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 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.
20.Representation of Functions in General.—We have seen that the notion of a function is wider than the notion of an analytical expression, and that the same function may be “represented” by one expression in one part of the domain of the argument and by some other expression in another part of the domain (§ 5). Thus there arises the general problem of the representation of functions. The function may be given by specifying the domain of the argument and the rule of calculation, or else the function may have to be determined in accordance with certain conditions; for example, it may have to satisfy in a prescribed domain an assigned differential equation. In either case the problem is to determine, when possible, a single analytical expression which shall have the same value as the function at all points in the domain of the argument. For the representation of most functions for which the problem can be solved recourse must be had to limiting processes. Thus we may utilize infinite series, or infinite products, or definite integrals; or again we may represent a function of one variable as the limit of an expression containing two variables in a domain in which one variable remains constant and another varies. An example of this process is afforded by the expression Lty= ∞xy / (x²y + 1), which represents a function of x vanishing at x = 0 and at all other values of x having the value of 1/x. The method of series falls under this more general process (cf. § 6). When the terms u1, u2, ... of a series are functions of a variable x, the sum snof the first n terms of the series is a function of x and n; and, when the series is convergent, its sum, which is Ltn= ∞ sn, can represent a function of x. In most cases the series converges for some values of x and not for others, and the values for which it converges form the “domain of convergence.” The sum of the series represents a function in this domain.
The apparently more general method of representation of a function of one variable as the limit of a function of two variables has been shown by R. Baire to be identical in scope with the method of series, and it has been developed by him so as to give a very complete account of the possibility of representing functions by analytical expressions. For example, he has shown that Riemann’s totally discontinuous function, which is equal to 1 when x is rational and to 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∫∞0e−ttx−1dt,or by the infinite productLtn=∞nx/x(1 + x)(1 + ½x) ...(1 +x).n − 1The second of these expressions avails for the representation of the function at all points at which x is not a negative integer.
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
∫∞0e−ttx−1dt,
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.
21.Power Series.—Taylor’s theorem leads in certain cases to a representation of a function by an infinite series. We have under certain conditions (§ 13)
and this becomes
provided that (α) a positive number k can be found so that at all points in the interval between a and a + k (except these points) ƒ(x) has continuous differential coefficients of all finite orders, and at a has progressive differential coefficients of all finite orders; (β) Cauchy’s form of the remainderRn, viz. [(x − a) / (n − 1)!] (1 − θ)n−1ƒn{a + θ(x − a)}, has the limit zero when n increases indefinitely, for all values of θ between 0 and 1, and for all values of x in the interval between a and a + k, except possibly a + k. When these conditions are satisfied, the series (1) represents the function at all points of the interval between a and a + k, except possibly a + k, and the function is “analytic” (§ 13) in this domain. Obvious modifications admit of extension to an interval between a and a − k, or between a − k and a + k. When a series of the form (1) represents a function it is called “the Taylor’s series for the function.”
Taylor’s series is a power series,i.e. a series of the form
Σ∞n=0an(x − a)n.
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.
As regards power series we have the following theorems:
1. If the power series converges at any point except a there is a number k which has the property that the series converges absolutely in the interval between a − k and a + k, with the possible exception of one or both end-points.
2. The power series represents a continuous function in its domain of convergence (the end-points may have to be excluded).
3. This function is analytic in the domain, and the power series representing it is the Taylor’s series for the function.
The theory of power series has been developed chiefly from the point of view of the theory of functions of complex variables.
22.Uniform Convergence.—We shall suppose that the domain of convergence of an infinite series of functions is an interval with the possible exception of isolated points. Let ƒ(x) be the sum of the series at any point x of the domain, and ƒn(x) the sum of the first n + 1 terms. The condition of convergence at a point a is that, after any positive number ε, however small, has been specified, it must be possible to find a number n so that |ƒm(a) − ƒp(a)| < ε for all values of m and p which exceed n. The sum, ƒ(a), is the limit of the sequence of numbers ƒn(a) atn = ∞. The convergence is said to be “uniform” in an interval if, after specification of ε, the same number n suffices at all points of the interval to make |ƒ(x) − ƒm(x)| < ε for all values of m which exceed n. The numbers n corresponding to any ε, however small, are all finite, but, when ε is less than some fixed finite number, they may have an infinite superior limit (§ 7); when this is the case there must be at least one point, a, of the interval which has the property that, whatever number N we take, ε can be taken so small that, at some point in the neighbourhood of a, n must be taken > N to make |ƒ(x) − fm(x)| < ε when m > n; then the series does not converge uniformly in the neighbourhood of a. The distinction may be otherwise expressed thus: Choose a first and ε afterwards, then the number n is finite; choose ε first and allow a to vary, then the number n becomes a function of a, which may tend to become infinite, or may remain below a fixed number; if such a fixed number exists, however small ε may be, the convergence is uniform.
For example, the series sin x − ½ sin 2x +1⁄3sin 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 seriesΣ∞n=0nx−(n + 1)x,n²x² + 1(n + 1)²x² + 1of 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.
For example, the series sin x − ½ sin 2x +1⁄3sin 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.
As regards series whose terms represent continuous functions we have the following theorems:
(1) If the series converges uniformly in an interval it represents a function which is continuous throughout the interval.
(2) If the series represents a function which is discontinuous in an interval it cannot converge uniformly in the interval.
(3) A series which does not converge uniformly in an interval may nevertheless represent a function which is continuous throughout the interval.
(4) A power series converges uniformly in any interval contained within its domain of convergence, the end-points being excluded.
(5) IfΣ∞r=0ƒr(x) = ƒ(x) converges uniformly in the interval between a and b
∫baƒ(x)dx =Σbr=0∫baƒr(x)dx,
or a series which convergesuniformlymay be integrated term by term.
(6) IfΣ∞r=0ƒ′r(x) converges uniformly in an interval, thenΣ∞r=0ƒr(x) converges in the interval, and represents a continuous differentiable function, φ(x); in fact we have
φ′(x) =Σ∞r=0ƒ′r(x),
or a series can be differentiated term by term if the series of derived functions converges uniformly.
A series whose terms represent functions which are not continuous throughout an interval may converge uniformly in the interval. IfΣ∞r=0ƒr(x) = ƒ(x), is such a series, and if all the functions ƒr(x) have limits at a, then ƒ(x) has a limit at a, which isΣ∞r=0Ltx=aƒr(x). A similar theorem holds for limits on the left or on the right.
23. Fourier’s Series.—An extensive class of functions admit of being represented by series of the form
and the rule for determining the coefficients an, bnof such a series, in order that it may represent a given function ƒ(x) in the interval between −c and c, was given by Fourier, viz. we have
The interval between −c and c may be called the “periodic interval,” and we may replace it by any other interval,e.g.that between 0 and 1, without any restriction of generality. When this is done the sum of the series takes the form
Ltn=∞∫10Σr = nr = −nƒ(z) cos {2rπ(z − x)}dz,
and this is
(ii.)
Fourier’s theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier’s time a function of this character was regarded as completely arbitrary.
By a discussion of the integral (ii.) based on the Second Theorem of the Mean (§ 15) it can be shown that, if 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 articleFourier’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.
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 articleFourier’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.
24.Representation of Continuous Functions by Series.—If the series for ƒ(x) formed by Fourier’s rule converges at the point a of the periodic interval, and if ƒ(x) is continuous at a, the sum of the series is ƒ(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier’s rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier’s series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as “terminated Fourier’s series” and “terminated power series,” according to the two following theorems:
(1) If ƒ(x) is continuous throughout the interval between 0 and 2π, and if any positive number ε however small is specified, it is possible to find an integer n, so that the difference between the value of ƒ(x) and the sum of the first n terms of the series for ƒ(x), formed by Fourier’s rule with periodic interval from 0 to 2π, shall be less than ε at all points of the interval. This result can be extended to a function which is continuous in any given interval.
(2) If ƒ(x) is continuous throughout an interval, and any positive number ε however small is specified, it is possible to find an integer n and a polynomial in x of the nth degree, so that the difference between the value of ƒ(x) and the value of the polynomial shall be less than ε at all points of the interval.
Again it can be proved that, if ƒ(x) is continuous throughout a given interval, polynomials in x of finite degrees can be found, so as to form an infinite series of polynomials whose sum is equal to ƒ(x) at all points of the interval. Methods of representation of continuous functions by infinite series of rational fractional functions have also been devised.
Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the seriesΣ∞0ancos (bnxπ), 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 x0as a limiting point, for which {ƒ(x′) − ƒ(x0)} / (x′ − x0) tends to become infinite with one sign when x′ − x0approaches zero through positive values, and infinite with the opposite sign when x′ − x0approaches 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 Σanis an absolutely convergent series of numbers, and φn(x) is an analytic function whose absolute value never exceeds unity.
Particular interest attaches to continuous functions which are not differentiable. Weierstrass gave as an example the function represented by the seriesΣ∞0ancos (bnxπ), 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 x0as a limiting point, for which {ƒ(x′) − ƒ(x0)} / (x′ − x0) tends to become infinite with one sign when x′ − x0approaches zero through positive values, and infinite with the opposite sign when x′ − x0approaches 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 Σanis an absolutely convergent series of numbers, and φn(x) is an analytic function whose absolute value never exceeds unity.
25.Calculations with Divergent Series.—When the series described in (1) and (2) of § 24 diverge, they may, nevertheless, be used for the approximate numerical calculation of the values of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the property of representing a function approximately when the expansion is not carried too far are called “asymptotic expansions.” Sometimes they are called “semi-convergent series”; but this term is avoided in the best modern usage, because it is often used to describe series whose convergence depends upon the order of the terms, such as the series 1 − ½ +1⁄3− ...
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) −Σnr = 0ƒr(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′, |Σmr = 0ƒr(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 + ½e−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 factlog {Γ(x)} = (x − ½) log x − x + ½ log 2π +ω(x),whereω(x) is the function defined by the definite integralω(x) =∫∞0{(1 − e−t)−1− t−1− ½} t−1e−txdt.The multiplier of e−txunder the sign of integration can be expanded in the power seriesB1−B2t2+B3t4− ...,2!4!6!where B1, B2, ... are “Bernoulli’s numbers” given by the formulaBm= 2.2m! (2π)−2mΣ∞r = 1(r−2m).When the series is integrated term by term, the right-hand member of the equation forω(x) takes the formB11−B21+B31− ...,1·2x3·4x35·6x5This series is divergent; but, if it is stopped at any term, the difference between the sum of the series so terminated and the value ofω(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.
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) −Σnr = 0ƒr(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′, |Σmr = 0ƒr(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 + ½e−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),
whereω(x) is the function defined by the definite integral
ω(x) =∫∞0{(1 − e−t)−1− t−1− ½} t−1e−txdt.
The multiplier of e−txunder 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)−2mΣ∞r = 1(r−2m).
When the series is integrated term by term, the right-hand member of the equation forω(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 ofω(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.
26.Interchange of the Order of Limiting Operations.—When we require to perform any limiting operation upon a function which is itself represented by the result of a limiting process, the question of the possibility of interchanging the order of the two processes always arises. In the more elementary problems of analysis it generally happens that such an interchange is possible; but in general it is not possible. In other words, the performance of the two processes in different orders may lead to two different results; or the performance of them in one of the two orders may lead to no result. The fact that the interchange is possible under suitable restrictions for a particular class of operations is a theorem to be proved.
Among examples of such interchanges we have the differentiation and integration of an infinite series term by term (§ 22), and the differentiation and integration of a definite integral with respect to a parameter by performing the like processes upon the subject of integration (§ 19). As a last example we may take the limit of the sum of an infinite series of functions at a point in the domain of convergence. Suppose that the seriesΣ∞0ƒr(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), andSéries divergentes(Paris, 1901), alsoFonctions 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), andSerie 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), andDiff.- 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” inActa Math. tt. ii., vii., andMath. 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), andStetigkeit u. irrationale Zahlen(Brunswick, 1872); Dirichlet, “Convergence des séries trigonométriques,”Crelle, Bd. iv.; P. Du Bois Reymond,Allgemeine Functionentheorie(Tübingen, 1882), and many memoirs inCrelleand inMath. Ann.; Heine, “Functionenlehre,”Crelle, Bd. lxxiv.; J. Pierpont,The Theory of Functions of a real Variable(Boston, 1905); F. Klein, “Allgemeine Functionsbegriff,”Math. Ann. Bd. xxii.; W.F. Osgood, “On Uniform Convergence,”Amer. J. of Math. vol. xix.; Pincherle, “Funzioni analitiche secondo Weierstrass,”Giorn. di mat. t. xviii.; Pringsheim, “Bedingungen d. Taylorschen Lehrsatzes,”Math. Ann. Bd. xliv.; Riemann, “Trigonometrische Reihe,”Ges. Werke(Leipzig, 1876); Schoenflies, “Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten,”Jahresber. d. deutschen Math.-Vereinigung, Bd. viii.; Study, Memoir on “Functions with Restricted Oscillation,”Math. Ann. Bd. xlvii.; Weierstrass, Memoir on “Continuous Functions that are not Differentiable,”Ges. math. Werke, Bd. ii. p. 71 (Berlin, 1895), and on the “Representation of Arbitrary Functions,” ibid. Bd. iii. p. 1; W.H. Young, “On Uniform and Non-uniform Convergence,”Proc. London Math. Soc.(Ser. 2) t. 6. Further information and very full references will be found in the articles by Pringsheim, Schoenflies and Voss in theEncyclopädie der math. Wissenschaften, Bde. i., ii. (Leipzig, 1898, 1899).
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 seriesΣ∞0ƒr(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), andSéries divergentes(Paris, 1901), alsoFonctions 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), andSerie 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), andDiff.- 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” inActa Math. tt. ii., vii., andMath. 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), andStetigkeit u. irrationale Zahlen(Brunswick, 1872); Dirichlet, “Convergence des séries trigonométriques,”Crelle, Bd. iv.; P. Du Bois Reymond,Allgemeine Functionentheorie(Tübingen, 1882), and many memoirs inCrelleand inMath. Ann.; Heine, “Functionenlehre,”Crelle, Bd. lxxiv.; J. Pierpont,The Theory of Functions of a real Variable(Boston, 1905); F. Klein, “Allgemeine Functionsbegriff,”Math. Ann. Bd. xxii.; W.F. Osgood, “On Uniform Convergence,”Amer. J. of Math. vol. xix.; Pincherle, “Funzioni analitiche secondo Weierstrass,”Giorn. di mat. t. xviii.; Pringsheim, “Bedingungen d. Taylorschen Lehrsatzes,”Math. Ann. Bd. xliv.; Riemann, “Trigonometrische Reihe,”Ges. Werke(Leipzig, 1876); Schoenflies, “Entwickelung d. Lehre v. d. Punktmannigfaltigkeiten,”Jahresber. d. deutschen Math.-Vereinigung, Bd. viii.; Study, Memoir on “Functions with Restricted Oscillation,”Math. Ann. Bd. xlvii.; Weierstrass, Memoir on “Continuous Functions that are not Differentiable,”Ges. math. Werke, Bd. ii. p. 71 (Berlin, 1895), and on the “Representation of Arbitrary Functions,” ibid. Bd. iii. p. 1; W.H. Young, “On Uniform and Non-uniform Convergence,”Proc. London Math. Soc.(Ser. 2) t. 6. Further information and very full references will be found in the articles by Pringsheim, Schoenflies and Voss in theEncyclopädie der math. Wissenschaften, Bde. i., ii. (Leipzig, 1898, 1899).
(A. E. H. L.)
II—Functions of Complex Variables
In the preceding section the doctrine of functionality is discussed with respect to real quantities; in this section the theory when complex or imaginary quantities are involved receives treatment. The following abstract explains the arrangement of the subject matter: (§ 1),Complex numbers, states what a complex number is; (§ 2),Plotting of simple expressions involving complex numbers, illustrates the meaning in some simple cases, introducing the notion of conformal representation and proving that an algebraic equation has complex, if not real, roots; (§ 3),Limiting operations, defines certain simple functions of a complex variable which are obtained by passing to a limit, in particular the exponential function, and the generalized logarithm, here denoted by λ(z); (§ 4),Functions of a complex variable in general, after explaining briefly what is to be understood by a region of the complex plane and by a path, and expounding a logical principle of some importance, gives the accepted definition of a function of a complex variable, establishes the existence of a complex integral, and proves Cauchy’s theorem relating thereto; (§ 5),Applications, considers the differentiation and integration of series of functions of a complex variable, proves Laurent’s theorem, and establishes the expansion of a function of a complex variable as a power series, leading, in (§ 6),Singular points, to a definition of the region of existence and singular points of a function of a complex variable, and thence, in (§ 7),Monogenic Functions, to what the writer believes to be the simplest definition of a function of a complex variable, that of Weierstrass; (§ 8),Some elementary properties of single valued functions, first discusses the meaning of a pole, proves that a single valued function with only poles is rational, gives Mittag-Leffler’s theorem, and Weierstrass’s theorem for the primary factors of an integral function, stating generalized forms for these, leading to the theorem of (§ 9),The construction of a monogenic function with a given region of existence, with which is connected (§10),Expression of a monogenic function by rational functions in a given region, of which the method is applied in (§ 11),Expression of(1 − z)−1by polynomials, to a definite example, used here to obtain (§ 12),An expansion of an arbitrary function by means of a series of polynomials, over a star region, also obtained in the original manner of Mittag-Leffler; (§ 13),Application of Cauchy’s theorem to the determination of definite integrals, gives two examples of this method; (§ 14),Doubly Periodic Functions, is introduced at this stage as furnishing an excellent example of the preceding principles. The reader who wishes to approach the matter from the point of view of Integral Calculus should first consult the section (§ 20) below, dealing withElliptic Integrals; (§ 15),Potential Functions, Conformal representation in general, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (§ 16),Multiple-valued Functions, Algebraic Functions, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (§ 17),Integrals of Algebraic Functions, enunciating Abel’s theorem; (§ 18),Indeterminateness of Algebraic Integrals, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (§ 19),Reversion of an algebraic integral, mentions a problem considered below in detail for an elliptic integral; (§ 20),Elliptic Integrals, considers the algebraic reduction of any elliptic integral to one of three standard forms, and proves that the function obtained by reversion is single-valued; (§ 21),Modular Functions, gives a statement of some of the more elementary properties of some functions of great importance, with a definition of Automorphic Functions, and a hint of the connexion with the theory of linear differential equations; (§ 22),A property of integral functions, deduced from the theory of modular functions, proves that there cannot be more than one value not assumed by an integral function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the periods; (§ 23),Geometrical applications of Elliptic Functions, shows that any plane curve of deficiency unity can be expressed by elliptic functions, and gives a geometrical proof of the addition theorem for the function ℜ(u); (§ 24),Integrals of Algebraic Functions in connexion with the theory of plane curves, discusses the generalization to curves of any deficiency; (§ 25),Monogenic Functions of several independent variables, describes briefly the beginnings of this theory, with a mention of some fundamental theorems: (§ 26),Multiply-Periodic Functions and the Theory of Surfaces, attempts to show the nature of some problems now being actively pursued.
Beside the brevity necessarily attaching to the account here given of advanced parts of the subject, some of the more elementary results are stated only, without proof, as, for instance: the monogeneity of an algebraic function, no reference being made, moreover, to the cases of differential equations whose integrals are monogenic; that a function possessing an algebraic addition theorem is necessarily an elliptic function (or a particular case of such); that any area can be conformally represented on a half plane, a theorem requiring further much more detailed consideration of the meaning ofareathan we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.
§ 1.Complex Numbers.—Complex numbers are numbers of the form x + iy, where x, y are ordinary real numbers, and i is a symbol imagined capable of combination with itself and the ordinary real numbers, by way of addition, subtraction, multiplication and division, according to the ordinary commutative, associative and distributive laws; the symbol i is further such that i² = −1.
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.
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.
§ 2.Plotting and Properties of Simple Expressions involving a Complex Number.—If we put ζ = (z-i)/(z + i), and, putting ζ = ξ + iη, take a new plane upon which ξ, η are rectangular co-ordinates, the equations ξ= (x² + y²− 1)/[x² + (y + 1)²], η = −2xy/[x² + (y + i)²] will determine, corresponding to any point of the first plane, a point of the second plane. There is the one exception of z = −i, that is, x = 0, y = −1, of which the corresponding point is at infinity. It can now be easily proved that as z describes the real axis in its plane the point ζ describes once a circle of radius unity, with centre at ζ = 0, and that there is a definite correspondence of point to point between points in the z-plane which are above the real axis and points of the ζ-plane which are interior to this circle; in particular z = i corresponds to ζ = 0.
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∂ξ=∂η,∂ξ= −∂η,∂²ξ+∂²ξ= 0;∂x∂x′∂y∂x′∂x²∂y²so 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 ]1/2varies 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 ζ0be the value of ζ corresponding to (x0, y0), and the expression of ζ − ζ0near 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 z1and 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 ζ0is 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.
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 ]1/2varies 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 ζ0be the value of ζ corresponding to (x0, y0), and the expression of ζ − ζ0near 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 z1and 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 ζ0is 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.
§ 3.Limiting Operations.—In order that a complex number ζ = ξ + iη may have a limit it is necessary and sufficient that each of ξ and η has a limit. Thus an infinite series w0+ w1+ w2+ ..., whose terms are complex numbers, is convergent if the real series formed by taking the real parts of its terms and that formed by the imaginary terms are both convergent. The series is also convergent if the real series formed by the moduli of its terms is convergent; in that case the series is said to be absolutely convergent, and it can be shown that its sum is unaltered by taking the terms in any other order. Generally the necessary and sufficient condition of convergence is that, for a given real positive ε, a number m exists such that for every n > m, and every positive p, the batch of terms wn+ wn+1+ ... + wn+pis less than ε in absolute value. If the terms depend upon a complex variable z, the convergence is calleduniformfor a range of values of z, when the inequality holds, for the same ε and m, for all the points z of this range.