VOLUME XI SLICE III Frost to Fyzabad (continued 6)

19. Differentiation and Integration.—The definition of partial differentiation of a function of several variables presents no difficulty. The most important theorems concerning differentiable functions are the “theorem of the total differential,” the theorem of the interchangeability of the order of partial differentiations, and the extension of Taylor’s theorem (see Infinitesimal Calculus).

With a view to the establishment of the notion of integration through a domain, we must define the “extent” of the domain. Take first a domain consisting of the point a and all the points x for which |x − a| < ½h, where h is a chosen positive number; the extent of this domain is hn, n being the number of variables; such a domain may be described as “square,” and the number h may be called its “breadth”; it is a homogeneous part of the numerical continuum of n dimensions, and its boundary consists of all the points for which |x − a| = ½h. Now the points of any domain, which does not extend to an infinite distance, may be assigned to a finite number m of square domains of finite breadths, so that every point of the domain is either within one of these square domains or on its boundary, and so that no point is within two of the square domains; also we may devise a rule by which, as the number m increases indefinitely, the breadths of all the square domains are diminished indefinitely. When this process is applied to a homogeneous part, H, of the numerical continuum Cn, then, at any stage of the process, there will be some square domains of which all the points belong to H, and there will generally be others of which some, but not all, of the points belong to H. As the number m is increased indefinitely the sums of the extents of both these categories of square domains will tend to definite limits, which cannot be negative; when the second of these limits is zero the domain H is said to be “measurable,” and the first of these limits is its “extent”; it is independent of the rule adopted for constructing the square domains and contracting their breadths. The notion thus introduced may be adapted by suitable modifications to continua of lower dimensions in Cn.

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 sn of the first n terms of the series is a function of x and n; and, when the series is convergent, its sum, which is 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

∫ ∞0 e−t tx−1 dt,

or by the infinite product

Ltn=∞ nx/x(1 + x)(1 + ½x) ... ( 1 + x ).

n − 1

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)

ƒ(x) = ƒ(a) + Σn−1r=1 (x − a)r ƒ(r)(a) + Rn;

r!

and this becomes

ƒ(x) = ƒ(a) + Σ∞r=1 (x − a)r ƒ(r)(a),

r!

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 remainder Rn, viz. [(x − a) / (n − 1)!] (1 − θ)n−1 ƒn {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=0 an (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.

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) at n = ∞. 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 + 1⁄3 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

Σ∞n=0 nx − (n + 1)x ,

n²x² + 1 (n + 1)²x² + 1

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) If Σ∞r=0 ƒr(x) = ƒ(x) converges uniformly in the interval between a and b

∫ ba ƒ(x)dx = Σbr=0 ∫ ba ƒr(x)dx,

or a series which converges uniformly may be integrated term by term.

(6) If Σ∞r=0 ƒ′r(x) converges uniformly in an interval, then Σ∞r=0 ƒr(x) converges in the interval, and represents a continuous differentiable function, φ(x); in fact we have

φ′(x) = Σ∞r=0 ƒ′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. If Σ∞r=0 ƒr(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 is Σ∞r=0 Lt x=a ƒr(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

a0 + Σ∞n=1 ( an cos nπx + bn sin nπx ),

c c

and the rule for determining the coefficients an, bn of 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

a0 = 1 ∫ c−c ƒ(x)dx, an= 1 ∫ c−c ƒ(x)cos nπx dx, bn= 1 ∫ c−c sin nπx dx.

2c c c c c

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

Lt n=∞ ∫ 10 Σr = nr = −n ƒ(z) cos {2rπ(z − x)}dz,

and this is

Lt n=∞ ∫ 10 ƒ(z) sin {(2n + 1) (z − x)π} dz.

sin {(z − x)π}

(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 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.

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 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.

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) − 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

B1 − B2 t2 + B3 t4 − ...,

2! 4! 6!

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

B1 1 − B2 1 + B3 1 − ...,

1·2 x 3·4 x3 5·6 x5

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.

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

Among examples of such interchanges we have the differentiation and integration of an infinite series term by term (§ 22), and the differentiation and integration of a definite integral with respect to a parameter by performing the like processes upon the subject of integration (§ 19). As a last example we may take the limit of the sum of an infinite series of functions at a point in the domain of convergence. Suppose that the series 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).

(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)−1 by polynomials, to a definite example, used here to obtain (§ 12), An expansion of an arbitrary function by means of a series of polynomials, over a star region, also obtained in the original manner of 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 with Elliptic Integrals; (§ 15), Potential Functions, Conformal representation in general, gives a sketch of the connexion of the theory of potential functions with the theory of conformal representation, enunciating the Schwarz-Christoffel theorem for the representation of a polygon, with the application to the case of an equilateral triangle; (§ 16), Multiple-valued Functions, Algebraic Functions, deals for the most part with algebraic functions, proving the residue theorem, and establishing that an algebraic function has a definite Order; (§ 17), Integrals of Algebraic Functions, enunciating Abel’s theorem; (§ 18), Indeterminateness of Algebraic Integrals, deals with the periods associated with an algebraic integral, establishing that for an elliptic integral the number of these is two; (§ 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 of area than we have given; while the character and properties, including the connectivity, of a Riemann surface have not been referred to. The theta functions are referred to only once, and the principles of the theory of Abelian Functions have been illustrated only by the developments given for elliptic functions.

§ 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.

§ 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 ∂y²

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.

§ 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+p is less than ε in absolute value. If the terms depend upon a complex variable z, the convergence is called uniform for a range of values of z, when the inequality holds, for the same ε and m, for all the points z of this range.

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.

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

Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be 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

| ƒ(z) − ƒ(z0) − F(z0) | < η,

z − z0

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

F(z0) = lim. ƒ(z) − ƒ(z0) = 1 ∫ dtƒ(t)

z − z0 2πi (t − z0)²

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.

§ 5. Applications.—The previous results have manifold applications.

(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

∫ zz0 S(z)dz = ∫ zz0 u0(z)dz + ∫ zz0 u1(z)dz + ... + ∫ zz0 un−1(z)dz + ∫ zz0 Rn(z)dz,

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,

ƒ(z0) = 1 ∫ ƒ(t)dt − 1 ∫ ƒ(t)dt ,

2πi Rt − z0 2πi rt − z0

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.

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

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.

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

Starting with a convergent power series, say in powers of z, this series can be arranged in powers of z − 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.

§ 8. Some Elementary Properties of Single Valued Functions.—A pole is a singular point of the function ƒ(z) which is not a singularity of the function 1/ƒ(z); this latter function is therefore, by the definition, capable of representation about this point, z0, by a series [ƒ(z)]−1 = Σan(z − z0)n. If herein a0 is not zero we can hence derive a representation for ƒ(z) as a power series about z0, contrary to the hypothesis that z0 is a singular point for this function. Hence a0 = 0; suppose also a1 = 0, a2 = 0, ... am−1 = 0, but am ± 0. Then [ƒ(z)]−1 = (z − z0)m[am + am+1 (z − z0) + ...], and hence (z − z0)mƒ(z) = am−1 + Σbn (z − z0)n, namely, the expression of ƒ(z) about z = z0 contains a finite number of negative powers of z − z0 and a (finite or) infinite number of positive powers. Thus a pole is always an isolated singularity.

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.

A single valued function is said to be an integral function when it has no singular points except z = ∞. Such is, for instance, an integral polynomial, which has z = ∞ for a pole, and the functions exp (z) which has z = ∞ as an essential singularity. A function which has no singular points for finite values of z other than poles is called a meromorphic function. If it also have a pole at z = ∞ it is a rational function; for then, if a1, ... as be its finite poles, of orders m1; m2, ... ms, the product (z − a1)m1 ... (z − as) msƒ(z) is an integral function with a pole at infinity, capable therefore, for large values of z, of an expression (z−1)−m Σ r=0 ar(z−1)r; thus (z − a1)m1 ... (z − as)msƒ(z) is capable of a form Σ r=0 brzr, but z−m Σ r=0 brzr remains finite for z = ∞. Therefore br+1 = br+2 = ... = 0, andƒ(z) is a rational function.

If for a single valued function F(z) every singular point in the finite part of the plane is isolated there can only be a finite number of these in any finite part of the plane, and they can be taken to 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

sin πx = Π ∞−∞ { (1 − x ) exp ( x ) },

πx n n

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

1 = xeCx Π ∞n=1 { (1 + x ) exp ( − x ) },

Γ(x) n n

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).

The two theorems given above, the one, known as Mittag-Leffler’s theorem, relating to the expression as a sum of simpler functions of a function whose singular points have the point z = ∞ as their only limiting point, the other, Weierstrass’s factor theorem, giving the expression of an integral function as a product of factors each with only one zero in the finite part of the plane, may be respectively generalized as follows:—

I. 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

Π ∞n=1 ( 1 − an − cn ) r n exp (gn),

z − cn

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

gn = rn Σ μ ns=1 1 ( an − cn ) s,

s z − cn

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.

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

§ 10. Expression of a Monogenic Function by means of Rational Functions in a given Region.—Suppose that we have a region R0 of the plane, as previously explained, for all the interior or boundary points of which z is finite, and let its boundary points, consisting of one or more closed polygonal paths, no two of which have a point in common, be called C0. Further suppose that all the points of this region, including the boundary points, are interior points of another region R, whose boundary is denoted by C. Let z be restricted to be within or upon the boundary of C0; let a, b, ... be finite points upon C or outside R. Then when b is near enough to a, the fraction (a − b)/(z − b) is arbitrarily small for all positions of z; say

| a − b | < ε, for |a − b| < η;

z − b

the rational function of the complex variable t,

1 [ 1 − ( a − b )n ],

t − a t − a

in which n is a positive integer, is not infinite at t = a, but has a pole at t = b. By taking n large enough, the value of this function, for all positions z of t belonging to R0, differs as little as may be desired from (t − a)−1. By taking a sum of terms such as

F = Σ Ap { 1 [ 1 − ( a − b ) n ] } p,

t − a t − b

we can thus build a rational function differing, in value, in R0, as little as may be desired from a given rational function

ƒ = Σ Ap(t − a)−p,

and differing, outside R or upon the boundary of R, from ƒ, in the fact that while ƒ is infinite at t = a, F is infinite only at t = b. By a succession of steps of this kind we thus have the theorem that, given a rational function of t whose poles are outside R or upon the boundary of R, and an arbitrary point c outside R or upon the boundary of R, which can be reached by a finite continuous path outside R from all the poles of the rational function, we can build another rational function differing in R0 arbitrarily little from the former, whose poles are all at the point c.

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

ƒ(z) = 1 ∫ ƒ(t)dt ,

2πi t − z

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

S = 1 Σ ƒ(ti) (ti+1 − ti) ,

2πi ti − z

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.