer.

As regards power series we have the following theorems:

1. If the power series converges at any point except a there is a number k which has the property that the series converges absolutely in the interval between a − k and a + k, with the possible exception of one or both end-points.

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

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

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

For example, the series sin x − ½ sin 2x + 1⁄3 sin 3x − ... is convergent for all real values of x, and, when π > x > −π its sum is ½x; but, when x is but a little less than π, the number of terms which must be taken in order to bring the sum at all near to the value of ½x is very large, and this number tends to increase indefinitely as x approaches π. This series does not converge uniformly in the neighbourhood of x = π. Another example is afforded by the series

of which the remainder after n terms is nx/(n²x² + 1). If we put x = 1/n, for any value of n, however great, the remainder is ½; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of x when x is near to zero—it must, in fact, be large compared with 1/x. The series does not converge uniformly in the neighbourhood of x = 0.

By a discussion of the integral (ii.) based on the Second Theorem of the Mean (§ 15) it can be shown that, if ƒ(x) has restricted oscillation in the interval (§ 11), the sum of the series is equal to ½{ƒ(x + 0) + ƒ(x − 0)} at any point x within the interval, and that it is equal to ½ {ƒ(+0) + ƒ(1 − 0} at each end of the interval. (See the article Fourier’s Series.) It therefore represents the function at any point of the periodic interval at which the function is continuous (except possibly the end-points), and has a definite value at each point of discontinuity. The condition of restricted oscillation includes all the functions contemplated in the statement of the theorem and some others. Further, it can be shown that, in any partial interval throughout which ƒ(x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier’s rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function ƒ(x) which tends to become infinite at a finite number of points a of the interval, provided (1) ƒ(x) tends to become determinately infinite at each of the points a, (2) the improper definite integral of ƒ(x) through the interval is convergent, (3) ƒ(x) has not an infinite number of discontinuities or of maxima or minima in the interval.

Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the series Σ ∞0 an cos (bn xπ), where a is positive and less than unity, and b is an odd integer exceeding (1 + 3⁄2π)/a. It can be shown that this series is uniformly convergent in every interval, and that the continuous function ƒ(x) represented by it has the property that there is, in the neighbourhood of any point x0, an infinite aggregate of points x′, having x0 as a limiting point, for which {ƒ(x′) − ƒ(x0)} / (x′ − x0) tends to become infinite with one sign when x′ − x0 approaches zero through positive values, and infinite with the opposite sign when x′ − x0 approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function ƒ(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F′(x) = ƒ(x); and, if ƒ(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Σanφn(x), where Σan is an absolutely convergent series of numbers, and φn(x) is an analytic function whose absolute value never exceeds unity.

In general, let ƒ0(x) + ƒ1(x) + ... be a series of functions which does not converge in a certain domain. It may happen that, if any number ε, however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value ƒ(a) of a certain function ƒ(x) is connected with the sum of the first n + 1 terms of the series by the relation |ƒ(a) − Σ nr = 0 ƒr(a)| < ε. It must 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′, |Σ mr = 0 ƒr(a)| > N. The divergent series ƒ0(x) + ƒ1(x) + ... is then an asymptotic expansion for the function f(x) in the domain.

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

n! = √(2π) ½nn + ½ e−n + θ/12n,

where θ is some number lying between 0 and 1. This formula is included in the asymptotic expansion for the Gamma function. We have in fact

log {Γ(x)} = (x − ½) log x − x + ½ log 2π + ω(x),

where ω(x) is the function defined by the definite integral

ω(x) = ∫ ∞0 {(1 − e−t)−1 − t−1 − ½} t−1 e−tx dt.

The multiplier of e−tx under the sign of integration can be expanded in the power series

where B1, B2, ... are “Bernoulli’s numbers” given by the formula

Bm = 2.2m! (2π)−2m Σ ∞r = 1 (r−2m).

When the series is integrated term by term, the right-hand member of the equation for ω(x) takes the form

This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value of ω(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. Poincaré. 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 É. 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.

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 Σ ∞0 ƒr(x) represents a function (ƒx) in an interval containing a point a, and that each of the functions ƒr(x) has a limit at a. If we first put x=a, and then sum the series, we have the value ƒ(a); if we first sum the series for any x, and afterwards take the limit of the sum at x = a, we have the limit of ƒ(x) at a; if we first replace each function ƒr(x) by its limit at a, and then sum the series, we may arrive at a value different from either of the foregoing. If the function ƒ(x) is continuous at a, the first and second results are equal; if the functions ƒr(x) are all continuous at a, the first and third results are equal; if the series is uniformly convergent, the second and third results are equal. This last case is an example of the interchange of the order of two limiting operations, and a sufficient, though not always a necessary, condition, for the validity of such an interchange will usually be found in some suitable extension of the notion of uniform convergence.

Authorities.—Among the more important treatises and memoirs connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905); O. Biermann, Analytische Functionen (Leipzig, 1887); É. Borel, Théorie des fonctions (Paris, 1898) (containing an introductory account of the Theory of Aggregates), and Séries divergentes (Paris, 1901), also Fonctions de variables réelles (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 Grösse (Leipzig, 1892), and Serie di Fourier (Pisa, 1880); A. Genocchi u. G. Peano, Diff.- u. Int.-Rechnung (Leipzig, 1899); J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898); A. Harnack, Diff. 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, Leçons sur l’intégration (Paris, 1904); M. Pasch, Diff.- u. Int.-Rechnung (Leipzig, 1882); E. Picard, Traité d’analyse (Paris, 1891); O. Stolz, Allgemeine Arithmetik (Leipzig, 1885), and Diff.- u. Int.-Rechnung (Leipzig, 1893-1899); J. Tannery, Théorie des fonctions (Paris, 1886); W.H. and G.C. Young, The Theory of Sets of Points (Cambridge, 1906); Brodén, “Stetige Functionen e. reellen Veränderlichen,” Crelle, Bd. cxviii.; G. Cantor, A series of memoirs on the “Theory of Aggregates” and on “Trigonometric series” in Acta Math. tt. ii., vii., and Math. Ann. Bde. iv.-xxiii.; Darboux, “Fonctions discontinues,” Ann. Sci. École normale sup. (2), t. iv.; Dedekind, Was sind u. was sollen d. Zahlen? (Brunswick, 1887), and Stetigkeit u. irrationale Zahlen (Brunswick, 1872); Dirichlet, “Convergence des séries trigonométriques,” Crelle, Bd. iv.; P. Du Bois Reymond, Allgemeine Functionentheorie (Tübingen, 1882), and many memoirs in Crelle and in Math. Ann.; Heine, “Functionenlehre,” Crelle, Bd. lxxiv.; J. Pierpont, The Theory of Functions of a real Variable (Boston, 1905); F. Klein, “Allgemeine Functionsbegriff,” Math. Ann. Bd. xxii.; W.F. Osgood, “On Uniform Convergence,” Amer. J. of Math. vol. xix.; Pincherle, “Funzioni analitiche secondo Weierstrass,” Giorn. di mat. t. xviii.; Pringsheim, “Bedingungen d. Taylorschen Lehrsatzes,” Math. Ann. Bd. xliv.; Riemann, “Trigonometrische Reihe,” Ges. Werke (Leipzig, 1876); Schoenflies, “Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten,” Jahresber. d. deutschen Math.-Vereinigung, Bd. viii.; Study, Memoir on “Functions with Restricted Oscillation,” Math. Ann. Bd. xlvii.; Weierstrass, Memoir on “Continuous Functions that are not Differentiable,” Ges. math. Werke, Bd. ii. p. 71 (Berlin, 1895), and on the “Representation of Arbitrary Functions,” ibid. Bd. iii. p. 1; W.H. Young, “On Uniform and Non-uniform Convergence,” Proc. London Math. Soc. (Ser. 2) t. 6. Further information and very full references will be found in the articles by Pringsheim, Schoenflies and Voss in the Encyclopädie der math. Wissenschaften, Bde. i., ii. (Leipzig, 1898, 1899).

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

Moreover, ζ being a rational function of z, both ξ and η are continuous differentiable functions of x and y, save when ζ is infinite; writing ζ = ƒ(x, y) = ƒ(z − iy, y), the fact that this is really independent of y leads at once to ∂f/∂x + i∂ƒ/∂y = 0, and hence to

so that ξ is not any arbitrary function of x, y, and when ξ is known η is determinate save for an additive constant. Also, in virtue of these equations, if ζ, ζ′ be the values of ζ corresponding to two near values of z, say z and z′, the ratio (ζ′ − ζ)/(z′ − z) has a definite limit when z′ = z, independent of the ultimate phase of z′ − z, this limit being therefore equal to ∂ζ/∂x, that is, ∂ξ/∂x + i∂η)/∂x. Geometrically this fact is interpreted by saying that if two curves in the z-plane intersect at a point P, at which both the differential coefficients ∂ξ/∂x, ∂η/∂x are not zero, and P′, P″ be two points near to P on these curves respectively, and the corresponding points of the ζ-plane be Q, Q′, Q″, then (1) 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 ξ, η in the one plane are related as are the axes of x, y. Thus any diagram of the z-plane becomes a diagram of the ζ-plane with the same angles; the magnification, however, which is equal to [(∂ξ/∂x)² + (∂ξ/∂y)² ]1/2 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 ζ is a polynomial in z,

ζ = p0zn + p1zn−1 + ... + pn;

H being an arbitrary real positive number, it can be shown that a radius R can be found such for every |z| > R we have |ζ| > H; consider the lower limit of |ζ| for |z| < R; as ξ² + η² is a real continuous function of x, y for |z| < R, there is a point (x, y), say (x0, y0), at which |ζ| is least, say equal to ρ, and therefore within a circle in the ζ-plane whose centre is the origin, of radius ρ, there are no points ζ representing values corresponding to |z| < R. But if ζ0 be the value of ζ corresponding to (x0, y0), and the expression of ζ − ζ0 near z0 = x0 + iy0, in terms of z − z0, be A(z − z0)m + B(z − z0)m + 1 + ..., where A is not zero, to two points near to (x0, y0), say (x1, y1) or z1 and z2 = z0 + (z1 − z0) (cos π/m + i sin π/m), will correspond two points near to ζ0, say ζ1, and 2ζ0 − ζ′1, situated so that ζ0 is between them. One of these must be within the circle (ρ). We infer then that ρ = 0, 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 − α, where α denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

The infinite series of most importance are those of which the general term is anzn, wherein an is a constant, and z is regarded as variable, n = 0, 1, 2, 3, ... Such a series is called a power series, if a real and positive number M exists such that for z = z0 and every n, |anz0n| < M, a condition which is satisfied, for instance, if the series converges for z = z0, then it is at once proved that the series converges absolutely for every z for which |z| < |z0|, and converges uniformly over every range |z| < r′ for which r′ < |z0|. 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 z0 on the circumference, as z approaches to z0 provided the series converges for z = z0, as can be shown without much difficulty. Within a common circle of convergence two power series Σ anzn, Σ bnzn can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If r1 be less than the radius of convergence of a series Σ an zn and for |z| = r1, the sum of the series be in absolute value less than a real positive quantity M, it can be shown that for |z| = r1 every term is also less than M in absolute value, namely, |an| < Mr1−n. If in every arbitrarily small neighbourhood of z=0 there be a point for which two converging power series Σan zn, Σbnzn agree in value, then the series are identical, or an = bn; thus also if Σanzn vanish at z = 0 there is a circle of finite radius about z = 0 as centre within which no other points are found for which the sum of the series is zero. Considering a power series ƒ(z) = Σanzn of radius of convergence R, if |z0| < R and we put z = z0 + t with |t| < R-|z0|, the resulting series Σan(z0 + t)n may be regarded as a double series in z0 and t, which, since |z0| + t < R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write ƒ(z) = Σ Antn; hence A0 = ƒ(z0), and we have [ƒ(z0 + t) − ƒ(z0)]/t = Σn=1 Antn−l, wherein the continuous series on the right reduces to A1 for t = 0; thus the ratio on the left has a definite limit when t = 0, equal namely to A1 or Σnanz0n − 1. In other words, the original series may legitimately be differentiated at any interior point z0 of its circle of convergence. Repeating this process we find ƒ(z0 + t) = Σtnƒ(n)(z0)/n!, where ƒ(n)(z0) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about z = 0 for Σanzn, we infer that for the series ƒ(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 ƒ(z).

Perhaps the simplest possible power series is ez = exp(z) = 1 + z2/2! + z3/3! + ... of which the radius of convergence is infinite. By multiplication we have exp(z)·exp(z1) = exp(z + z1). In particular when x, y are real, and z = x + iy, exp(z) = exp(x)exp(iy). Now the functions

U0 = sin y, V0 = 1 − cos y, U1 = y − sin y,

V1 = 1⁄2y² − 1 + cos y, U2 = 1⁄6y³ − y + sin y, V2 = 1⁄24y4 − 1⁄2y2 + 1 − cos y, ...

all vanish for y = 0, and the differential coefficient of any one after the first is the preceding one; as a function (of a real variable) is increasing when its differential coefficient is positive, we infer, for y positive, that each of these functions is positive; proceeding to a limit we hence infer that

cos y = 1 − 1⁄2y² + 1⁄24y4 − ...,   sin y = y − 1⁄6y³ + 1⁄120y5 − ...,

for positive, and hence, for all values of y. We thus have exp(iy) = cos y + i sin y, and exp (z) = exp (x)·(cos y + i sin y). In other words, the modulus of exp (z) is exp (x) and the phase is y. Hence also

exp(z + 2πi) = exp(x) [cos (y + 2π) + i sin(y + 2π)],

which we express by saying that exp (z) has the period 2πi, and hence also the period 2kπi, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x = 0, we at once prove that exp (z) has no other periods.

Taking in the plane of z an infinite strip lying between the lines y = 0, y = 2π and plotting the function ζ = exp (z) upon a new plane, it follows at once from what has been said that every complex value of ζ arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation ζ = exp(z) thus defines z, regarded as depending upon ζ, with only an additive ambiguity 2kπi, where k is an integer. We write z = λ(ζ); when ζ is real this becomes the logarithm of ζ; in general λ(ζ) = log |ζ| + i ph (ζ) + 2kπi, where k is an integer; and when ζ describes a closed circuit surrounding the origin the phase of ζ increases by 2π, or k increases by unity. Differentiating the series for ζ we have dζ/dz = ζ, so that z, regarded as depending upon ζ, is also differentiable, with dz/dζ = ζ− 1. On the other hand, consider the series ζ − 1 − ½(ζ− 1)2 + 1⁄3(ζ − 1)3 − ...; it converges when ζ = 2 and hence converges for |ζ − 1| < 1; its differential coefficient is, however, 1 − (ζ − 1) + (ζ − 1)2 − ..., that is, (1 + ζ − 1)− 1. Wherefore if φ(ζ) denote this series, for |ζ − 1| < 1, the difference λ(ζ) − φ(ζ), regarded as a function of ξ and η, has vanishing differential coefficients; if we take the value of λ(ζ) which vanishes when ζ = 1 we infer thence that for |ζ − 1| < 1, λ(ζ) = Σn=1 [(−1)(n−1)/n (ζ − 1)n. It is to be remarked that it is impossible for ζ while subject to |ζ − 1| < 1 to make a circuit about the origin. For values of ζ for which |ζ − 1| ≮ 1, we can also calculate λ(ζ) with the help of infinite series, utilizing the fact that λ(ζζ′) = λ(ζ) + λ(ζ′).

The function λ(ζ) is required to define ζa when ζ and a are complex numbers; this is defined as exp [aλ(ζ)], that is as Σn=0 an [λ(ζ)]n/n!. When a is a real integer the ambiguity of λ(ζ) is immaterial here, since exp [aλ(ζ) + 2kaπi] = exp [aλ(ζ)]; when a is of the form 1/q, where q is a positive integer, there are q values possible for ζ1/q, of the form exp [1/q λ(ζ)] exp (2kπi/q), with k = 0, 1, ... q − 1, all other values of k leading to one of these; the qth power of any one of these values is ζ; when a = p/q, where p, q are integers without common factor, q being positive, we have ζp/q = (ζ1/q)p. The definition of the symbol ζa 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 ii; the number i is of modulus unity and phase ½π; thus λ(i) = i (½π + 2kπ); thus

ii = exp (−½π − 2kπ) = exp (−½π) exp (−2kπ),

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 = ½[exp (iz) + exp (−iz)] and sin z = −½i [exp (iz) − exp(−iz)]. It will be found that these obey the ordinary relations holding when z is real, except that their moduli are not inferior to unity. For example, cos i = 1 + 1/2! + 1/4! + ... is obviously greater than unity.

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

There is a logical principle also which must be referred to. We frequently have cases where, about every interior or boundary, point z0 of a certain region a circle can be put, say of radius r0, such that for all points z of the region which are interior to this circle, for which, that is, |z − z0| < r0, a certain property holds. Assuming that to r0 is given the value which is the upper limit for z0, of the possible values, we may call the points |z − z0| < r0, the neighbourhood belonging to or proper to z0, and may speak of the property as the property (z, z0). The value of r0 will in general vary with z0; what is in most cases of importance is the question whether the lower limit of r0 for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (z, z0) is of a kind which we may call extensive; such, namely, that if it holds, for some position of z0 and all positions of z, within a certain region, then the property (z, z1) holds within a circle of radius R about any interior point z1 of this region for all points z for which the circle |z − z1| = R is within the region. Also in this case r0 varies continuously with z0. (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, (1) for some point z0 within or upon the boundary of the sub-region, (2) for every point z within or upon the boundary of the sub-region.

We prove these statements (A), (B) in reverse order. To prove (B) let a region for which the property (z, z0) holds for all points z and some point z0 of the region, be called suitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say ζ; after a certain stage all these will be interior to the proper region of ζ; 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 ƒ(x, y). It may have a finite upper limit H for the region, so that no point (x, y) exists for which ƒ(x, y) > H, but points (x, y) exist for which ƒ(x, y) > H − ε, however small ε may be; if not we say that its upper limit is infinite. There is then at least one point of the region such that, for points of the region within a circle about this point, the upper limit of ƒ(x, y) is H, however small the radius of the circle be taken; for if not we can put about every point of the region a circle within which the upper limit of ƒ(x, y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function ƒ(x, y) is the radius r0 of the neighbourhood proper to any point z0, spoken of above. We can hence prove the statement (A) above.

Suppose the property (z, z0) extensive, and, if possible, that the lower limit of r0 is zero. Let then ζ be a point such that the lower limit of r0 is zero for points z0 within a circle about ζ however small; let r be the radius of the neighbourhood proper to ζ; take z0 so that |z0-ζ| < ½r; the property (z, z0), being extensive, holds within a circle, centre z0, of radius r − |z0 − ζ|, which is greater than |z0 − ζ|, and increases to r as |z0 − ζ| diminishes; this being true for all points z0 near ζ, the lower limit of r0 is not zero for the neighbourhood of ζ, contrary to what was supposed. This proves (A). Also, as is here shown that r0 ⋝ r − |z0 − ζ|, may similarly be shown that r ⋝ r0 − |z0 − ζ|. Thus r0 differs arbitrarily little from r when |z0 − ζ| is sufficiently small; that is, r0 varies continuously with z0. Next suppose the function ƒ(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, η 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 |ƒ(x, y) −ƒ(x0, y0)| < ½η, and therefore (x′, y′) being any other point interior to this circle, |ƒ(x′, y′) − ƒ(x, y)| < η. We can then apply the result (A) obtained above, taking for the neighbourhood proper to any point z0 the circular area within which, for any two points (x, y), (x′, y′), we have |ƒ(x′, x′) − ƒ(x, y)| < η. This is clearly an extensive property. Thus, a number r is assignable, greater than zero, such that, for any two points (x, y), (x′, y′) within a circle |z − z0| = r about any point z0, we have |ƒ(x′, y′) − ƒ(x, y)| < η, and, in particular, |ƒ(x, y) −ƒ(x0, y0)| < η, where η is an arbitrary real positive quantity agreed upon beforehand.

Take now any path in the region, whose extreme points are z0, z, and let z1, ... zn−1 be intermediate points of the path, in order; denote the continuous function ƒ(x, y) by ƒ(z), and let ƒr denote any quantity such that |ƒr − ƒ(zr)| ⋜ |ƒ(zr+1) − ƒ(zr)|; consider the sum

(z1 − z0)ƒ0 + (z2 − z1)ƒ1 + ... + (z − zn−1)ƒn−1.

By the definition of a path we can suppose, n being large enough, that the intermediate points z1, ... zn − 1 are so taken that if zi, zi + 1 be any two points intermediate, in order, to zr and zr + 1, we have |zi + i-zi| < |zr+1 − zr|; we can thus suppose |z1 − z0|, |z2 − z1|, ... |z − zn−1|all 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, 1, zr, 2, ... zr, m−1 be intermediate in order to zr and zr+1, and |ƒr, i − ƒ(zr, i)| < |ƒ(zr, i+1) − ƒ(zr, i)|, the difference between Σ(zr+1 − zr)ƒr and

Σ { (zr, 1-zr)ƒr, 0 + (zr, 2 − zr, 1)ƒr, 1 + ... + (zr+1 − zr, m−1)ƒr, m−1 },

which is equal to

Σr Σi (zr, i+1 − zr, i) (ƒr, i − ƒr),

is, when |zr+1 − zr| is small enough, to ensure |ƒ(zr+1) − ƒ(zr)| < η, less in absolute value than

Σ2η Σ |zr, i+1 − zr, i|,

which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 2ηS, and is therefore arbitrarily small.

The limit in question is called ∫ zz0 ƒ(z)dz. In particular when ƒ(z) = 1, it is obvious from the definition that its value is z − z0; when ƒ(z) = z, by taking ƒr = ½(zr+1 − zr), it is equally clear that its value is ½(z² − z0²); these results will be applied immediately.

Suppose now that to every interior and boundary point z0 of a certain region there belong two definite finite numbers ƒ(z0), F(z0), such that, whatever real positive quantity η may be, a real positive number ε exists for which the condition

which we describe as the condition (z, z0), is satisfied for every point z, within or upon the boundary of the region, satisfying the limitation |z − z0| < ε. Then ƒ(z0) is called a differentiable function of the complex variable z0 over this region, its differential coefficient being F(z0). The function ƒ(z0) is thus a continuous function of the real variables x0, y0, where z0 = x0 + iy0, over the region; it will appear that F(z0) is also continuous and in fact also a differentiable function of z0.

Supposing η to be retained the same for all points z0 of the region, and σ0 to be the upper limit of the possible values of ε for the point z0, it is to be presumed that σ0 will vary with z0, and it is not obvious as yet that the lower limit of the values of σ0 as z0 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, z0), above, is satisfied for all points z, within or upon the boundary of this sub-region, for an appropriate position of z0, 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 ƒ(z), the integral ∫ zz1 ƒ(z)dz has the same value by whatever path within the region we pass from z1 to z. This we prove by showing that when taken round a closed path in the region the integral ∫ƒ(z)dz vanishes. Consider first a triangle over which the condition (z, z0) holds, for some position of z0 and every position of z, within or upon the boundary of the triangle. Then as

ƒ(z) = ƒ(z0) + (z − z0) F(z0) + ηθ(z − z0), where |θ| < 1,

we have

∫ƒ(z)dz = [ƒ(z0) − z0 F(z0)] ∫dz + F(z0) ∫zdz + η∫θ(z − z0)dz,

which, as the path is closed, is η ∫θ(z − z0)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 ηap, where a is the greatest side of the triangle and p is its perimeter; if Δ be the area of the triangle, we have Δ = ½ab sin C > (α/π) ba, where α is the least angle of the triangle, and hence a(a + b + c) < 2a(b + c) < 4πΔ/α; the integral ∫ƒ(z)dz round the perimeter of the triangle is thus < 4πηΔ/α. Now consider any region made up of triangles, as before explained, in each of which the condition (z, z0) holds, as in the triangle just taken. The integral ∫ƒ(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 4πηK/α, where K is the whole area of the region, and α is the smallest angle of the component triangles. However small η 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: (1) 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 ƒ(z) at any interior point of a region is expressible in terms of the value of ƒ(z) at the boundary points. For consider in the original region the function ƒ(z)/(z − z0), where z0 is an interior point: this satisfies the same conditions as ƒ(z) except in the immediate neighbourhood of z0. Taking out then from the original region a small regular polygonal region with z0 as centre, the theorem holds for the remaining portion. Proceeding to the limit when the polygon becomes a circle, it appears that the integral ∫ dzƒ(z)/(z − z0) round the boundary of the original region is equal to the same integral taken counter-clockwise round a small circle having z0 as centre; on this circle, however, if z − z0 = rE(iθ), dz/(z − z0) = idθ, and ƒ(z) differs arbitrarily little from f(z0) if r is sufficiently small; the value of the integral round this circle is therefore, ultimately, when r vanishes, equal to 2πiƒ(z0). Hence ƒ(z0) = 1/2πi ∫ (dtƒ(t)/(t − z0), where this integral is round the boundary of the original region. From this it appears that

also round the boundary of the original region. This form shows, however, that F(z0) is a continuous, finite, differentiable function of z0 over the whole interior of the original region.

(1) 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(2) = u0(z) + u1(z) + ... + un−1(z) + Rn(z), where |Rn(z)| < ε for all points of the path, we have

wherein, in absolute value, ∫ zz0 Rn(z)dz < εL, if L be the length of the path. Thus the series may be integrated, and the resulting series is also uniformly convergent.

(2) If ƒ(x, y) be definite, finite and continuous at every point of a region, and over any closed path in the region ∫ƒ(x, y)dz = 0, then ψ(z) = ∫ zz0 ƒ(x, y)dz, for interior points z0, z, is a differentiable function of z, having for its differential coefficient the function ƒ(x, y), which is therefore also a differentiable function of z at interior points.

(3) Hence if the series u0(z) + u1(z) + ... to ∞ 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 by (1/2πi) ∫ 2πi/(t − z)², is obtainable by differentiating the series. This theorem, unlike (1), does not hold for functions of a real variable.

(4) If the region of definition of a differentiable function ƒ(z) include the region bounded by two concentric circles of radii r, R, with centre at the origin, and z0 be an interior point of this region,

where the integrals are both counter-clockwise round the two circumferences respectively; putting in the first (t − z0)−1 = Σn=0 z0n/tn+1, and in the second (t − z0)−1 = − Σn=0 tn/z0n+1, we find ƒ(z0) = Σ ∞−∞ Anz0n, wherein An = (1/2πi) ∫ [ƒ(t)/tn+1] dt, taken round any circle, centre the origin, of radius intermediate between r and R. Particular cases are: (α) when the region of definition of the function includes the whole interior of the outer circle; then we may take r = 0, the coefficients An for which n < 0 all vanish, and the function ƒ(z0) is expressed for the whole interior |z0| < R by a power series Σ ∞0 Anz0n. In other words, about every interior point c of the region of definition a differentiable function of z is expressible by a power series in z − c; a very important result.

(β) 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 zmƒ(z) has a finite limit when |z| diminishes to zero, all the coefficients An for which n < −m vanish, and we have

f(z0) = A−mz0−m + A−m+1z0−m+1 + ... + A−1z0−1 + A0 + A1z0 ... to ∞.

Such a case occurs, for instance, when ƒ(z) = cosec z, the number m being unity.

About every interior point z0 of the region of existence the function may be represented by a power series in z − z0, and the series converges and represents the function over any circle centre at z0 which contains no singular point in its interior. This has been proved above. And it can be similarly proved, putting z = 1/ζ, that if the region of existence of the function contains all points of the plane for which |z| > R, then the function is representable for all such points by a power series in z− 1 or ζ; in such case we say that the region of existence of the function contains the point z = ∞. A series in z− 1 has a finite limit when |z| = ∞; a series in z cannot remain finite for all points z for which |z| > R; for if, for |z| = R, the sum of a power series Σanzn in z is in absolute value less than M, we have |an| < Mr−n, and therefore, if M remains finite for all values of r however great, an = 0. Thus the region of existence of a function if it contains all finite points of the plane cannot contain the point z = ∞; such is, for instance, the case of the function exp (z) = Σzn/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 |z| < 1 by the series Σ n=0 zm, where m = n², the series Σ n=0zm where m = n!, and the series Σ n=1 zm/(m + 1)(m + 2) where m = an, a being a positive integer, although in the last case the series actually converges for every point of the circle of convergence |z| = 1. 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 − z0; as z0 approaches to a singular point of the function, lying on the circle of convergence, the radii of convergence of these derived series in z − z0 diminish to zero; when, however, a circle can be put about z0, 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 − z0 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 − z0 converging beyond the original circle gives what is known as an analytical 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.

Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − z0, about any point z0 interior to its circle of convergence, and the new series converges certainly for |z − z0| < r − |z0|, if r be the original radius of convergence. If for every position of z0 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 z0 the derived series converges for |z − z0| < r − |z0| + D, then it can be shown that for points z, interior to the original circle, lying in the annulus r − |z0| < |z − z0| < r − |z0| + D, the value represented by the derived series agrees with that represented by the original series. If for another point z1 interior to the original circle the derived series converges for |z − z1| < r − |z1| + E, and the two circles |z − z0| = r − |z0| + D, |z − z1| = r − |z1| + E have interior points common, lying beyond |z| = 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 a monogenic function, the epithet having reference to the single origin, by one power series, of the expressions representing the function; it is also sometimes called a monogenic analytical function, or simply an analytical 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, that ∫ƒ(z)dz = 0 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 his Théorie 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.

The integral ∫ƒ(z)dz taken by a closed circuit about the pole not containing any other singularity is at once seen to be 2πiA1, where A1 is the coefficient of (z − z0)−1 in the expansion of ƒ(z) at the pole; this coefficient has therefore a certain uniqueness, and it is called the residue of ƒ(z) at the pole. Considering a region in which there are no other singularities than poles, all these being interior points, the integral (1/2πi) ∫ ƒ(z)dz round 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 an essential 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 Σ ∞−∞ an(z − z0)n; it thus cannot remain finite in the immediate neighbourhood of the point. The point is necessarily an isolated essential singularity also of the function {ƒ(z) − A}−1 for if this were expressible by a power series about the point, so would also the function ƒ(z) be; as {ƒ(z) − A}− 1 approaches infinity, so does ƒ(z) approach the arbitrary value A. Similar remarks apply to the point z = ∞, the function being regarded as a function of ζ = 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.

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 be a1, a2, a3, ... with |a1| ⋜ |a2| ⋜ |a3| ... and limit |an| = ∞. About as the function is expressible as Σ ∞−∞ An(z − as)n; let ƒs(z) = Σ 1−∞ An(z − as)n be the sum of the negative powers in this expansion. Assuming z = 0 not to be a singular point, let ƒs(z) be expanded in powers of z, in the form Σ n=0 Cnzn, and μs be chosen so that Fs(z) = ƒs(z) − Σ μs−11 Cnzn = Σ ∞μs Cnzn is, for |z| < rs < |as|, less in absolute value than the general term εs of a fore-agreed convergent series of real positive terms. Then the series φ(z) = Σ ∞s=1 Fs(z) converges uniformly in any finite region of the plane, other than at the points as, and is expressible about any point by a power series, and near as, φ(z) − fs(z) is expressible by a power series in z − as. Thus F(z) − φ(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 |Fs(z)| < εs is imposed only to render the series ΣFs(z) uniformly convergent; this condition may in particular cases be satisfied by a series Σ Gs(z) where Gs(z) = ƒs(z) − Σ νs−11 Cnzn and νs < μs. An example of the theorem is the function π cot πz − z− 1 for which, taking at first only half the poles, ƒs(z) = 1/(z − s); in this case the series Σ Fs(z) where Fs(z) = (z − s)−1 + s−1 is uniformly convergent; thus π cot πz − z−1 − Σ ∞−∞ [(z − s)−1 + s−1], where s = 0 is excluded from the summation, is an integral function. It can be proved that this integral function vanishes.

Considering an integral function ƒ(z), if there be no finite positions of z for which this function vanishes, the function λ[ƒ(z)] is at once seen to be an integral function, φ(z), or ƒ(z) = exp[φ(z)]; if however great R may be there be only a finite number of values of z for which ƒ(z) vanishes, say z = a1, ... am, then it is at once seen that ƒ(z) = exp [φ(z)]. (z − a1)h1...(z − am)hm, where φ(z) is an integral function, and h1, ... hm are positive integers. If, however, ƒ(z) vanish for z = a1, a2 ... where |a1| ⋜ |a2| ⋜ ... and limit |an| = ∞, and if for simplicity we assume that z − 0 is not a zero and all the zeros a1, a2, ... are of the first order, we find, by applying the preceding theorem to the function [1 / ƒ(z)] [dƒ(z) / dz], that ƒ(z) = exp [φ(z)] Π ∞n=1 {(1 − z/an) exp φn(z)}, where φ(z) is an integral function, and φn(z) is an integral polynomial of the form φn(z) = z/an + z2/2an2 + ... + zs/sans. The number s may be the same for all values of n, or it may increase indefinitely with n; it is sufficient in any case to take s = n. In particular for the function sinπx/πx, we have

where n = 0 is excluded from the product. Or again we have

where C is a constant, and Γ(x) is a function expressible when x is real and positive by the integral ∫ ∞0 e−t tx−1dt.

There exist interesting investigations as to the connexion of the value of s above, the law of increase of the modulus of the integral function ƒ(z), and the law of increase of the coefficients in the series ƒ(z) = Σ anzn as n increases (see the bibliography below under Integral 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).

I. If a1, a2, a3, ... 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 points a1, a2, ..., and with every point ai there be associated a polynomial in (z − ai)−1, say gi; then there exists a single valued function whose region of existence excludes only the points (a) and the points (c), having in a point ai a pole whereat the expansion consists of the terms gi, together with a power series in z − ai; the function is expressible as an infinite series of terms gi − γi, where γi is also a rational function.

II. With a similar aggregate (a), with limiting points (c), suppose with every point ai there is associated a positive integer ri. Then there exists a single valued function whose region of existence excludes only the points (c), vanishing to order ri at the point ai, but not elsewhere, expressible in the form

where with every point an is associated a proper point cn of (c), and

μn 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 = R(1 − n−1) exp (iπ √2·n), when (c) consists of the points of the circle |z| = R, the product expression above denotes different monogenic functions in the different regions, not continuable into one another.

Now any monogenic function ƒ(t) whose region of definition includes C and the interior of R can be represented at all points z in R0 by

where the path of integration is C. This integral is the limit of a sum

where the points ti are upon C; and the proof we have given of the existence of the limit shows that the sum S converges to ƒ(z) uniformly in regard to z, when z is in R0, so that we can suppose, when the subdivision of C into intervals ti+1 − ti, has been carried sufficiently far, that

|S − ƒ(z)| < ε,

for all points z of R0, where ε is arbitrary and agreed upon beforehand. The function S is, however, a rational function of z with poles upon C, that is external to R0. We can thus find a rational function differing arbitrarily little from S, and therefore arbitrarily little from ƒ(z), for all points z of R0, with poles at arbitrary positions outside R0 which can be reached by finite continuous curves lying outside R from the points of C.

In particular, to take the simplest case, if C0, C be simple closed polygons, and Γ 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 ƒ(z) for all points of R0 whose poles are at one finite point c external to Γ. By a transformation of the form t − c = r−1, with the appropriate change in the rational function, we can suppose this point c to be at infinity, in which case the rational function becomes a polynomial. Suppose ε1, ε2, ... to be an indefinitely continued sequence of real positive numbers, converging to zero, and Pr to be the polynomial such that, within C0, |Pr − ƒ(z)| < εr; then the infinite series of polynomials

P1(z) + {P2(z) − P1(z)} + {P3(z) − P2(z)} + ...,

whose sum to n terms is Pn(z), converges for all finite values of z and represents ƒ(z) within C0.

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 Γ of a region, we can suppose the poles of the rational function, constructed to approximate to ƒ(z) within R0, to be at points of Γ. A series of rational functions of the form

H1(z) + {H2(z) − H1(z)} + {H3(z) − H2(z)} + ...

then, as before, represents ƒ(z) within R0. And R0 may be taken to coincide as nearly as desired with the interior of the region bounded by Γ.

Let c be an arbitrary real positive quantity; putting the complex variable ζ = ξ + iη, enclose the points ζ = l, ζ = 1 + c by means of (i.) the straight lines η = ±a, from ξ = l to ξ = 1 + c, (ii.) a semicircle convex to ζ = 0 of equation (ξ − 1)2 + η2 = a2, (iii.) a semicircle concave to ζ = 0 of equation (ξ − 1 − c)2 + η2 = a2. The quantities c and a are to remain fixed. Take a positive integer r so that 1/r (c/a) is less than unity, and put σ = 1/r (c/a). Now take

c1 = 1 + c/r, c2 = 1 + 2c/r, ... cr = 1 + c;

if n1, n2, ... nr, be positive integers, the rational function

is finite at ζ = 1, and has a pole of order n1 at ζ = c1; the rational function

is thus finite except for ζ = c2, where it has a pole of order n1n2; finally, writing

the rational function

has a pole only at ζ = 1 + c, of order n1n2 ... nr.

The difference (1 − ζ)−1 − U is of the form (1 − ζ)−1P, where P, of the form

1 − (1 − ρ1) (1 − ρ2)...(1 − ρk),

in which there are equalities among ρ1, ρ2, ... ρk, is of the form

Σρ1 − Σρ1ρ2 + Σρ1ρ2ρ3 − ...;

therefore, if |ri| = |ρi|, we have

now, so long as ζ is without the closed curve above described round ζ = 1, ζ = 1 + c, we have

and hence

Take an arbitrary real positive ε, and μ, a positive number, so that εmu − 1 < εa, then a value of n1 such that σn1 < μ/(1 + μ) and therefore σn1/(1 − σn1 < μ, and values for n2, n3 ... such that σn2 < 1/n1 σ2n1, σn3 < 1/n1n2 σ3n1, ... σnr < 1/(n1 ... nr−1) σnr n1; then, as 1 + x < ex, we have

and therefore less than

a−1 {exp (σn1 + σ2n1 + ... + σnr n1) − 1},

which is less than

and therefore less than ε.

The rational function U, with a pole at ζ = 1 + c, differs therefore from (1 − ζ)−1, for all points outside the closed region put about ζ = 1, ζ = l + c, by a quantity numerically less than ε. So long as a remains the same, r and σ will remain the same, and a less value of ε will require at most an increase of the numbers n1, n2, ... nr; but if a be taken smaller it may be necessary to increase r, and with this the complexity of the function U.

Now put

thereby the points ζ = 0, 1, 1 + c become the points z = 0, 1, ∞, the function (1 − z)−1 being given by (1 − z)−1 = c(c + 1)−1 (1 − ζ)−1 + (c + 1)−1; the function U becomes a rational function of z with a pole only at z = ∞, that is, it becomes a polynomial in z, say [(c + 1)/c] H − 1/c, where H is also a polynomial in z, and

the lines η = ±a become the two circles expressed, if z = x + iy, by

the points (η = 0, ξ = 1 − a), (η = 0, ξ = 1 + c + a) become respectively the points (y = 0, x = c(1 − a)/(c + a), (y = 0, x = −c(l + c + a)/a), whose limiting positions for a = 0 are respectively (y = 0, x = 1), (y = 0, x = −∞). The circle (x + c)² + y² = c(c + 1)y/a can be written

where μ = ½c(c + 1)/a; its ordinate y, 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 1 ⋜ x ⋜ ∞, we can take a quantity a, and hence, with an arbitrary c, determine a number r; then corresponding to an arbitrary εs, we can determine a polynomial Ps, such that, for all points interior to the region, we have

|(1 − z−1) − Ps| < εs;

thus the series of polynomials

P1 + (P2 − P1) + (P3 − P2) + ...,

constructed with an arbitrary aggregate of real positive numbers ε1, ε2, ε3, ... with zero as their limit, converges uniformly and represents (1 − z)−1 for the whole region considered.

§ 12. Expansion of a Monogenic Function in Polynomials, over a Star Region.—Now consider any monogenic function ƒ(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 point x in this region R we can then write

where ƒ(x) represents a monogenic branch of the function, in case it be not everywhere single valued, and t is on the boundary of the region. Describe now another region R0 lying entirely within R, and let x be restricted to be within R0 or upon its boundary; then for any point t on the boundary of R, the points z of the plane for which zt− 1 is real and positive and equal to or greater than 1, being points for which |z| = |t| or |z| > |t|, are without the region R0, and not infinitely near to its boundary points. Taking then an arbitrary real positive ε we can determine a polynomial in xt− 1, say P(xt−1), such that for all points x in R0 we have

|(1 − xt−1)−1 − P(xt−1)| < ε;

the form of this polynomial may be taken the same for all points t on the boundary of R, and hence, if E be a proper variable quantity of modulus not greater than ε,

where L is the length of the path of integration, the boundary of R, and M is a real positive quantity such that upon this boundary |t−1 ƒ(t)| < M. If now

P (xt−1) = c0 + c1xt−1 + ... + cm xm t−m,

and

this gives

|ƒ(x) − {c0μ0 + c1μ1x + ... + cmμmxm}| ⋜ εLM/2π,

where the quantities μ0, μ1, μ2, ... are the coefficients in the expansion of ƒ(x) about the origin.

If then an arbitrary finite region be constructed of the kind explained, excluding the barriers joining the singular points of ƒ(x) to x = ∞, it is possible, corresponding to an arbitrary real positive number σ, to determine a number m, and a polynomial Q(x), of order m, such that for all interior points of this region

|ƒ(x) − Q(x)| < σ.

Hence as before, within this region ƒ(x) can be represented by a series of polynomials, converging uniformly; when ƒ(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 of t, φ(t) be capable of expression as a power series in t − x about a point x, for |t − x| ⋜ ρ, and for all points of this circle |φ(t)| < g, we know that |φ(n)(x)| < gρ−n(n!). Hence, taking |z| < 1⁄3ρ, and, for any assigned positive integer μ, taking m so that for n > m we have (μ + n)μ < (3⁄2)n, we have

and therefore

where

Now draw barriers as before, directed from the origin, joining the singular point of φ(z) to z = ∞, take a finite region excluding all these barriers, let ρ be a quantity less than the radii of convergence of all the power series developments of φ(z) about interior points of this region, so chosen moreover that no circle of radius ρ with centre at an interior point of the region includes any singular point of φ(z), let g be such that |φ(z)| < g for all circles of radius ρ whose centres are interior points of the region, and, x being any interior point of the region, choose the positive integer n so that 1/n |x| < 1⁄3ρ; then take the points a1 = x/n, a2 = 2x/n, a3 = 3x/n, ... an = x; it is supposed that the region is so taken that, whatever x may be, all these are interior points of the region. Then by what has been said, replacing x, z respectively by 0 and x/n, we have

with

αμ < g/ρμ 2m1,

provided (μ + m1 + 1)μ < (2⁄3)m1 + 1; in fact for μ ⋜ 2n2n−2 it is sufficient to take m1 = n2n; by another application of the same inequality, replacing x, z respectively by a1 and x/n, we have

where

|β′μ| < g / ρμ 2m2

provided (μ + m2 + 1)μ < (3⁄2)m2 + 1; we take m2 = n2n − 2, supposing μ < 2n2n−4. So long as λ2 ⋜ m2 ⋜ n2n−2 and μ < 2n2n−4 we have μ + λ2 < 2n2n−2, and we can use the previous inequality to substitute here for φ(μ + λ2) (a1). When this is done we find

where |βμ| < 2g/ρμ 2m2, the numbers m1, m2 being respectively n2n and n2n−2.

Applying then the original inequality to φ(μ) (a3) = φ(μ) (a2 + x/n), and then using the series just obtained, we find a series for φ(μ) (a3). This process being continued, we finally obtain

where h = λ1 + λ2 + ... + λn, K = λ1! λ2! ... λn!, m1 = n2n, m2 = n2n−2, ..., mn= n², |ε| < 2g/2mn.

By this formula φ(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.

We are to evaluate an integral ∫ ba ƒ(x)dx; we form a closed contour of which the portion of the real axis from x = a to x = b forms a part, and consider the integral ∫ƒ(z)dz round this contour, supposing that the value of this integral can be determined along the curve forming the completion of the contour. The contour being supposed such that, within it, ƒ(z) is a single valued and finite function of the complex variable z save at a finite number of isolated interior points, the contour integral is equal to the sum of the values of ∫ƒ(z)dz taken round these points. Two instances will suffice to explain the method. (1) The integral ∫ ∞0 [(tan x)/x] dx is convergent if it be understood to mean the limit when ε, ζ, σ, ... all vanish of the sum of the integrals

Now draw a contour consisting in part of the whole of the positive and negative real axis from x = −nπ to x = +nπ, where n is a positive integer, broken by semicircles of small radius whose centres are the points x = ±½π, x = ±¾π, ... , the contour containing also the lines x = nπ and x = −nπ for values of y between 0 and nπ tan α, where α is a small fixed angle, the contour being completed by the portion of a semicircle of radius nπ sec α which lies in the upper half of the plane and is terminated at the points x = ±nπ, y = nπ tan α. Round this contour the integral ∫ [(tan z / z)] dz has the value zero. The contributions to this contour integral arising from the semicircles of centres −½(2s − 1)π, + ½(2s − 1)π, supposed of the same radius, are at once seen to have a sum which ultimately vanishes when the radius of the semicircles diminishes to zero. The part of the contour lying on the real axis gives what is meant by 2 ∫ nπ0 [(tan x / x)] dx. The contribution to the contour integral from the two straight portions at x = ±nπ is

where i tan iy, = −[exp(y) − exp(−y)]/[exp(y) + exp(−y)], is a real quantity which is numerically less than unity, so that the contribution in question is numerically less than

Finally, for the remaining part of the contour, for which, with R = nπ sec α, we have z = R(cos θ + i sin θ) = RE(iθ), we have

when n and therefore R is very large, the limit of this contribution to the contour integral is thus

− ∫ π−αα dθ = − (π − 2α).

Making n very large the result obtained for the whole contour is

where ε is numerically less than unity. Now supposing α to diminish to zero we finally obtain

(2) For another case, to illustrate a different point, we may take the integral

wherein a is real quantity such that 0 < a < 1, and the contour consists of a small circle, z = rE(iθ), terminated at the points x = r cos α, y = ± r sin α, where α is small, of the two lines y = ± r sin α for r cos α ⋜ x ⋜ R cos β, where R sin β = r sin α, and finally of a large circle z = RE(iφ), terminated at the points x = R cos β, y = ±R sin β. We suppose α and β both zero, and that the phase of z is zero for r cos a ⋜ x ⋜ R cos β, y = r sin α = R sin β. Then on r cos α ⋜ x ⋜ R cos β, y = −r sin α, the phase of z will be 2π, and zα − 1 will be equal to xα − 1 exp [2πi(a − 1)], where x is real and positive. The two straight portions of the contour will thus together give a contribution

It can easily be shown that if the limit of zƒ(z) for z = 0 is zero, the integral ∫ƒ(z)dz taken round an arc, of given angle, of a small circle enclosing the origin is ultimately zero when the radius of the circle diminishes to zero, and if the limit of zƒ(z) for z = ∞ is zero, the same integral taken round an arc, of given angle, of a large circle whose centre is the origin is ultimately zero when the radius of the circle increases indefinitely; in our case with ƒ(z) = zα−1/(1 + z), we have zƒ(z) = za/(1 + z), which, for 0 < a < 1, diminishes to zero both for z = 0 and for z = ∞. Thus, finally the limit of the contour integral when r = 0, R = ∞ is

Within the contour ƒ(z) is single valued, and has a pole at z = 1; at this point the phase of z is π and za−1 is exp [iπ(a − 1)] or − exp(iπa); this is then the residue of ƒ(z) at z = −1; we thus have

that is

Before passing to this it may be convenient to make here a few remarks as to the periodicity of (single valued) monogenic functions. To say that ƒ(z) is periodic is to say that there exists a constant ω such that for every point z of the interior of the region of existence of ƒ(z) we have ƒ(z + ω) = ƒ(z). This involves, considering all existing periods ω = ρ + iσ, that there exists a lower limit of ρ² + σ² other than zero; for otherwise all the differential coefficients of ƒ(z) would be zero, and ƒ(z) a constant; we can then suppose that not both ρ and σ are numerically less than ε, where ε > σ. Hence, if g be any real quantity, since the range (−g, ... g) contains only a finite number of intervals of length ε, and there cannot be two periods ω = ρ + iσ such that με ⋜ ρ < (μ + 1)ε, νε ⋜ σ < (ν + 1)ε, where μ, ν are integers, it follows that there is only a finite number of periods for which both ρ and σ are in the interval (−g ... g). Considering then all the periods of the function which are real multiples of one period ω, and in particular those periods λω wherein 0 < λ ⋜ 1, there is a lower limit for λ, greater than zero, and therefore, since there is only a finite number of such periods for which the real and imaginary parts both lie between −g and g, a least value of λ, say λ0. If Ω = λ0ω and λ = Mλ0 + λ′, where M is an integer and 0 ⋜ λ′ < λ0, any period λω is of the form MΩ + λ′ω; since, however, Ω, MΩ and λω are periods, so also is λ′ω, and hence, by the construction of λ0, we have λ′ = 0; thus all periods which are real multiples of ω are expressible in the form MΩ where M is an integer, and Ω a period.

If beside ω the functions have a period ω′ which is not a real multiple of ω, consider all existing periods of the form μω + νω′ wherein μ, ν are real, and of these those for which 0 ⋜ μ ⋜ 1, 0 < ν ⋜ 1; as before there is a least value for ν, actually occurring in one or more periods, say in the period Ω′ = μ0ω + ν0ω′; now take, if μω + νω′ be a period, ν = N′ν0 + ν′, where N′ is an integer, and 0 ⋜ ν′ < ν0; thence μω + νω′ = μω + N′(Ω′ − μ0ω) + ν′ω′; take then μ − Nμ0 = Nλ0 + λ′, where N is an integer and λ0 is as above, and 0 ⋜ λ′ < λ0; we thus have a period NΩ + N′Ω′ + λ′ω + ν′ω′, and hence a period λ′ω + ν′ω′, wherein λ′ < λ0, ν′ < ν0; hence ν′ = 0 and λ′ = 0. All periods of the form μω + νω′ are thus expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods and N, N′ are integers. But in fact any complex quantity, P + iQ, and in particular any other possible period of the function, is expressible, with μ, ν real, in the form μω + νω′; for if ω = ρ + iσ, ω′ = ρ′ + iσ′, this requires only P = μρ + νρ′, Q = μσ + νσ′, equations which, since ω′/ω is not real, always give finite values for μ and ν.

It thus appears that if a single valued monogenic function of z be periodic, either all its periods are real multiples of one of them, and then all are of the form MΩ, where Ω is a period and M is an integer, or else, if the function have two periods whose ratio is not real, then all its periods are expressible in the form NΩ + N′Ω′, where Ω, Ω′ are periods, and N, N′ are integers. In the former case, putting ζ = 2πiz/Ω, and the function ƒ(z) = φ(ζ), the function φ(ζ) has, like exp (ζ), the period 2πi, and if we take t = exp (ζ) or ζ = λ(t) the function is a single valued function of t. If then in particular ƒ(z) is an integral function, regarded as a function of t, it has singularities only for t = 0 and t = ∞, and may be expanded in the form Σ ∞−∞ an tn.

Taking the case when the single valued monogenic function has two periods ω, ω′ whose ratio is not real, we can form a network of parallelograms covering the plane of z whose angular points are the points c + mω + m′ω′, wherein c is some constant and m, m′ are all possible positive and negative integers; choosing arbitrarily one of these parallelograms, and calling it the primary parallelogram, all the values of which the function is at all capable occur for points of this primary parallelogram, any point, z′, of the plane being, as it is called, congruent to a definite point, z, of the primary parallelogram, z′ − z being of the form mω + m′ω′, where m, m′ are integers. Such a function cannot be an integral function, since then, if, in the primary parallelogram |ƒ(z)| < M, it would also be the case, on a circle of centre the origin and radius R, that |ƒ(z)| < M, and therefore, if Σan zn be the expansion of the function, which is valid for an integral function for all finite values of z, we should have |an| < MR−n, which can be made arbitrarily small by taking R large enough. The function must then have singularities for finite values of z.

We consider only functions for which these are poles. Of these there cannot be an infinite number in the primary parallelogram, since then those of these poles which are sufficiently near to one of the necessarily existing limiting points of the poles would be arbitrarily near to one another, contrary to the character of a pole. Supposing the constant c used in naming the corners of the parallelograms so chosen that no pole falls on the perimeter of a parallelogram, it is clear that the integral 1/(2πi) ∫ƒ(z) dz round the perimeter of the primary parallelogram vanishes; for the elements of the integral corresponding to two such opposite perimeter points as z, z + ω (or as z, z + ω′) are mutually destructive. This integral is, however, equal to the sum of the residues of ƒ(z) at the poles interior to the parallelogram. Which sum is therefore zero. There cannot therefore be such a function having only one pole of the first order in any parallelogram; we shall see that there can be such a function with two poles only in any parallelogram, each of the first order, with residues whose sum is zero, and that there can be such a function with one pole of the second order, having an expansion near this pole of the form (z-a)−2 + (power series in z − a).

Considering next the function φ(z) = [ƒ(z)]−1 dƒ(z)/dz, it is easily seen that an ordinary point of ƒ(z) is an ordinary point of φ(z), that a zero of order m for ƒ(z) in the neighbourhood of which ƒ(z) has a form, (z − a)m multiplied by a power series, is a pole of φ(z) of residue m, and that a pole of ƒ(z) of order n is a pole of φ(z) of residue −n; manifestly φ(z) has the two periods of ƒ(z). We thus infer, since the sum of the residues of φ(z) is zero, that for the function ƒ(z), the sum of the orders of its vanishing at points belonging to one parallelogram, Σm, is equal to the sum of the orders of its poles, Σn; which is briefly expressed by saying that the number of its zeros is equal to the number of its poles. Applying this theorem to the function ƒ(z) − A, where A is an arbitrary constant, we have the result, that the function ƒ(z) assumes the value A in one of the parallelograms as many times as it becomes infinite. Thus, by what is proved above, every conceivable complex value does arise as a value for the doubly periodic function ƒ(z) in any one of its parallelograms, and in fact at least twice. The number of times it arises is called the order of the function; the result suggests a property of rational functions.

Consider further the integral ∫ z [ƒ′(z)/ƒ(z)] dz, where ƒ′(z) = dƒ(z)/dz taken round the perimeter of the primary parallelogram; the contribution to this arising from two opposite perimeter points such as z and z + ω is of the form −ω ∫ z [ƒ′(z)/ƒ(z)] dz, which, as z increases from z0 to z0 + ω′, gives, if λ denote the generalized logarithm, − ω {λ [ƒ(z0 + ω′)] − λ[ƒ(z0)]}, that is, since ƒ(z0 + ω′) = ƒ(z0), gives 2πiNω, where N is an integer; similarly the result of the integration along the other two opposite sides is of the form 2πiN′ω′, where N′ is an integer. The integral, however, is equal to 2πi times the sum of the residues of zƒ′(z) / ƒ(z) at the poles interior to the parallelogram. For a zero, of order m, of ƒ(z) at z = a, the contribution to this sum is 2πima, for a pole of order n at z = b the contribution is −2πinb; we thus infer that Σma − Σnb = Nω + N′ω′; this we express in words by saying that the sum of the values of z where ƒ(z) = 0 within any parallelogram is equal to the sum of the values of z where ƒ(z) = ∞ save for integral multiples of the periods. By considering similarly the function ƒ(z) − A where A is an arbitrary constant, we prove that each of these sums is equal to the sum of the values of z where the function takes the value A in the parallelogram.

For this consider first the network of parallelograms whose corners are the points Ω = mω + m′ω′, where m, m′ take all positive and negative integer values; putting a small circle about each corner of this network, let P be a point outside all these circles; this will be interior to a parallelogram whose corners in order may be denoted by z0, z0 + ω, z0 + ω + ω′, z0 + ω′; we shall denote z0, z0 + ω by A0, B0; this parallelogram Π0 is surrounded by eight other parallelograms, forming with Π0 a larger parallelogram Π1, of which one side, for instance, contains the points z0 − ω − ω′, z0 − ω′, z0 − ω′ + ω, z0 − ω′ + 2ω, which we shall denote by A1, B1, C1, D1. This parallelogram Π1 is surrounded by sixteen of the original parallelograms, forming with Π1 a still larger parallelogram Π2 of which one side, for instance, contains the points z0 − 2ω − 2ω′, z0 − ω − 2ω′, z0 − 2ω′, z0 + ω − 2ω′, z0 + 2ω − 2ω′, z0 + 3ω − 2ω′, which we shall denote by A2, B2, C2, D2, E2, F2. And so on. Now consider the sum of the inverse cubes of the distances of the point P from the corners of all the original parallelograms. The sum will contain the terms

and three other sets of terms, each infinite in number, formed in a similar way. If the perpendiculars from P to the sides A0B0, A1B1C1, A2B2C2D2E2, and so on, be p, p + q, p + 2q and so on, the sum S0 is at most equal to

of which the general term is ultimately, when n is large, in a ratio of equality with 2q−3 n−2, so that the series S0 is convergent, as we know the sum Σn−2 to be; this assumes that p ≠ 0; if P be on A0B0 the proof for the convergence of S0 − 1/PA03, is the same. Taking the three other sums analogous to S0 we thus reach the result that the series

φ(z) = −2Σ (z − Ω)−3,

where Ω is mω + m′ω′, and m, m′ are to take all positive and negative integer values, and z is any point outside small circles described with the points Ω as centres, is absolutely convergent. Its sum is therefore independent of the order of its terms. By the nature of the proof, which holds for all positions of z outside the small circles spoken of, the series is also clearly uniformly convergent outside these circles. Each term of the series being a monogenic function of z, the series may therefore be differentiated and integrated outside these circles, and represents a monogenic function. It is clearly periodic with the periods ω, ω′; for φ(z + ω) is the same sum as φ(z) with the terms in a slightly different order. Thus φ(z + ω) = φ(z) and φ(z + ω′) = φ(z).

Consider now the function

where, for the subject of integration, the area of uniform convergence clearly includes the point z = 0; this gives

and

wherein Σ′ is a sum excluding the term for which m = 0 and m′ = 0. Hence ƒ(z + ω) − ƒ(z) and ƒ(z + ω′) − ƒ(z) are both independent of z. Noticing, however, that, by its form, ƒ(z) is an even function of z, and putting z = −½ω, z = −½ω′ respectively, we infer that also ƒ(z) has the two periods ω and ω′. In the primary parallelogram Π0, however, ƒ(z) is only infinite at z = 0 in the neighbourhood of which its expansion is of the form z−2 + (power series in z). Thus ƒ(z) is such a doubly periodic function as was to be constructed, having in any parallelogram of periods only one pole, of the second order.

To prove the last of these results, we write, for |z| < |Ω|,

and hence, if Σ′Ω−2n = σn, since Σ′Ω−(2n−1) = 0, we have, for sufficiently small z greater than zero,

ƒ(z) = z−2 + 3σ2·z2 + 5σ3·z4 + ...

and

φ(z) = −2z−3 + 6σ2·z + 20σ3·z3 + ...;

using these series we find that the function

F(z) = [φ(z)]² − 4[ƒ(z)]³ + 60σ2ƒ(z) + 140σ3

contains no negative powers of z, being equal to a power series in z² beginning with a term in z². The function F(z) is, however, doubly periodic, with periods ω, ω′, and can only be infinite when either ƒ(z) or φ(z) is infinite; this follows from its form in ƒ(z) and φ(z); thus in one parallelogram of periods it can be infinite only when z = 0; we have proved, however, that it is not infinite, but, on the contrary, vanishes, when z = 0. Being, therefore, never infinite for finite values of z it is a constant, and therefore necessarily always zero. Putting therefore ƒ(z) = ζ and φ(z) = dζ/dz we see that

Historically it was in the discussion of integrals such as

∫ dζ (4ζ³ − 60σ2·ζ − 140σ3)−1/2,

regarded as a branch of Integral Calculus, that the doubly periodic functions arose. As in the familiar case

z = ∫ ζ0 (1 − ζ²)−1/2 dζ,

where ζ = sin z, it has proved finally to be simpler to regard ζ as a function of z. We shall come to the other point of view below, under § 20, Elliptic Integrals.

Consider the expression

where As = ƒ(as), ζ is an abbreviation for ƒ(z) and η for φ(z), and (ζ, 1)m, (ζ, 1)m−2, denote integral polynomials in ζ, of respective orders m and m − 2, so that there are 2m unspecified, homogeneously entering, constants in the numerator. It is supposed that no one of the points a1, ... am is one of the points mω + m′ω′ where f(z) = ∞. The function Φ(z) is a monogenic function of z with the periods ω, ω′, becoming infinite (and having singularities) only when (1) ζ = ∞ or (2) one of the factors ζ-As is zero. In a period parallelogram including z = 0 the first arises only for z = 0; since for ζ = ∞, η is in a finite ratio to ζ3/2; the function Φ(z) for ζ = ∞ is not infinite provided the coefficient of ζm in (ζ, 1)m is not zero; thus Φ(z) is regular about z = 0. When ζ − As = 0, that is ƒ(z) = f(as), we have z = ±as + mω + m′ω′, and no other values of z, m and m′ being integers; suppose the unspecified coefficients in the numerator so taken that the numerator vanished to the first order in each of the m points −a1, −a2, ... −am; that is, if φ(as) = Bs, and therefore φ(−as) = −Bs, so that we have the m relations

(As, 1)m − Bs(As, 1)m−2 = 0;

then the function Φ(z) will only have the m poles a1, ... am. Denoting further the m zeros of F(z) by a1′, ... am′, putting ƒ(as′) = As′, φ(as′) = Bs′, suppose the coefficients of the numerator of Φ(z) to satisfy the further m − 1 conditions

(As′, 1)m + Bs′ (As′, 1)m−2 = 0

for s = 1, 2, ... (m − 1). The ratios of the 2m coefficients in the numerator of Φ(z) can always be chosen so that the m + (m − 1) linear conditions are all satisfied. Consider then the ratio

F(z) / Φ(z);

it is a doubly periodic function with no singularity other than the one pole am′. It is therefore a constant, the numerator of Φ(z) vanishing spontaneously in am′. We have

F(z) = AΦ(z),

where A is a constant; by which F(z) is expressed rationally in terms of ƒ(z) and φ(z), as was desired.

When z = 0 is a pole of F(z), say of order r, the other poles, each of the first order, being a1, ... am, similar reasoning can be applied to a function

where h, k are such that the greater of 2h − 2m, 2k + 3 − 2m is equal to r; the case where some of the poles a1, ... am are multiple is to be met by introducing corresponding multiple factors in the denominator and taking a corresponding numerator. We give a solution of the general problem below, of a different form.

One important application of the result is the theorem that the functions ƒ(z + t), φ(z + t), which are such doubly periodic function of z as have been discussed, can each be expressed, so far as they depend on z, rationally in terms of ƒ(z) and φ(z), and therefore, so far as they depend on z and t, rationally in terms of ƒ(z), ƒ(t), φ(z) and φ(t). It can in fact be shown, by reasoning analogous to that given above, that

This shows that if F(z) be any single valued monogenic function which is doubly periodic and of meromorphic character, then F(z + t) is an algebraic function of F(z) and F(t). Conversely any single valued monogenic function of meromorphic character, F(z), which is such that F(z + t) is an algebraic function of F(z) and F(t), can be shown to be a doubly periodic function, or a function obtained from such by degeneration (in virtue of special relations connecting the fundamental constants).

The functions ƒ(z), φ(z) above are usually denoted by ℜ(z), ℜ′(z); further the fundamental differential equation is usually written

(ℜ′z)² = 4(ℜz)³ − g2ℜz − g3,

and the roots of the cubic on the right are denoted by e1, e2, e3; for the odd function, ℜ′z, we have, for the congruent arguments −½ωand ½ω, ℜ′ (½ω) = −ℜ′ (−½ω) = −ℜ′ (½ω), and hence ℜ′ (½ω) = 0; hence we can take e1 = ℜ (½ω), e2 = ℜ (½ω + ½ω′), e3 = ℜ (½ω). It can then be proved that [ℜ(z) − e1] [ℜ (z + ½ω) − e1] = (e1 − e2) (e1 − e3), with similar equations for the other half periods. Consider more particularly the function ℜ(z) − e1; like ℜ(z) it has a pole of the second order at z = 0, its expansion in its neighbourhood being of the form z−2 (1 − e1z2 + Az4 + ...); having no other pole, it has therefore either two zeros, or a double zero in a period parallelogram (ω, ω′). In fact near its zero ½ω its expansion is (x − ½ω) ℜ′ (½ω) + ½(z − ½ω)² ℜ″ (½ω) + ...; we have seen that ℜ′ (½ω) = 0; thus it has a zero of the second order wherever it vanishes. Thus it appears that the square root [ℜ(z) − e1]1/2, if we attach a definite sign to it for some particular value of z, is a single valued function of z; for it can at most have two values, and the only small circuits in the plane which could lead to an interchange of these values are those about either a pole or a zero, neither of which, as we have seen, has this effect; the function is therefore single valued for any circuit. Denoting the function, for a moment, by ƒ1(z), we have ƒ1(z + ω) = ±ƒ1(z), ƒ1(z + ω′) = ±ƒ1(z); it can be seen by considerations of continuity that the right sign in either of these equations does not vary with z; not both these signs can be positive, since the function has only one pole, of the first order, in a parallelogram (ω, ω′); from the expansion of ƒ1(z) about z = 0, namely z− 1 (1 − ½e1z² + ...), it follows that ƒ1(z) is an odd function, and hence ƒ1 (−½ω′) = −ƒ1 (½ω′), which is not zero since [ƒ1 (½ω′)]² = e3 − e1, so that we have ƒ1 (z + ω′) = −ƒ1(z); an equation f1(z + ω) = −ƒ1(z) would then give ƒ1(z + ω + ω′) = ƒ1(z), and hence ƒ1(½ω + ½ω′) = ƒ1(−½ω − ½ω′), of which the latter is −ƒ1(½ω + ½ω′); this would give ƒ1(½ω + ½ω′) = 0, while [ƒ1(½ω + ½ω′)]² = e2 − e1. We thus infer that ƒ1(z + ω) = ƒ1(z), ƒ1(z + ω′) = −ƒ1(z), ƒ1(z + ω + ω′) = −ƒ1(z). The function ƒ1(z) is thus doubly periodic with the periods ω and 2ω′; in a parallelogram of which two sides are ω and 2ω′ it has poles at z = 0, z = ω′ each of the first order, and zeros of the first order at z = ½ω, z = ½ω + ω′; it is thus a doubly periodic function of the second order with two different poles of the first order in its parallelogram (ω, 2ω′). We may similarly consider the functions ƒ2(z) = [ℜ(z) − e2]1/2, ƒ3(z) = [ℜ(z) − e3]1/2; they give

Taking u = z (e1 − e3)1/2, with a definite determination of the constant (e1 − e3)1/2, it is usual, taking the preliminary signs so that for z = 0 each of zƒ1(z), zƒ2(z), zƒ3(z) is equal to +1, to put

thus sn(u) is an odd doubly periodic function of the second order with the periods 4K, 2iK, having poles of the first order at u = iK′, u = 2K + iK′, and zeros of the first order at u = 0, u = 2K; similarly cn(u), dn(u) are even doubly periodic functions whose periods can be written down, and sn²(u) + cn²(u) = 1, k²sn²(u) + dn²(u) = 1; if x = sn(u) we at once find, from the relations given here, that

if we put x = sinφ we have

and if we call φ the amplitude of u, we may write φ = am(u), x = sin·am(u), which explains the origin of the notation sn(u). Similarly cn(u) is an abbreviation of cos·am(u), and dn(u) of Δam(u), where Δ(φ) meant (1 − k²sin²φ)1/2. The addition equation for each of the functions ƒ1(z), ƒ2(z), ƒ3(z) is very simple, being

where f1′(z) means dƒ1(z)/dz, which is equal to −ƒ2(z)·ƒ3(z), and ƒ²(z) means [ƒ(z)]2. This may be verified directly by showing, if R denote the right side of the equation, that ∂R/∂z = ∂R/∂t; this will require the use of the differential equation

[ƒ1′(z)]2 = [ƒ12(z) + e1 − e2] [ƒ12(z) + e1 − e3],

and in fact we find

hence it will follow that R is a function of z + t, and R is at once seen to reduce to ƒ(z) when t = 0. From this the addition equation for each of the functions sn(u), cn(u), dn(u) can be deduced at once; if s1, c1, d1, s2, c2, d2 denote respectively sn(u1), cn(u1), dn(u1), sn(u2), cn(u2), dn(u2), they can be put into the forms

where

D = 1 − k2s12s22.

The introduction of the function ƒ1(z) is equivalent to the introduction of the function ℜ(z; ω, 2ω′) constructed from the periods ω, 2ω′ as was ℜ(z) from ω and ω′; denoting this function by ℜ1(z) and its differential coefficient by ℜ′1(z), we have in fact

as we see at once by considering the zeros and poles and the limit of zƒ1(z) when z = 0. In terms of the function ℜ1(z) the original function ℜ(z) is expressed by

ℜ(z) = ℜ1(z) + ℜ1(z + ω′) − ℜ1(ω′),

as a consideration of the poles and expansion near z = 0 will show.

A function having ω, ω′ for periods, with poles at two arbitrary points a, b and zeros at a′, b′, where a′ + b′ = a + b save for an expression mω + m′ω′, in which m, m′ are integers, is a constant multiple of

if the expansion of this function near z = a be

λ(z − a)−1 + μ + Σ n=1 μn (z − a)n,

the expansion near z = b is

−λ (z − b)− 1 + μ + Σ n=1 (−1)n μn (z − b)n,

as we see by remarking that if z′ − b = −(z − a) the function has the same value at z and z′; hence the differential equation satisfied by the function is easily calculated in terms of the coefficients in the expansions.

From the function ℜ(z) we can obtain another function, termed the Zeta-function; it is usually denoted by ζ(z), and defined by

for which as before we have equations

where 2η, 2η′ are certain constants, which in this case do not both vanish, since else ζ(z) would be a doubly periodic function with only one pole of the first order. By considering the integral

∫ ζ(z)dz

round the perimeter of a parallelogram of sides ω, ω′ containing z = 0 in its interior, we find ηω′ − η′ω = 1, so that neither of η, η′ is zero. We have ζ′(z) =−ℜ(z). From ζ(z) by means of the equation

we determine an integral function σ(z), termed the Sigma-function, having a zero of the first order at each of the points z = Ω; it can be seen to satisfy the equations

By means of these equations, if a1 + a2 + ... + am = a′1 + a′2 + ... + a′m, it is readily shown that

is a doubly periodic function having a1, ... am as its simple poles, and a′1, ... a′m as its simple zeros. Thus the function σ(z) has the important property of enabling us to write any meromorphic doubly periodic function as a product of factors each having one zero in the parallelogram of periods; these form a generalization of the simple factors, z − a, which have the same utility for rational functions of z. We have ζ(z) = σ′(z)/σ(z).

The functions ζ(z), ℜ(z) may be used to write any meromorphic doubly periodic function F(z) as a sum of terms having each only one pole; for if in the expansion of F(z) near a pole z = a the terms with negative powers of z − a be

A1(z − a)−1 + A2(z − a)−2 + ... + Am+1(z − a)−(m+1),

then the difference

will not be infinite at z = a. Adding to this a sum of further terms of the same form, one for each of the poles in a parallelogram of periods, we obtain, since the sum of the residues A is zero, a doubly periodic function without poles, that is, a constant; this gives the expression of F(z) referred to. The indefinite integral ∫F(z)dz can then be expressed in terms of z, functions ℜ(z − a) and their differential coefficients, functions ζ(z − a) and functions logσ(z − a).

Conversely, let U be any continuous real function on the circumference, U0 being the value of it at a point P0 of the circumference, and describe a small circle with centre at P0 cutting the given circle in A and B, so that for all points P of the arc AP0B we have |U − U0| < ε, where ε is a given small real quantity. Describe a further circle, centre P0 within the former, cutting the given circle in A′ and B′, and let Q be restricted to lie in the small space bounded by the arc A′P0B′ and this second circle; then for all positions of P upon the greater arc AB of the original circle QP² is greater than a definite finite quantity which is not zero, say QP² > D². Consider now the integral

which we evaluate as the sum of two, respectively along the small arc AP0B and the greater arc AB. It is easy to verify that, for the whole circumference,

Hence we can write

If the finite angle between QA and QB be called Φ and the finite angle AOB be called Θ, the sum of the first two components is numerically less than

If the greatest value of |(U − U0)| on the greater arc AB be called H, the last component is numerically less than

of which, when the circle, of centre P0, passing through A′B′ is sufficiently small, the factor a² − r² is arbitrarily small. Thus it appears that u′ is a function of the position of Q whose limit, when Q, interior to the original circle, approaches indefinitely near to P0, is U0. From the form

since the inclination of QP to a fixed direction is, when Q varies, P remaining fixed, a solution of the differential equation

where z, = x + iy, is the point Q, we infer that u′ is a differentiable function satisfying this equation; indeed, when r < a, we can write

= a0 + a1x + b1y + a2 (x² − y²) + 2b2xy + ...,

where

In this series the terms of order n are sums, with real coefficients, of the various integral polynomials of dimension n which satisfy the equation ∂²ψ/∂x² + ∂²ψ/∂y²; the series is thus the real part of a power series in z, and is capable of differentiation and integration within its region of convergence.

Conversely we may suppose a function, P, defined for the interior of a finite region R of the plane of the real variables x, y, capable of expression about any interior point x0, y0 of this region by a power series in x − x0, y − y0, with real coefficients, these various series being obtainable from one of them by continuation. For any region R0 interior to the region specified, the radii of convergence of these power series will then have a lower limit greater than zero, and hence a finite number of these power series suffice to specify the function for all points interior to R0. Each of these series, and therefore the function, will be differentiable; suppose that at all points of R0 the function satisfies the equation

we then call it a monogenic potential function. From this, save for an additive constant, there is defined another potential function by means of the equation

The functions P, Q, being given by a finite number of power series, will be single valued in R0, and P + iQ will be a monogenic function of z within R0· In drawing this inference it is supposed that the region R0 is such that every closed path drawn in it is capable of being deformed continuously to a point lying within R0, that is, is simply connected.

Suppose in particular, c being any point interior to R0, that P approaches continuously, as z approaches to the boundary of R, to the value log r, where r is the distance of c to the points of the perimeter of R. Then the function of z expressed by

ζ = (z − c) exp (−P − iQ)

will be developable by a power series in (z − z0) about every point z0 interior to R0, and will vanish at z = c; while on the boundary of R it will be of constant modulus unity. Thus if it be plotted upon a plane of ζ the boundary of R will become a circle of radius unity with centre at ζ=0, this latter point corresponding to z=c. A closed path within R0, passing once round z=c, will lead to a closed path passing once about ζ = 0. Thus every point of the interior of R will give rise to one point of the interior of the circle. The converse is also true, but is more difficult to prove; in fact, the differential coefficient dζ/dz does not vanish for any point interior to R. This being assumed, we obtain a conformal representation of the interior of the region R upon the interior of a circle, in which the arbitrary interior point c of R corresponds to the centre of the circle, and, by utilizing the arbitrary constant arising in determining the function Q, an arbitrary point of the boundary of R corresponds to an arbitrary point of the circumference of the circle.

There thus arises the problem of the determination of a real monogenic potential function, single valued and finite within a given arbitrary region, with an assigned continuous value at all points of the boundary of the region. When the region is circular this problem is solved by the integral 1/π ∫ Udω − 1/π ∫ Udθ previously given. When the region is bounded by the outermost portions of the circumferences of two overlapping circles, it can hence be proved that the problem also has a solution; more generally, consider a finite simply connected region, whose boundary we suppose to consist of a single closed path in the sense previously explained, ABCD; joining A to C by two non-intersecting paths AEC, AFC lying within the region, so that the original region may be supposed to be generated by the overlapping regions AECD, CFAB, of which the common part is AECF; suppose now the problem of determining a single valued finite monogenic potential function for the region AECD with a given continuous boundary value can be solved, and also the same problem for the region CFAB; then it can be shown that the same problem can be solved for the original area. Taking indeed the values assigned for the original perimeter ABCD, assume arbitrarily values for the path AEC, continuous with one another and with the values at A and C; then determine the potential function for the interior of AECD; this will prescribe values for the path CFA which will be continuous at A and C with the values originally proposed for ABC; we can then determine a function for the interior of CFAB with the boundary values so prescribed. This in its turn will give values for the path AEC, so that we can determine a new function for the interior of AECD. With the values which this assumes along CFA we can then again determine a new function for the interior of CFAB. And so on. It can be shown that these functions, so alternately determined, have a limit representing such a potential function as is desired for the interior of the original region ABCD. There cannot be two functions with the given perimeter values, since their difference would be a monogenic potential function with boundary value zero, which can easily be shown to be everywhere zero. At least two other methods have been proposed for the solution of the same problem.

A particular case of the problem is that of the conformal representation of the interior of a closed polygon upon the upper half of the plane of a complex variable t. It can be shown without much difficulty that if a, b, c, ... be real values of t, and α, β, γ, ... be n real numbers, whose sum is n − 2, the integral

z = ∫ (t − a)α−1 (t − b)β−1 ... dt,

as t describes the real axis, describes in the plane of z a polygon of n sides with internal angles equal to απ, βπ, ..., and, a proper sign being given to the integral, points of the upper half of the plane of t give rise to interior points of the polygon. Herein the points a, b, ... of the real axis give rise to the corners of the polygon; the condition Σα = n − 2 ensures merely that the point t = ∞ does not correspond to a corner; if this condition be not regarded, an additional corner and side is introduced in the polygon. Conversely it can be shown that the conformal representation of a polygon upon the half plane can be effected in this way; for a polygon of given position of more than three sides it is necessary for this to determine the positions of all but three of a, b, c, ...; three of them may always be supposed to be at arbitrary positions, such as t = 0, t = 1, t = ∞.

As an illustration consider in the plane of z = x + iy, the portion of the imaginary axis from the origin to z = ih, where h is positive and less than unity; let C be this point z = ih; let BA be of length unity along the positive real axis, B being the origin and A the point z = 1; let DE be of length unity along the negative real axis, D being also the origin and E the point z = − 1; let EFA be a semicircle of radius unity, F being the point z = i. If we put ζ = [(z² + h²)/(1 + h²z²)]1/2, with ζ = 1 when z = 1, the function is single valued within the semicircle, in the plane of z, which is slit along the imaginary axis from the origin to z = ih; if we plot the value of ζ upon another plane, as z describes the continuous curve ABCDE, ζ will describe the real axis from ζ = 1 to ζ = − 1, the point C giving ζ = 0, and the points B, D giving the points ζ = ±h. Near z = 0 the expansion of ζ is ζ − h = z² (1 − h4 / 2h) + ..., or ζ + h = −z² (1 − h4 / 2h) + ...; in either case an increase of ½π in the phase of z gives an increase of π in the phase of ζ − h or ζ + h. Near z = ih the expansion of ζ is ζ = (z − ih)1/2 [2ih/(1 − h4)]1/2 + ..., and an increase of 2π in the phase of z − ih also leads to an increase of π in the phase of ζ. Then as z describes the semicircle EFA, ζ also describes a semicircle of radius unity, the point z = i becoming ζ = i. There is thus a conformal representation of the interior of the slit semicircle in the z-plane, upon the interior of the whole semicircle in the ζ-plane, the function

z = [(ζ² − h²) / (1 − h²ζ²)]1/2

being single valued in the latter semicircle. By means of a transformation t = (ζ + 1)² / (ζ − 1)², the semicircle in the plane of ζ can further be conformably represented upon the upper half of the whole plane of t.

As another illustration we may take the conformal representation of an equilateral triangle upon a half plane. Taking the elliptic function ℜ(u) for which ℜ′²(u) = 4ℜ³(u) − 4, so that, with ε = exp (2⁄3πi), we have e1 = 1, e2 = ε², e3 = ε, the half periods may be taken to be

drawing the equilateral triangle whose vertices are O, of argument O, A of argument ω, and B of argument ω + ω′ = −ε²ω, and the equilateral triangle whose angular points are O, B and C, of argument ω′, let E, of argument 1⁄3(2ω + ω′), and D, of argument 1⁄3(ω + 2ω′), be the centroids of these triangles respectively, and let BE, OE, AE cut OA, AB, BO in K, L, H respectively, and BD, OD, CD cut OC, BC, OB in F, G, H respectively; then if u = ξ + iη be any point of the interior of the triangle OEH and v = εu0 = ε(ξ − iη) be any point of the interior of the triangle OHD, the points respectively of the ten triangles OEK, EKA, EAL, ELB, EBH, DHB, DBG, DGC, DCF, DFO are at once seen to be given by −εv, ω + εu, ω − η²v, ω + ω′ + ε²u, ω + ω′ − v, ω + ω′ − u, ω + ω′ + εv, ω′ − εu, ω′ + ε²v, −ε²u. Further, when u is real, since the term − 2(u + mω + m′ε²ω)−3, which is the conjugate complex of −2(u + mω + m′ε²ω)3, arises in the infinite sum which expresses ℜ′(u), namely as −2(u + μω + μ′εω)−3, where μ = m − m′, μ′ = −m′, it follows that ℜ′(u) is real; in a similar way we prove that ℜ′(u) is pure imaginary when u is pure imaginary, and that ℜ′(u) = ℜ′(εu) = ℜ′(ε²u), as also that for v = εu0, ℜ′(v) is the conjugate complex of ℜ′(u). Hence it follows that the variable

t = ½ iℜ′(u)

takes each real value once as u passes along the perimeter of the triangle ODE, being as can be shown respectively ∞, 1, 0, − 1 at O, D, H, E, and takes every complex value of imaginary part positive once in the interior of this triangle. This leads to

u = 1⁄3 i ∫ ∞t (t2 − 1)−2/3dt

in accordance with the general theory.

It can be deduced that τ = t2 represents the triangle ODH on the upper half plane of τ, and ζ = (i − τ−1)1/2 represents similarly the triangle OBD.

The commonest case of the occurrence of multiple valued functions is that in which the function s satisfies an algebraic equation ƒ(s, z) = p0sn + p1sn−1 + ... + pn = 0, wherein p0, p1, ... pn are integral polynomials in z. Assuming ƒ(s, z) incapable of being written as a product of polynomials rational in s and z, and excepting values of z for which the polynomial coefficient of sn vanishes, as also the values of z for which beside ƒ(s, z) = 0 we have also ∂f(s, z)/∂s = 0, and also in general the point z = ∞, the roots of this equation about any point z=c are given by n power series in z − c. About a finite point z = c for which the equation ∂f(s, z)/∂s = 0 is satisfied by one or more of the roots s of ƒ(s, z) = 0, the n roots break up into a certain number of cycles, the r roots of a cycle being given by a set of power series in a radical (z − c)1/r, these series of the cycle being obtainable from one another by replacing (z − c)1/r by ω(z − r)1/r, where ω, equal to exp (2πih/r), is one of the rth roots of unity. Putting then z − c = tr we may say that the r roots of a cycle are given by a single power series in t, an increase of 2π in the phase of t giving an increase of 2πr in the phase of z − c. This single series in t, giving the values of s belonging to one cycle in the neighbourhood of z = c when the phase of z − c varies through 2πr, is to be looked upon as defining a single place among the aggregate of values of z and s which satisfy ƒ(s, z) = 0; two such places may be at the same point (z = c, s = d) without coinciding, the corresponding power series for the neighbouring points being different. Thus for an ordinary value of z, z = c, there are n places for which the neighbouring values of s are given by n power series in z − c; for a value of z for which ∂f(s, z)/∂s = 0 there are less than n places. Similar remarks hold for the neighbourhood of z = ∞; there may be n places whose neighbourhood is given by n power series in z− 1 or fewer, one of these being associated with a series in t, where t = (z−1)1/r; the sum of the values of r which thus arise is always n. In general, then, we may say, with t of one of the forms (z − c), (z − c)1/r, z−1, (z−1)1/r. that the neighbourhood of any place (c, d) for which ƒ(c, d) = 0 is given by a pair of expressions z = c + P(t), s = d + Q(t), where P(t) is a (particular case of a) power series vanishing for t = 0, and Q(t) is a power series vanishing for t = 0, and t vanishes at (c, d), the expression z − c being replaced by z−1 when c is infinite, and similarly the expression s − d by s−1 when d is infinite. The last case arises when we consider the finite values of z for which the polynomial coefficient of sn vanishes. Of such a pair of expressions we may obtain a continuation by writing t = t0 + λ1τ + λ2τ² + ..., where τ is a new variable and λ1 is not zero; in particular for an ordinary finite place this equation simply becomes t = t0 + τ. It can be shown that all the pairs of power series z = c + P(t), s = d + Q(t) which are necessary to represent all pairs of values of z, s satisfying the equation ƒ(s, z) = 0 can be obtained from one of them by this process of continuation, a fact which we express by saying that the equation ƒ(s, z) = 0 defines a monogenic algebraic construct. With less accuracy we may say that an irreducible algebraic equation ƒ(s, z) = 0 determines a single monogenic function s of z.

Any rational function of z and s, where ƒ(s, z) = 0, may be considered in the neighbourhood of any place (c, d) by substituting therein z = c + P(t), s = d + Q(t); the result is necessarily of the form tmH(t), where H(t) is a power series in t not vanishing for t=0 and m is an integer. If this integer is positive, the function is said to vanish to order m at the place; if this integer is negative, = −μ, the function is infinite to order μ at the place. More generally, if A be an arbitrary constant, and, near (c, d), R(s, z) −A is of the form tmH(t), where m is positive, we say that R(s, z) becomes m times equal to A at the place; if R(s, z) is infinite of order μ at the place, so also is R(s, z) − A. It can be shown that the sum of the values of m at all the places, including the places z = ∞, where R(s, z) vanishes, which we call the number of zeros of R(s, z) on the algebraic construct, is finite, and equal to the sum of the values of μ where R(s, z) is infinite, and more generally equal to the sum of the values of m where R(s, z) = A; this we express by saying that a rational function R(s, z) takes any value (including ∞) the same number of times on the algebraic construct; this number is called the order of the rational function.

That the total number of zeros of R(s, z) is finite is at once obvious, these values being obtainable by rational elimination of s between ƒ(s, z) = 0, R(s, z) = 0. That the number is equal to the total number of infinities is best deduced by means of a theorem which is also of more general utility. Let R(s, z) be any rational function of s, z, which are connected by ƒ(s, z) = 0; about any place (c, d) for which z = c + P(t), s = d + Q(t), expand the product

in powers of t and pick out the coefficient of t−1. There is only a finite number of places of this kind. The theorem is that the sum of these coefficients of t−1 is zero. This we express by

The theorem holds for the case n=1, that is, for rational functions of one variable z; in that case, about any finite point we have z − c = t, and about z = ∞ we have z−1 = t, and therefore dz/dt = −t−2; in that case, then, the theorem is that in any rational function of z,

the sum ΣA1 of the sum of the residues at the finite poles is equal to the coefficient of 1/z in the expansion, in ascending powers of 1/z, about z = ∞; an obvious result. In general, if for a finite place of the algebraic construct associated with ƒ(s, z) = 0, whose neighbourhood is given by z = c + tr, s = d + Q(t), there be a coefficient of t−1 in R(s, z) dz/dt, this will be r times the coefficient of t−r in R(s, z) or R[d + Q(t), c + tr], namely will be the coefficient of t−r in the sum of the r series obtainable from R [d + Q(t), c + tr] by replacing t by ωt, where ω is an rth root of unity; thus the sum of the coefficients of t−1 in R(s, z) dz/dt for all the places which arise for z = c, and the corresponding values of s, is equal to the coefficient of (z − c)−1 in R(s1, z) + R(s2z) + ... + R(sn, z), where s1, ... sn are the n values of s for a value of z near to z = c; this latter sum Σ R(si, z) is, however, a rational function of z only. Similarly, near z = ∞, for a place given by z−1=tr, s = d + Q(t), or s−1 = Q(t), the coefficient of t−1 in R(s, z) dz/dt is equal to −r times the coefficient of tr in R[d + Q(t), t−r], that is equal to the negative coefficient of z−l in the sum of the r series R[d + Q(ωt), t−r], so that, as before, the sum of the coefficients of t−1 in R(s, z) dz/dt at the various places which arise for z = ∞ is equal to the negative coefficient of z− 1 in the same rational function of z, Σ R(si, z). Thus, from the corresponding theorem for rational functions of one variable, the general theorem now being proved is seen to follow.

Apply this theorem now to the rational function of s and z,

at a zero of R(s, z) near which R(s, z) = tmH(t), we have

where λ denotes the generalized logarithmic function, that is equal to

mt−1 + power series in t;

similarly at a place for which R(s, z) = t−μK(t); the theorem

thus gives Σm = Σμ, or, in words, the total number of zeros of R(s, z) on the algebraic construct is equal to the total number of its poles. The same is therefore true of the function R(s, z) − A, where A is an arbitrary constant; thus the number in question, being equal to the number of poles of R(s, z) − A, is equal also to the number of times that R(s, z) = A on the algebraic construct.

We have seen above that all single valued doubly periodic meromorphic functions, with the same periods, are rational functions of two variables s, z connected by an equation of the form s² = 4z³ + Az + B. Taking account of the relation connecting these variables s, z with the argument of the doubly periodic functions (which was above denoted by z), it can then easily be seen that the theorem now proved is a generalization of the theorem proved previously establishing for a doubly periodic function a definite order. There exists a generalization of another theorem also proved above for doubly periodic functions, namely, that the sum of the values of the argument in one parallelogram of periods for which a doubly periodic function takes a given value is independent of that value; this generalization, known as Abel’s Theorem, is given § 17 below.

For instance, the integral ∫ z1 z−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about z = 0, which is equal to 2πi; if we put u = ∫ z1 z−1dz and consider z as a function of u, then we must regard this function as unaffected by the addition of 2πi to its argument u; we know in fact that z = exp (u) and is a single valued function of u, with the period 2πi. Or again the integral ∫ z0 (1 + z²)−1dz is liable to an additive indeterminateness equal to the value obtained by a closed path about either of the points z = ±i; thus if we put u = ∫ z0 (1 + z²)−1dz, the function z of u is periodic with period π, this being the function tan (u). Next we take the integral u = ∫ (z)(0) (1 − z²)−1/2dz, agreeing that the upper and lower limits refer not only to definite values of z, but to definite values of z each associated with a definite determination of the sign of the associated radical (1 − z²)−1/2. We suppose 1 + z, 1 − z each to have phase zero for z = 0; then a single closed circuit of z = −1 will lead back to z = 0 with (l − z²)1/2 = −1; the additive indeterminateness of the integral, obtained by a closed path which restores the initial value of the subject of integration, may be obtained by a closed circuit containing both the points ±1 in its interior; this gives, since the integral taken about a vanishing circle whose centre is either of the points z = ±1 has ultimately the value zero, the sum

where, in each case, (1 − z²)1/2 is real and positive; that is, it gives

or 2π. Thus the additive indeterminateness of the integral is of the form 2kπ, where k is an integer, and the function z of u, which is sin (u), has 2π for period. Take now the case

adopting a definite determination for the phase of each of the factors z − a, z − b, z − c, z − d at the arbitrary point z0, and supposing the upper limit to refer, not only to a definite value of z, but also to a definite determination of the radical under the sign of integration. From z0 describe a closed loop about the point z = a, consisting, suppose, of a straight path from z0 to a, followed by a vanishing circle whose centre is at a, completed by the straight path from a to z0. Let similar loops be imagined for each of the points b, c, d, no two of these having a point in common. Let A denote the value obtained by the positive circuit of the first loop; this will be in fact equal to twice the integral taken from z0 along the straight path to a; for the contribution due to the vanishing circle is ultimately zero, and the effect of the circuit of this circle is to change the sign of the subject of integration. After the circuit about a, we arrive back at z0 with the subject of integration changed in sign; let B, C, D denote the values of the integral taken by the loops enclosing respectively b, c and d when in each case the initial determination of the subject of integration is that adopted in calculating A. If then we take a circuit from z0 enclosing both a and b but not either c or d, the value obtained will be A − B, and on returning to z0 the subject of integration will have its initial value. It appears thus that the integral is subject to an additive indeterminateness equal to any one of the six differences such as A − B. Of these there are only two linearly independent; for clearly only A − B, A − C, A − D are linearly independent, and in fact, as we see by taking a closed circuit enclosing all of a, b, c, d, we have A − B + C − D = 0; for there is no other point in the plane beside a, b, c, d about which the subject of integration suffers a change of sign, and a circuit enclosing all of a, b, c, d may by putting z = 1/ζ be reduced to a circuit about ζ = 0 about which the value of the integral is zero. The general value of the integral for any position of z and the associated sign of the radical, when we start with a definite determination of the subject of integration, is thus seen to be of the form u0 + m(A − B) + n(A − C), where m and n are integers. The value of A − B is independent of the position of z0, being obtainable by a single closed positive circuit about a and b only; it is thus equal to twice the integral taken once from a to b, with a proper initial determination of the radical under the sign of integration. Similar remarks to the above apply to any integral ∫ H(z)dz, in which H(z) is an algebraic function of z; in any such case H(z) is a rational function of z and a quantity s connected therewith by an irreducible rational algebraic equation ƒ(s, z) = 0. Such an integral ƒK(z, s)dz is called an Abelian Integral.

To each value of z belong two values of s, of opposite sign; starting, for some particular value of z, with a definite one of these two values, the sign to be attached to s for any other value of z will be determined by the path of integration for z. When z is in the neighbourhood of any finite value z0 for which the radical s is not zero, if we put z − z0 = t, we can find s − s0 = a power series in t, say s=s0 + Q(t); when z is in the neighbourhood of a value, a, for which s vanishes, if we put z = a + t², we shall obtain s = tQ(t), where Q(t) is a power series in t; when z is very large and s² is a quartic polynomial in z, if we put z−1 = t, we shall find s−1 = t²Q(t); when z is very large and s² is a cubic polynomial in z, if we put z−1 = t², we shall find s−l = t³Q(t). By means of substitutions of these forms the character of the integral ∫ R(z, s)dz may be investigated for any position of z; in any case it takes a form ∫ [Ht−m + Kt−m+1 + ... + Pt−1 + R + St + ... ]dt involving only a finite number of negative powers of t in the subject of integration. Consider first the particular case ∫ s−1dz; it is easily seen that neither for any finite nor for infinite values of z can negative powers of t enter; the integral is everywhere finite, and is said to be of the first kind; it can, moreover, be shown without difficulty that no integral ∫ R(z, s)dz, save a constant multiple of ∫ s−1dz, has this property. Consider next, s² being of the form a0z4 + 4a1z³ + ..., wherein a0 may be zero, the integral ∫ (a0z² + 2a1z) s−1dz; for any finite value of z this integral is easily proved to be everywhere finite; but for infinite values of z its value is of the form At−1 + Q(t), where Q(t) is a power series; denoting by √a0 a particular square root of a0 when a0 is not zero, the integral becomes infinite for z = ∞ for both signs of s, the value of A being + √a0 or − √a0 according as s is √a0·z² (1 + [2a1/a0] z−1 + ... ) or is the negative of this; hence the integral J1 = ∫ ( [a0z² + 2a1z]/s + √a0) dz becomes infinite when z is infinite, for the former sign of s, its infinite term being 2√a0·t−1 or 2a0·z, but does not become infinite for z infinite for the other sign of s. When a0 = 0 the signs of s for z = ∞ are not separated, being obtained one from the other by a circuit of z about an infinitely large circle, and the form obtained represents an integral becoming infinite as before for z = ∞, its infinite part being 2√a1·t−1 or 2√a1·√z. Similarly if z0 be any finite value of z which is not a root of the polynomial ƒ(z) to which s² is equal, and s0 denotes a particular one of the determinations of s for z=z0, the integral

wherein ƒ′(z) = dƒ(z)/dz, becomes infinite for z = z0, s = s0, but not for z = z0, s = −s0. its infinite term in the former case being the negative of 2s0(z − z0). For no other finite or infinite value of z is the integral infinite. If z = θ be a root of ƒ(z), in which case the corresponding value of s is zero, the integral

becomes infinite for z=0, its infinite part being, if z − θ = t², equal to −[ƒ′(θ)]½ t−1: and this integral is not elsewhere infinite. In each of these cases, of the integrals J1, J2, J3, the subject of integration has been chosen so that when the integral is written near its point of infinity in the form ∫[At−2 + Bt−1 + Q(t)] dt, the coefficient B is zero, so that the infinity is of algebraic kind, and so that, when there are two signs distinguishable for the critical value of z, the integral becomes infinite for only one of these. An integral having only algebraic infinities, for finite or infinite values of z, is called an integral of the second kind, and it appears that such an integral can be formed with only one such infinity, that is, for an infinity arising only for one particular, and arbitrary, pair of values (s, z) satisfying the equation s² = ƒ(z), this infinity being of the first order. A function having an algebraic infinity of the mth order (m > 1), only for one sign of s when these signs are separable, at (1) z = ∞, (2) z = z0, (3) z = a, is given respectively by (s d/dz)m−1 J1, (s d/dz)m−1 J2, (s d/dz)m−1 J3, as we easily see. If then we have any elliptic integral having algebraic infinities we can, by subtraction from it of an appropriate sum of constant multiples of J1, J2, J3 and their differential coefficients just written down, obtain, as the result, an integral without algebraic infinities. But, in fact, if J, J1 denote any two of the three integrals J1, J2, J3, there exists an equation AJ + BJ′ + Cƒs−1dz = rational function of s, z, where A, B, C are properly chosen constants. For the rational function

is at once found to become infinite for (z0, s0), not for (z0, −s0), its infinite part for the first point being 2s/(z − z0), and to become infinite for z infinitely large, and one sign of s only when these are separable, its infinite part there being 2z √a0 or 2 √a1 √z when a0 = 0. It does not become infinite for any other pair (z, s) satisfying the relation s2 = ƒ(z); this is in accordance with the easily verified equation

and there exists the analogous equation

Consider now the integral

this is at once found to be infinite, for finite values of z, only for (z0, s0), its infinite part being log (z − z0), and for z = ∞, for one sign of s only when these are separable, its infinite part being −log t, that is −log z when a0 ≠ 0, and −log (z1/2) when a0 = 0. And, if ƒ(θ) = 0, the integral

is infinite at z = θ, s = 0 with an infinite part log t, that is log (z − θ)1/2, is not infinite for any other finite value of z, and is infinite like P for z = ∞. An integral possessing such logarithmic infinities is said to be of the third kind.

Hence it appears that any elliptic integral, by subtraction from it of an appropriate sum formed with constant multiples of the integral J3 and the rational functions of the form (s d/dz)m−1 J1 with constant multiples of integrals such as P or P1, with constant multiples of the integral u = ∫s−1dz, and with rational functions, can be reduced to an integral H becoming infinite only for z = ∞, for one sign of s only when these are separable, its infinite part being of the form A log t, that is, A log z or A log (z1/2). Such an integral H = ∫R(z, s)dz does not exist, however, as we at once find by writing R(z, s) = P(z) + sQ(z), where P(z), Q(z) are rational functions of z, and examining the forms possible for these in order that the integral may have only the specified infinity. An analogous theorem holds for rational functions of z and s; there exists no rational function which is finite for finite values of z and is infinite only for z = ∞ for one sign of s and to the first order only; but there exists a rational function infinite in all to the first order for each of two or more pairs (z, s), however they may be situated, or infinite to the second order for an arbitrary pair (z, s); and any rational function may be formed by a sum of constant multiples of functions such as

and their differential coefficients.

The consideration of elliptic integrals is therefore reducible to that of the three

respectively of the first, second and third kind. Now the equation s2 = a0z4 + ... = a0 (z − θ) (z − φ) (z − ψ) (z − χ), by putting

y = 2s (z − θ)−2 [a0 (θ − φ) (θ − ψ) (θ − χ) ]−1/2

is at once reduced to the form y2 = 4x3 − g2x − g3 = 4(x − e1) (x − e2) (x − e3), say; and these equations enable us to express s and z rationally in terms of x and y. It is therefore sufficient to consider three elliptic integrals

Of these consider the first, putting

where the limits involve not only a value for x, but a definite sign for the radical y. When x is very large, if we put x−1 = t2, y−1 = 2t3 (1 − ¼ g2t4 − ¼ g3t6)−1/2, we have

u = ∫ t0 (1 + 1⁄8 g2t4 + ... ) dt = t + 1⁄40 g2t5 + ...,

whereby a definite power series in u, valid for sufficiently small value of u, is found for t, and hence a definite power series for x, of the form

x = u−2 + 1⁄20 g2u2 + ...

Let this expression be valid for 0 < |u| < R, and the function defined thereby, which has a pole of the second order for u=0, be denoted by φ(u). In the range in question it is single valued and satisfies the differential equation

[φ′(u)]2 = 4[φ(u)]3 − g2φ(u) − g3;

in terms of it we can write x = φ(u), y = − φ′(u), and, φ′(u) being an odd function, the sign attached to y in the original integral for x = ∞ is immaterial. Now for any two values u, v in the range in question consider the function

it is at once seen, from the differential equation, to be such that ∂F/∂u = ∂F/∂v; it is therefore a function of u + v; supposing |u + v| < R we infer therefore, by putting v = 0, that

By repetition of this equation we infer that if u1, ... un be any arguments each of which is in absolute value less than R, whose sum is also in absolute value less than R, then φ(u1 + ... + un) is a rational function of the 2n functions φ(us), φ′(us); and hence, if |u| < R, that

where H is some rational function of the arguments φ(u/n), φ′(u/n). In fact, however, so long as |u/n| < R, each of the functions φ(u/n), φ′(u/n) is single valued and without singularity save for the pole at u=0; and a rational function of single valued functions, each of which has no singularities other than poles in a certain region, is also a single valued function without singularities other than poles in this region. We infer, therefore, that the function of u expressed by H [φ(u/n), φ′(u/n)] is single valued and without singularities other than poles so long as |u| < nR; it agrees with φ(u) when |u| < R, and hence furnishes a continuation of this function over the extended range |u| < nR. Moreover, from the method of its derivation, it satisfies the differential equation [φ′(u)]2 = 4[φ(u)]3 − g2φ(u) − g3. This equation has therefore one solution which is a single valued monogenic function with no singularities other than poles for any finite part of the plane, having in particular for u = 0, a pole of the second order; and the method adopted for obtaining this near u=0 shows that the differential equation has no other such solution. This, however, is not the only solution which is a single valued meromorphic function, a the functions φ(u + α), wherein α is arbitrary, being such. Taking now any range of values of u, from u = 0, and putting for any value of u, x = φ(u), y = −φ′(u), so that y2=4x3-g2x-g3, we clearly have

conversely if x0 = φ(u0), y0 = −φ′(u0) and ξ, η be any values satisfying η2 = 4ξ2 − g2ξ − g3, which are sufficiently near respectively to x0, y0, while v is defined by

then ξ, η are respectively φ(v) and −φ′(v); for this equation leads to an expansion for ξ − x0 in terms of v = u0 and only one such expansion, and this is obtained by the same work as would be necessary to expand φ(v) when v is near to u0; the function φ(u) can therefore be continued by the help of this equation, from v = u0, provided the lower limit of |ξ − x0| necessary for the expansions is not zero in the neighbourhood of any value (x0, y0). In fact the function φ(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 of u which is outside these circles, the lower limit of the radii of convergence of the expansions of φ(u) is greater than zero; the same will therefore be the case for the lower limit of the radii |ξ − x0| necessary for the continuations spoken of above provided that the values of (ξ, η) considered do not lead to infinitely increasing values of v; there does not exist, however, any definite point (ξ0, η0) in the neighbourhood of which the integral ∫ (ξ, η)(x0, y0) dξ/η increases indefinitely, it is only by a path of infinite length that the integral can so increase. We infer therefore that if (ξ, η) be any point, where η2 = 4ξ3 − g2ξ − g3, and v be defined by

then ξ = φ(v) and η = −φ′(v). Thus this equation determines (ξ, η) without ambiguity. In particular the additive indeterminatenesses of the integral obtained by closed circuits of the point of integration are periods of the function φ(u); by considerations advanced above it appears that these periods are sums of integral multiples of two which may be taken to be

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 (x0, y0), other than one enclosing two of the points (e1, 0), (e2, 0), (e3, 0), (∞, ∞), which leads back to the initial point (x0, y0), which is impossible. On the whole, therefore, it appears that the function φ(u) agrees with the function ℜ(u) previously discussed, and the discussion of the elliptic integrals can be continued in the manner given under § 14, Doubly Periodic Functions.

It is easy to see that the series for ℜ(u), u−2 + Σ′[(u + mω + m′ω′)2 − (mω + m′ω′)2], is unaffected by replacing ω, ω′ by two quantities Ω, Ω′ equal respectively to pω + qω′, p′ω′ + q′ω′, where p, q, p′, q′ are any integers for which pq′ − p′q = ±1; further it can be proved that all substitutions with integer coefficients Ω = pω + qω′, Ω′ = p′ω + q′ω′, wherein pq′ − p′q = 1, can be built up by repetitions of the two particular substitutions (Ω = −ω′, Ω′ = ω), (Ω = ω, Ω′ = ω + ω′). Consider the function of the ratio ω′/ω expressed by

h = −ℜ (½ω′) / ℜ(½ω);

it is at once seen from the properties of the function ℜ(u) that by the two particular substitutions referred to we obtain the corresponding substitutions for h expressed by

h′ = 1/h,   h′ = 1 − h;

thus, by all the integer substitutions Ω = pω + qω′, Ω′ = p′ω + q′ω′, in which pq′ − p′q = 1, the function h can only take one of the six values h, 1/h, 1 − h, 1/(1 − h), h/(h − 1), (h − 1)/h, which are the roots of an equation in θ,

the function of τ, = ω′/ω, expressed by the right side, is thus unaltered by every one of the substitutions τ′ = (p′ + q′τ / p + qτ), wherein p, q, p′, q′ are integers having pq′ − p′q = 1. If the imaginary part σ, of τ, which we may write τ = ρ + iσ, is positive, the imaginary part of τ′, which is equal to σ(pq′ − p′q)/[(p + qρ)2 + q2σ2], is also positive; suppose σ to be positive; it can be shown that the upper half of the infinite plane of the complex variable τ can be divided into regions, all bounded by arcs of circles (or straight lines), no two of these regions overlapping, such that any substitution of the kind under consideration, τ′ = (p′ + q′τ)/(p + qτ) leads from an arbitrary point τ, of one of these regions, to a point τ′ of another; taking τ = ρ + iσ, one of these regions may be taken to be that for which −½ < ρ < ½, ρ2 + σ2 > 1, together with the points for which ρ is negative on the curves limiting this region; then every other region is obtained from this so-called fundamental region by one and only one of the substitutions τ = (p′ + q′τ)/(p + qτ), and hence by a definite combination of the substitutions τ′ = −1/τ, τ′ = 1 + τ. Upon the infinite half plane of τ, the function considered above,

is a single valued monogenic function, whose only essential singularities are the points τ′ = (p′ + q′τ)/(p + qτ) for which τ = ∞, namely those for which τ′ is any real rational value; the real axis is thus a line over which the function z(τ) cannot be continued, having an essential singularity in every arc of it, however short; in the fundamental region, z(τ) has thus only the single essential singularity, r = ρ + iσ, where σ = ∞; in this fundamental region z(τ) takes any assigned complex value just once, the relation z(τ′) = z(τ) requiring, as can be shown, that τ′ is of the form (p′ + q′τ)/(p + qτ), in which p, q, p′, q′ are integers with pq′ − p′q = 1; the function z(τ) has thus a similar behaviour in every other of the regions. The division of the plane into regions is analogous to the division of the plane, in the case of doubly periodic functions, into parallelograms; in that case we considered only functions without essential singularities, and in each of the regions the function assumed every complex value twice, at least. Putting, as another function of τ, J(τ) = z(τ) [z(τ) − 1], it can be shown that J(τ) = 0 for τ = exp (2⁄3πi), that J(τ) = 1 for τ = i, these being values of τ on the boundary of the fundamental region; like z(τ) it has an essential singularity for τ = ρ + iσ, σ = + ∞. In the theory of linear differential equations it is important to consider the inverse function τ(J); this is infinitely many valued, having a cycle of three values for circulation of J about J = 0 (the circuit of this point leading to a linear substitution for τ of period 3, such as τ′ = −(1 + τ)−1), having a cycle of two values about J = 1 (the circuit leading to a linear substitution for τ of period 2, such as τ′ = −τ−1), and having a cycle of infinitely many values about J = ∞ (the circuit leading to a linear substitution for τ which is not periodic, such as τ′ = 1 + τ). These are the only singularities for the function τ(J). Each of the functions

beside many others (see below), is a single valued function of τ, and is expressible without ambiguity in terms of the single valued function of τ,

It should be remarked, however, that η(τ) is not unaltered by all the substitutions we have considered; in fact

η(−τ−1) = (−iτ) ½η (τ),   η(1 + τ) = exp (1⁄12 iπ) η(τ).

The aggregate of the substitutions τ′ = (p′ + q′τ)/(p + qτ), wherein p, q, p′, q′ are integers with pq′ − p′q = 1, represents a Group; the function J(τ), unaltered by all these substitutions, is called a Modular Function. More generally any function unaltered by all the substitutions of a group of linear substitutions of its variable is called an Automorphic Function. A rational function, of its variable h, of this character, is the function (1 − h + h2)3 h−2(1 − h)−2 presenting itself incidentally above; and there are other rational functions with a similar property, the group of substitutions belonging to any one of these being, what is a very curious fact, associable with that of the rotations of one of the regular solids, about an axis through its centre, which bring the solid into coincidence with itself. Other automorphic functions are the double periodic functions already discussed; these, as we have seen, enable us to solve the algebraic equation y2 = 4x3 − g2x − g3 (and in fact many other algebraic equations, see below, under § 23, Geometrical Applications of Elliptic Functions) in terms of single valued functions x = ℜ(u), y = −ℜ′(u). A similar utility, of a more extended kind, belongs to automorphic functions in general; but it can be shown that such functions necessarily have an infinite number of essential singularities except for the simplest cases.

The modular function J(τ) considered above, unaltered by the group of linear substitutions τ′ = (p′ + q′τ) / (p + qτ), where p, q, p′, q′ are integers with pq′ − p′q = 1, may be taken as the independent variable x of a differential equation of the third order, of the form

where s′ = ds/dx, &c., of which the dependent variable s is equal to τ. A differential equation of this form is satisfied by the quotient of two independent integrals of the linear differential equation of the second order satisfied by the hypergeometric functions. If the solution of the differential equation for s be written s(α,β,γ, x), we have in fact τ = s(½, 1⁄3, 0, J). If we introduce also the function of τ given by

we similarly have τ = s(0, 0, 0, λ); this function λ is a single valued function of τ, which is also a modular function, being unaltered by a group of integral substitutions also of the form τ′ = (p′ + q′τ)/(p + qτ), with pq′ − p′q = 1, but with the restriction that p′ and q are even integers, and therefore p and q′ are odd integers. This group is thus a subgroup of the general modular group, and is in fact of the kind called a self-conjugate subgroup. As in the general case this subgroup is associated with a subdivision of the plane into regions of which any one is obtained from a particular region, called the fundamental region, by a particular one of the substitutions of the subgroup. This fundamental region, putting τ = ρ + iσ, may be taken to be that given by −1 < ρ < 1, (ρ + ½)2 + σ2 > ¼, (ρ − ½)2 + σ2 > ¼, and is built up of six of the regions which arose for the general modular group associated with J(τ). Within this fundamental region, λ takes every complex value just once, except the values λ = 0, 1, ∞, which arise only at the angular points τ = 0, τ = ∞, τ = − 1 and the equivalent point τ = 1; these angular points are essential singularities for the function λ(τ). For λ(τ) as for J(τ), the region of existence is the upper half plane of τ, there being an essential singularity in every length of the real axis, however short.

If, beside the plane of τ, we take a plane to represent the values of λ, the function τ = s(0, 0, 0, λ) being considered thereon, the values of τ belonging to the interior of the fundamental region of the τ-plane considered above, will require the consideration of the whole of the λ-plane taken once with the exception of the portions of the real axis lying between −∞ and 0 and between 1 and +∞, the two sides of the first portion corresponding to the circumferences of the τ-plane expressed by (ρ + ½)2 + σ2 = ¼, (ρ − ½)2 + σ2 = ¼, while the two sides of the latter portion, for which λ is real and > 1, correspond to the lines of the τ-plane expressed by ρ = ±1. The line for which λ is real, positive and less than unity corresponds to the imaginary axis of the τ-plane, lying in the interior of the fundamental region. All the values of τ = s(0, 0, 0, λ) may then be derived from those belonging to the fundamental region of the τ-plane by making λ describe a proper succession of circuits about the points λ = 0, λ = 1; any such circuit subjects τ to a linear substitution of the subgroup of τ considered, and corresponds to a change of τ from a point of the fundamental region to a corresponding point of one of the other regions.

A similar theorem can be proved in regard to the values assumed by the function φ(z) for points z of modulus greater than R, however great R may be, also with the help of modular functions. In general terms it may be stated that it is a very exceptional thing for an integral function not to assume every complex value an infinite number of times.

Another application of modular functions is to prove that the function s(α, β, γ, λ) is a single valued function of τ = s(0, 0, 0, λ); for, putting τ′ = (τ − i)/(τ + i), the values of τ′ which correspond to the singular points λ = 0, 1, ∞ of s(α, β, γ, λ), though infinite in number, all lie on the circumference of the circle |τ′| = 1, within which therefore s(α, β, γ, x) is expressible in a form Σ ∞n=0 anτ′n. More generally any monogenic function of λ which is single valued save for circuits of the points λ = 0, 1, ∞, is a single valued function of τ = s(0, 0, 0, λ). Identifying λ with the square of the modulus in Legendre’s form of the elliptical integral, we have τ = iK′/K, where

functions such as λ1/4, (1 − λ)1/4, [λ(1 − λ)]1/4, which have only λ = 0, 1, ∞ as singular points, were expressed by Jacobi as power series in q = eiπτ, and therefore, at least for a limited range of values of τ, as single valued functions of τ; it follows by the theorem given that any product of a root of λ and a root of 1 − λ is a single valued function of τ. More generally the differential equation

may be solved by expressing both the independent and dependent variables as single valued functions of a single variable τ, the expression for the independent variable being x = λ(τ).

For a general position of (x, y) upon ƒ the equations φ1(x′, x′)/φ(x′, x′) = φ1(x, y)/φ(x, y), φ2(x′, x′)/φ(x′, x′) = φ2(x, y)/φ(x, y), subject to ƒ(x′, x′) = 0, will have the same number of solutions (x′, x′); if their only solution is x′ = x, x′ = y, then to any position (ξ, η) of F will conversely correspond only one position (x, y) of ƒ. If these equations have another solution beside (x, y), then any curve λφ + λ1φ1 + λ2φ2 = 0 which passes (through the double points of ƒ and) through the n − 2 points of ƒ constituted by the fixed n− 3 points and a point (x0, y0), will necessarily pass through a further point, say (x0′, y0′), and will have only one further intersection with ƒ; such a curve, with the n − 2 assigned points, beside the double points, of ƒ, will be of the form μψ + μ1ψ1 + ... = 0, where μ2, μ3, ... are generally zero; considering the curves ψ + tψ1 = 0, for variable t, one of these passes through a further arbitrary point of ƒ, by choosing t properly, and conversely an arbitrary value of t determines a single further point of ƒ; the co-ordinates of the points of ƒ are thus rational functions of a parameter t, which is itself expressible rationally by the co-ordinates of the point; it can be shown algebraically that such a curve has not ½(n − 3)n but ½(n − 3)n + 1 double points. We may therefore assume that to every point of F corresponds only one point of ƒ, and there is a birational transformation between these curves; the coefficients in this transformation will involve rationally the co-ordinates of the n− 3 fixed points taken upon ƒ, that is, at the least, by taking these to be consecutive points, will involve the co-ordinates of one point of ƒ, and will not be rational in the coefficients of ƒ unless we can specify a point of ƒ whose co-ordinates are rational in these. The curve F is intersected by a straight line aξ + bη + c = 0 in as many points as the number of unspecified intersections of ƒ with aφ + bφ1 + cφ2 = 0, that is, 3; or F will be a cubic curve, without double points.

Such a cubic curve has at least one point of inflection Y, and if a variable line YPQ be drawn through Y to cut the curve again in P and Q, the locus of a point R such that YR is the harmonic mean of YP and YQ, is easily proved to be a straight line. Take now a triangle of reference for homogeneous co-ordinates XYZ, of which this straight line is Y = 0, and the inflexional tangent at Y is Z = 0; the equation of the cubic curve will then be of the form

ZY² = aX³ + bX²Z + cXZ² + dZ³;

by putting X equal to λX + μZ, that is, choosing a suitable line through Y to be X = 0, and choosing λ properly, this is reduced to the form

ZY² = 4X³ − g2XZ² − g3Z³,

of which a representation is given, valid for every point, in terms of the elliptic functions ℜ(u), ℜ′(u), by taking X = Zℜ(u), Y = Zℜ′(u). The value of u belonging to any point is definite save for sums of integral multiples of the periods of the elliptic functions, being given by

where (∞) denotes the point of inflection.

It thus appears that the co-ordinates of any point of a plane curve, ƒ, of order n with ½(n − 3)n double points are expressible as elliptic functions, there being, save for periods, a definite value of the argument u belonging to every point of the curve. It can then be shown that if a variable curve, φ, of order m be drawn, passing through the double points of the curve, the values of the argument u at the remaining intersections of φ with ƒ, have a sum which is unaffected by variation of the coefficients of φ, save for additive aggregates of the periods. In virtue of the birational transformation this theorem can be deduced from the theorem that if any straight line cut the cubic y² = 4x³ − g2x − g3, in points (u1), (u2), (u3), the sum u1 + u2 + u3 is zero, or a period; or the general theorem is a corollary from Abel’s theorem proved under § 17, Integrals of Algebraic Functions. To prove the result directly for the cubic we remark that the variation of one of the intersections (x, y) of the cubic with the straight line y = mx + n, due to a variation δm, δn in m and n, is obtained by differentiation of the equation for the three abscissae, namely the equation

F(x) = 4x³ − g2x − g3 − (mx + n)² = 0,

and is thus given by

and the sum of three such fractions as that on the right for the three roots of F(x) = 0 is zero; hence u1 + u2 + u3 is independent of the straight line considered; if in particular this become the inflexional tangent each of u1, u2, u3 vanishes. It may be remarked in passing that x1 + x2 + x3 = ¼m², and hence is ¼ {(y1 − y2)/(x1 − x2)}²; so that we have another proof of the addition equation for the function ℜ(u). From this theorem for the cubic curve many of its geometrical properties, as for example those of its inflections, the properties of inscribed polygons, of the three kinds of corresponding points, and the theory of residuation, are at once obvious. And similar results hold for the curve of order n with ½(n − 3)n double points.

The algebraical integrals ∫ R(x, y)dx associated with this may as before be divided into those of the first kind, which have no infinities, those of the second kind, possessing only algebraical infinities, and those of the third kind, for which logarithmic infinities enter. Here there is a certain number, p, greater than unity, of linearly independent integrals of the first kind; and this number p is unaltered by any birational transformation of the fundamental equation ƒ(x, y) = 0; a rational function can be constructed with poles of the first order at p + 1 arbitrary positions (x, y), satisfying ƒ(x, y) = 0, but not with a fewer number unless their positions are chosen properly, a property we found for the case p = 1; and p is the number of linearly independent curves of order n − 3 passing through the double points of the curve of order n expressed by ƒ(x, y) = 0. Again any integral of the second kind can be expressed as a sum of p integrals of this kind, with poles of the first order at arbitrary positions, together with rational functions and integrals of the first kind; and an integral of the second kind can be found with one pole of the first order of arbitrary position, and an integral of the third kind with two logarithmic infinities, also of arbitrary position; the corresponding properties for p = 1 are proved above.

There is, however, a difference of essential kind in regard to the inversion of integrals of the first kind; if u = ∫R(x, y)dx be such an integral, it can be shown, in common with all algebraic integrals associated with ƒ(x, y) = 0, to have 2p linearly independent additive constants of indeterminateness; the upper limit of the integral cannot therefore, as we have shown, be a single valued function of the value of the integral. The corresponding theorem, if ∫Ri(x, y)dx denote one of the integrals of the first kind, is that the p equations

∫ Ri (x1, y1)dx1 + ... + ∫ Ri (xp, yp)dxp = ui,

determine the rational symmetric functions of the p positions (x1, y1), ... (xp, yp) as single valued functions of the p variables, u1, ... up. It is thus necessary to enter into the theory of functions of several independent variables; and the equation ƒ(x, y) = 0 is thus not, in this way, capable of solution by single valued functions of one variable. That solution in fact is to be sought with the help of automorphic functions, which, however, as has been remarked, have, for p > 1, an infinite number of essential singularities.

A pole is a point (u(0)1, ... u(0)p) in the neighbourhood of which the function is expressible as a quotient of converging power series in u1 − u(0)1 ... up − u(0)p; of these the denominator series D must vanish at (u(0)1, ... u(0)p), since else the fraction is expressible as a power series and the point is not a singular point, but the numerator series N must not also vanish at (u(0)1, ... u(0)p), or if it does, it must be possible to write D = MD0, N = MN0, where M is a converging power series vanishing at (u(0)1, ...u(0)p), and N0 is a converging power series, in (u1 − u(0)1 ... up − u(0)p), not so vanishing. A polar point of indetermination is a point about which the function can be expressed as a quotient of two converging power series, both of which vanish at the point. As in such a simple case as (Ax + By)/ (ax + by), about x = 0, y = 0, it can be proved that then the function can be made to approach to any arbitrarily assigned value by making the variables u1, ... up approach to u(0)1, ... u(0)p by a proper path. It is the necessary existence of such polar points of indetermination, which in case p > 2 are not merely isolated points, which renders the theory essentially more difficult than that of functions of one variable. An essential singularity is any which does not come under one of the two former descriptions and includes very various possibilities. A point at infinity in this theory is one for which any one of the variables u1, ... up is indefinitely great; such points are brought under the preceding definitions by means of the convention that for u(0)i = ∞, the difference ui − u(0)i is to be understood to stand for u−1i. This being so, a single valued function of u1, ... up without essential singularities for infinite or finite values of the variables can be shown, by induction, to be, as in the case of p = 1, necessarily a rational function of the variables. A function having no singularities for finite values of all the variables is as before called an integral function; it is expressible by a power series converging for all finite values of the variables; a single valued function having for finite values of the variables no singularities other than poles or polar points of indetermination is called a meromorphic function; as for p = 1 such a function can be expressed as a quotient of two integral functions having no common zero point other than the points of indetermination of the function; but the proof of this theorem is difficult.

The single valued functions which occur, as explained above, in the inversion of algebraic integrals of the first kind, for p > 1, are meromorphic. They must also be periodic, unaffected that is when the variables u1, ... up are simultaneously increased each by a proper constant, these being the additive constants of indeterminateness for the p integrals ∫ Ri(x, y)dx arising when (x, y) makes a closed circuit, the same for each integral. The theory of such single valued meromorphic periodic functions is simpler than that of meromorphic functions of several variables in general, as it is sufficient to consider only finite values of the variables; it is the natural extension of the theory of doubly periodic functions previously discussed. It can be shown to reduce, though the proof of this requires considerable developments of which we cannot speak, to the theory of a single integral function of u1, ... up, called the Theta Function. This is expressible as a series of positive and negative integral powers of quantities exp (c1u1), exp (c2u2), ... exp (cpup), wherein c1, ... cp are proper constants; for p = 1 this theta function is essentially the same as that above given under a different form (see § 14, Doubly Periodic Functions), the function σ(u). In the case of p = 1, all meromorphic functions periodic with the same two periods have been shown to be rational functions of two of them connected by a single algebraic equation; in the same way all meromorphic functions of p variables, periodic with the same sets of simultaneous periods, 2p sets in all, can be shown to be expressible rationally in terms of p + 1 such periodic functions connected by a single algebraic equation. Let x1, ... xp, y denote p + 1 such functions; then each of the partial derivatives dxi/∂ui will equally be a meromorphic function of the same periods, and so expressible rationally in terms of x1, ... xp, y; thus there will exist p equations of the form

dxi = R1du1 + ... + Rpdup,

and hence p equations of the form

dui = Hi, 1dx1 + ... + Hi, pdxp,

wherein Hi, j are rational functions of x1, ... xp, y, these being connected by a fundamental algebraic (rational) equation, say ƒ(x1, ... xp, y) = 0. This then is the generalized form of the corresponding equation for p = 1.

In recent years developments have been made which point to a similar unity of conception as possible for surfaces, or indeed for algebraic constructs of any number of dimensions. These developments have been in two directions, at first followed independently, but now happily brought into the most intimate connexion. On the analytical side, E. Picard has considered the possibility of classifying integrals of the form ∫(Rds + Sdy), belonging to a surface ƒ(x, y, z) = 0, wherein R and S are rational functions of x, y, z, according as they are (1) everywhere finite, (2) have poles, which then lie along curves upon the surface, or (3) have logarithmic infinities, also then lying along curves, and has brought the theory to a high degree of perfection. On the geometrical side A. Clebsch and M. Noether, and more recently the Italian school, have considered the geometrical characteristics of a surface which are unaltered by birational transformation. It was first remarked that for surfaces of order n there are associated surfaces of order n − 4, having properties in relation thereto analogous to those of curves of order n − 3 for a plane curve of order n; if such a surface ƒ(x, y, z) = 0 have a double curve with triple points triple also for the surface, and φ(x, y, z) = 0 be a surface of order n − 4 passing through the double curve, the double integral

is everywhere finite; and, the most general everywhere finite integral of this form remains invariant in a birational transformation of the surface ƒ, the theorem being capable of generalization to algebraic constructs of any number of dimensions. The number of linearly independent surfaces of order n − 4, possessing the requisite particularity in regard to the singular lines and points of the surface, is thus a number invariant by birational transformation, and the equality of these numbers for two surfaces is a necessary condition of their being capable of such transformation. The number of surfaces of order m having the assigned particularity in regard to the singular points and lines of the fundamental surface can be given by a formula for a surface of given singularity; but the value of this formula for m = n − 4 is not in all cases equal to the actual number of surfaces of order n − 4 with the assigned particularity, and for a cone (or ruled surface) is in fact negative, being the negative of the deficiency of the plane section of the cone. Nevertheless this number for m = n − 4 is also found to be invariant for birational transformation. This number, now denoted by pa, is then a second invariant of birational transformation. The former number, of actual surfaces of order n − 4 with the assigned particularity in regard to the singularities of the surface, is now denoted by pg. The difference pg − pa, which is never negative, is a most important characteristic of a surface. When it is zero, as in the case of the general surface of order n, and in a vast number of other ordinary cases, the surface is called regular.

On a plane algebraical curve we may consider linear series of sets of points, obtained by the intersection with it of curves λφ + λ1φ1 + ... = 0, wherein λ, λ1, ... are variable coefficients; such a series consists of the sets of points where a rational function of given poles, belonging to the construct ƒ(x, y) = 0, has constant values. And we may consider series of sets of points determined by variable curves whose coefficients are algebraical functions, not necessarily rational functions, of parameters. Similarly on a surface we may consider linear systems of curves, obtained by the intersection with the given surface of variable surfaces λφ + λ1φ1 + ... = 0, and may consider algebraic systems, of which the individual curve is given by variable surfaces whose coefficients are algebraical, not necessarily rational, functions of parameters. Of a linear series upon a plane curve there are two numbers manifestly invariant in birational transformation, the order, which is the number of points forming a set of the series, and the dimension, which is the number of parameters λ1/λ, λ2/λ, ... entering linearly in the equation of the series. The series is complete when it is not contained in a series of the same order but of higher dimension. So for a linear system of curves upon a surface, we have three invariants for birational transformation; the order, being in the number of variable intersections of two curves of the system, the dimension, being the number of linear parameters λ1/λ, λ2/λ, ... in the equation for the system, and the deficiency of the individual curves of the system. Upon any curve of the linear system the other curves of the system define a linear series, called the characteristic series; but even when the linear system is complete, that is, not contained in another linear system of the same order and higher dimension, it does not follow that the characteristic series is complete; it may be contained in a series whose dimension is greater by pg − pa than its own dimension. When this is so it can be shown that the linear system of curves is contained in an algebraic system whose dimension is greater by pg − pa than the dimension of the linear system. The extra p = pg − pa variable parameters so entering may be regarded as the independent co-ordinates of an algebraic construct ƒ(y, x1, ... xp) = 0; this construct has the property that its co-ordinates are single valued meromorphic functions of p variables, which are periodic, possessing 2p systems of periods; the p variables are expressible in the forms

ui = ∫ R1(x, y) dx1 + ... + Rp(x, y) dxp,

wherein Ri(x, y) denotes a rational function of x1, ... xp and y. The original surface has correspondingly p integrals of the form ∫(R dx + S dy), wherein R, S are rational in x, y, z, which are everywhere finite; and it can be shown that it has no other such integrals. From this point of view, then, the number p, = pg − pa is, for a surface, analogous to the deficiency of a plane curve; another analogy arises in the comparison of the theorems: for a plane curve of zero deficiency there exists no algebraic series of sets of points which does not consist of sets belonging to a linear series; for a surface for which pg − pa = 0 there exists no algebraic system of curves not contained in a linear system.

But whereas for a plane curve of deficiency zero, the co-ordinates of the points of the curve are rational functions of a single parameter, it is not necessarily the case that for a surface having pg − pa = 0 the co-ordinates of the points are rational functions of two parameters; it is necessary that pg − pa = 0, but this is not sufficient. For surfaces, beside the pg linearly independent surfaces of order n − 4 having a definite particularity at the singularities of the surface, it is useful to consider surfaces of order k(n − 4), also having each a definite particularity at the singularities, the number of these, not containing the original surface as component, which are linearly independent, is denoted by Pk. It can then be stated that a sufficient condition for a surface to be rational consists of the two conditions pa = 0, P2 = 0. More generally it becomes a problem to classify surfaces according to the values of the various numbers which are invariant under birational transformation, and to determine for each the simplest form of surface to which it is birationally equivalent. Thus, for example, the hyperelliptic surface discussed by Humbert, of which the co-ordinates are meromorphic functions of two variables of the simplest kind, with four sets of periods, is characterized by pg = 1, pa = −1; or again, any surface possessing a linear system of curves of which the order exceeds twice the deficiency of the individual curves diminished by two, is reducible by birational transformation to a ruled surface or is a rational surface. But beyond the general statement that much progress has already been made in this direction, of great interest to the student of the theory of functions, nothing further can be added here.

Bibliography.—The learner will find a lucid introduction to the theory in E. Goursat, Cours d’analyse mathématique, t. ii. (Paris, 1905), or, with much greater detail, in A.R. Forsyth, Theory of Functions of a Complex Variable (2nd ed., Cambridge, 1900); for logical rigour in the more difficult theorems, he should consult W.F. Osgood, Lehrbuch der Functionentheorie, Bd. i. (Leipzig, 1906-1907); for greater precision in regard to the necessary quasi-geometrical axioms, beside the indications attempted here, he should consult W.H. Young, The Theory of Sets of Points (Cambridge, 1906), chs. viii.-xiii., and C. Jordan, Cours d’analyse, t. i. (Paris, 1893), chs. i., ii.; a comprehensive account of the Theory of Functions of Real Variables is by E.W. Hobson (Cambridge, 1907). Of the theory regarded as based after Weierstrass upon the theory of power series, there is J. Harkness and F. Morley, Introduction to the Theory of Analytic Functions (London, 1898), an elementary treatise; for the theory of the convergence of series there is also T.J. I’A. Bromwich, An Introduction to the Theory of Infinite Series (London, 1908); but the student should consult the collected works of Weierstrass (Berlin, 1894 ff.), and the writings of Mittag-Leffler in the early volumes of the Acta mathematica; earlier expositions of the theory of functions on the basis of power series are in C. Méray, Leçons nouvelles sur l’analyse infinitésimale (Paris, 1894), and in Lagrange’s books on the Theory of Functions. An account of the theory of potential in its applications to the present theory is found in most treatises; in particular consult E. Picard, Traité d’analyse, t. ii. (Paris, 1893). For elliptic functions there is an introductory book, P. Appell and E. Lacour, Principes de la théorie des fonctions elliptiques et applications (Paris, 1897), beside the treatises of G.H. Halphen, Traité des fonctions elliptiques et de leurs applications (three parts, Paris, 1886 ff.), and J. Tannery et J. Molk, Éléments de la théorie des fonctions elliptiques (Paris, 1893 ff.); a book, A.G. Greenhill, The Applications of Elliptic Functions (London, 1892), shows how the functions enter in problems of many kinds. For modular functions there is an extensive treatise, F. Klein and R. Fricke, Theorie der elliptischen Modulfunctionen (Leipzig, 1890); see also the most interesting smaller volume, F. Klein, Über das Ikosaeder (Leipzig, 1884) (also obtainable in English). For the theory of Riemann’s surface, and algebraic integrals, an interesting introduction is P. Appeil and E. Goursat, Théorie des fonctions algébriques et de leurs intégrales; for Abelian functions see also H. Stahl, Theorie der Abel’schen Functionen (Leipzig, 1896), and H.F. Baker, An Introduction to the Theory of Multiply Periodic Functions (Cambridge, 1907), and H.F. Baker, Abel’s Theorem and the Allied Theory, including the Theory of the Theta Functions (Cambridge, 1897); for theta functions of one variable a standard work is C.G. Jacobi, Fundamenta nova, &c. (Königsberg, 1828); for the general theory of theta functions, consult W. Wirtinger, Untersuchungen über Theta-Functionen (Leipzig, 1895). For a history of the theory of algebraic functions consult A. Brill and M. Noether, Die Entwicklung der Theorie der algebraischen Functionen in älterer und neuerer Zeit, Bericht der deutschen Mathematiker-Vereinigung (1894); and for a special theory of algebraic functions, K. Hensel and G. Landsberg, Theorie der algebraischen Function u.s.w. (Leipzig, 1902). The student will, of course, consult also Riemann’s and Weierstrass’s Ges. Werke. For the applications to geometry in general an important contribution, of permanent value, is E. Picard and G. Simart, Théorie des fonctions algébriques de deux variables indépendantes (Paris, 1897-1906). This work contains, as Note v. t. ii. p. 485, a valuable summary by MM. Castelnuovo and Enriques, Sur quelques résultats nouveaux dans la théorie des surfaces algébriques, containing many references to the numerous memoirs to be found, for the most part, in the transactions of scientific societies and the mathematical journals of Italy.

Beside the books above enumerated there exists an unlimited number of individual memoirs, often of permanent importance and only imperfectly, or too elaborately, reproduced in the pages of the volumes in which the student will find references to them. The German Encyclopaedia of Mathematics, and the Royal Society’s Reference Catalogue of Current Scientific Literature, Pure Mathematics, published yearly, should also be consulted.