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have

a=ƒ_ƒ(x)dx, a2=1ƒ_ƒ(x) cos"**dx, b.= f(x) sin***dx.

The interval between-c and c may be called the periodic interval," and we may replace it by any other interval, e.g. that between o and 1, without any restriction of generality. When

this is done the sum of the series takes the form

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

nx

For example, the series sin x- sin 2x+ sin 3x-...is convergent for all real values of x, and, when >x>- its sum' is x; but, when x is but a little less than , the number of terms which must be taken in order to bring the sum at all near to the value of x is very large, and this number tends to increase indefinitely as x approaches T. This series does not converge uniformly in the neighbourhood of x=. Another example is afforded by the series (n+1)x n=on2x2+1 (n+1)2+1, of which the remainder after ʼn terms is nx/(n2x2+1). If we put x=1/n, for any value of n, however great, the remainder is; and the number of terms required to be taken to make the remainder tend to zero depends upon the value of 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_2_ƒ,(x)=f(x) converges uniformly in the interval between a and b

7-0

Sf(x)dx-f.fr(x)dx,

or a series which converges unformly may be integrated term by

term.

(6) If £ f'r(x) converges uniformly in an interval, then

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f(x) converges in the interval, and represents a continuous differentiable function, (x); in fact we have

$'(x) =
20f'r(x),

or a series can be differentiated term by term if the series of derived functions converges uniformly.

0

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

L'2" f(8) COS|2rπ(3—x)} dz,

uS's ((2) sin ((2n+1)(z = x) x} dz.

(2) sin (=-x)}

(ii.)

Li Fourier's theorem is that, if the periodic interval can be divided into a finite number of partial intervals within each of which the function is ordinary (§ 14), the series represents the function within each of those partial intervals. In Fourier's time a function of this character was regarded as completely arbitrary. By a discussion of the integral (ii.) based on the Second Theorem of the Mean (§ 15) it can be shown that, if f(x) has restricted oscilla. tion in the interval (§ 11), the sum of the series is equal to f(x+0)+ f(x-0)) at any point x within the interval, and that it is equal to f(+0)+f(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 f(x) is continuous, the series converges uniformly, and that no series of the form (i), with coefficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same periodic interval. The result can be extended to a function f(x) which tends to become infinite at a finite number of points a of the at each of the points a, (2) the improper definite integral of f(x) interval, provided (1) f(x) tends to become determinately infinite through the interval is convergent, (3) f(x) has not an infinite number of discontinuities or of maxima or minima in the interval.

24. Representation of Continuous Functions by Series.-If the series for f(x) formed by Fourier's rule converges at the point a of the periodic interval, and if f(x) is continuous at a, the sum of the series is f(a); but it has been proved by P. du Bois Reymond that the function may be continuous at a, and yet the series formed by Fourier's rule may be divergent at a. Thus some continuous functions do not admit of representation by Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either of two forms, which may be described as " terminated Fourier's series" and "terminated power series," according to the two following theorems:

(1) If f(x) is continuous throughout the intervai between o and 27, and if any positive number however small is specified, it is possible to find an integer n, so that the difference between the value of f(x) and the sum of the first n terms of the series for f(x), formed by Fourier's rule with periodic interval from o to 2, shall be less than e at all points of the interval. This result can be extended to a function which is continuous in any given interval.

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

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

23. Fourier's Series.-An extensive class of functions admit functions have also been devised. of being represented by series of the form

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(i.) and the rule for determining the coefficients a, b, of such a series, in order that it may represent a given function f(x) in

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and that the continuous function f(x) represented by it has the property that there is, in the neighbourhood of any point xs, an infinite aggregate of points x', having xo as a limiting point, for which f(x)-f(x)}/(x-x) tends to become infinite with one sign when x-xo approaches zero through positive values, and infinite with the opposite sign when x-xe approaches zero through negative values. Accordingly the function is not differentiable at any point. The definite integral of such a function f(x) through the interval between a fixed point and a variable point x, is a continuous differentiable function F(x), for which F(x)=f(x); and, if f(x) is one-signed throughout any interval F(x) is monotonous throughout that interval, but yet F(x) cannot be represented by a curve. In any interval, however small, the tangent would have to take the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, it can be shown that all functions which are everywhere continuous and nowhere differentiable are capable of representation by series of the form Za..(x), where 2a, is an absolutely convergent series of numbers, and (x) is an analytic function whose absolute value never exceeds unity.

I

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-+}-. . . In .neral, let f(x)+fi(x)+..... be a series of functions which does not converge in a certain domain. It may happen that, if any number e, however small, is first specified, a number n can afterwards be found so that, at a point a of the domain, the value f(a) of a certain function f(x) is connected with the sum of the first n+1 terms of the series by the relation [ƒ(a) – £ƒ,(a)|<e. 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 '. 2f.(a)|>N. The divergent series fo(x)+f,(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.

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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.

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

AUTHORITIES. Among the more important treatises and memoirs connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905); O. Biermann, Analytische Functionen (Leipzig, 1887); E. Borel, 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); Theory of Fourier's Series (Cambridge, 1907); C. Jordan, Cours E. W. Hobson, The Theory of Functions of a real Variable and the 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

The multiplier of es under the sign of integration can be expanded (Paris, 1886); W. H. and G. C. Young, The Theory of Sets of Points in the power series

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in Acta

(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 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

When the series is integrated term by term, the right-hand member des séries trigonométriques," Crelle, Bd. iv.; P. Du Bois Reymond, of the equation for (x) takes the form

BI BI B, 1

+

1.2 x 3.4 x 5.6 x3

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Allgemeine Functionentheorie (Tübingen, 1882), and many memoirs in Crelle and in Math. Ann.; Heine, Functionenlehre," Crelle, Bd. Ixxiv.; 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 Differ entiable," 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.)

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 (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 asymp-
totic 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 E. Borel. The great merit of
asymptotic expansions is that they admit of addition, subtraction,
multiplication and division, term by term, in the same way as
absolutely convergent series, and they admit also of integration
term by term; that is to say, the results of such operations are

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II.-FUNCTIONS OF COMPLEX VARIABLES

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 (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 arca 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 =-1.

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(s); (§ 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; (8 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 (89), 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 (1−2) ̄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-variable 2=x+iy, the distance OP be called r, and the positive 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 § 2. Plotting and Properties of Simple Expressions involving associated with an algebraic integral, establishing that for an a Complex Number.-If we put }=(2−i)/(z+i), and, putting elliptic integral the number of these is two; (§ 19), Reversion of 5+in, take a new plane upon which, are rectangu an algebraic integral, mentions a problem considered below in lar co-ordinates, the equations = (x+1)/[x2+(y+1)3], detail for an elliptic integral; (§ 20), Elliptic Integrals, considers n = −2xy/[x2+(y+1)2] will determine, corresponding to any the algebraic reduction of any elliptic integral to one of three point of the first plane, a point of the second plane. There is standard forms, and proves that the function obtained by the one exception of z=-i, that is, x=o, y=-1, of which the reversion is single-valued; (§ 21), Modular Functions, gives a corresponding point is at infinity. It can now be easily proved statement of some of the more elementary properties of some that as z describes the real axis in its plane the point describes functions of great importance, with a definition of Automorphic once a circle of radius unity, with centre at =o, and that there Functions, and a hint of the connexion with the theory of linear is a definite correspondence of point to point between points differential equations; (§ 22), A property of integral functions, in the z-plane which are above the real axis and points of the deduced from the theory of modular functions, proves that there-plane which are interior to this circle; in particular 2=i cannot be more than one value not assumed by an integral corresponds to } = 0. function, and gives the basis of the well-known expression of the modulus of the elliptic functions in terms of the ratio of the

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=0, 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 angle less than 2 between Ox and OP be called , we may write z=r(cos + sin 0); then is called the modulus or absolute value of 2 and often denoted by and is called the phase or amplitude of 2, and often denoted by ph (3); strictly the phase is ambiguous by additive multiples of 2. If '=x'+iy be represented by P', the complex argument +2 is represented by a point P obtained by drawing from P' a line equal to and parallel to OP; the geoof the plane; as, for instance, the equation +2=2+2 involves metrical representation involves for its validity certain properties 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 moduli, and is greater than (or equal to) the difference of their generally less than (and is never greater than) the sum of their moduli; the former statement thus holds for the sum of any number of complex numbers. We shall write E(10) for cos +i sin ; it is at once verified that E(ia). E(is) =E[i(a+8)], so that the phase of a product of complex quantities is obtained by addition of their respective phases.

Moreover, being a rational function of 2, both έ and are continuous differentiable functions of x and y, save when is infinite;

writing } = f(x, y) = f(z—iy, y), the fact that this is really independent be in absolute value less than a real positive quantity M, it can be of y leads at once to aflax+iaflay=o, and hence to

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=0;

so that is not any arbitrary function of x, y, and when & is known ཟླ is determinate save for an additive constant. Also, in virtue of these equations, if , be the values of corresponding to two near values of z, say z and z', the ratio (-5)/(-2) has a definite limit when z=2, independent of the ultimate phase of 2-2, this limit being therefore equal to 05/0x, that is, a/dx+id/dx. Geometrically this fact is interpreted by saying that if two curves in the -plane intersect at a point P, at which both the differential coefficients ag/ax, əŋ/əx are not zero, and P', P" be two points near to P on these curves respectively, and the corresponding points of the J-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'00, (3) the rotation from PP' to PP" is in the same sense as from QQ' to QQ", it being understood that the axes of E, 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 5-plane with the same angles; the magnification,

however, which is equal to [(3)+(35)] 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 is a polynomial 5=Poz" + P12n-1+...+kui

in 2,

H being an arbitrary real positive number, it can be shown that a radius R can be found such for every ||> R we have||> H; consider the lower limit of |} | for |2| <R; as + is a real continuous function of x, y for z <R, there is a point (x, y), say (x, y), at which is least, say equal to p, and therefore within a circle in the -plane whose centre is the origin, of radius p, there are no points representing values corresponding to | z|<Ŕ. But if to be the value of corresponding to (x, y), and the expression of - near zo=x+1ye, in terms of 2-20, be A(-20) + B(-2)+1+..., where A is not zero, to two points near to (xo, yo),

say (11, 31) or ♬ and 2=20+(-20) (costi sin), will corre

spond two points near to fe, say 1, and 250-'1, situated so that fo is between them. One of these must be within the circle (p). We infer then that po, and have proved that every polynomial in z vanishes for some value of 2, and can therefore be written as a product of factors of the form z-a, where a denotes a complex number. This proposition alone suffices to suggest the importance of complex numbers.

...

3. Limiting Operations.-In order that a complex number =+in may have a limit it is necessary and sufficient that each of and has a limit. Thus an infinite series wo+w1+we+ 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 e, a number m exists such that for every. >m, and every positive p, the batch of terms w2+wa+1+ ++, is less than e 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 e and m, for all the points z of this range.

The infinite series of most importance are those of which the general term is a,, wherein a, is a constant, and z is regarded as variable, no, 1, 2, 3,... Such a series is called a power series. If a real and positive number M exists such that for z=20 and every <M, a condition which is satisfied, for instance, if the series converges for z=zo, then it is at once proved that the series converges absolutely for every z for which ||<20, and converges uniformly over every range || <' for which '< to 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 is, however, continuous with the sum for a point zo on the circumference, as z approaches to so provided the series converges for z=20, as can be shown without much difficulty. Within a common circle of convergence two power series Zanz, Ebaz" can be multiplied together according to the ordinary rule, this being a consequence of a theorem for absolutely convergent series. If 7 be less than the radius of convergence of a series Ea," and for ||=7, the sum of the series

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shown that for[]=7, every term is also less than M in absolute value, namely, a <M. If in every arbitrarily small neighbourhood of z=o there be a point for which two converging power series Ea.2", Ebaz" agree in value, then the series are identical, or a,b,; thus also if Za 2 vanish at z=o there is a circle of finite radius about so as centre within which no other points are found for which the sum of the series is zero. Considering a power series f(z) = Zanz" of radius of the resulting series Ea.(20+) may be regarded as a double series convergence R, if <R and we put z=2+ with <R-1201. in to and t, which, since |zo|+<R, is absolutely convergent; it may then be arranged according to powers of t. Thus we may write f(2) =EA"; hence Aof(20), and we have [f(0+1)-f(20)]/t= Ant, wherein the continuous series on the right reduces to A for 1=0; thus the ratio on the left has a definite limit when t=0, equal namely to A, or Ena,zo In other words, the original series may legitimately be differentiated at any interior point zo of its circle of convergence. Repeating this process we find f(z+1) = 2!"f")(zo)/!!!, where f(zo) is the nth differential coefficient. Repeating for this power series, in t, the argument applied about 2=0 for La., we infer that for the series f(z) every point which reduces it to zero is an isolated point, and of such points only a finite number lie within a circle which is within the circle of convergence of f(z). 3/31+ of which the radius of convergence is infinite. By Perhaps the simplest possible power series'is e = exp (2) = 1 +22/2! + multiplication we have exp (2).exp (21) = exp (2+2). In particular when x, y are real, and z=x+iy, exp (z) = exp (x) exp (iy). Now the functions Uosin y, Vo=1-cos y, U1-y-sin y,

V, 4y2-1+cos y, U2=y-y+sin y, V2-y-y+1-cos y.... all vanish for y=o, and the differential coefficient of any one after increasing when its differential coefficient is positive, we infer, for the first is the preceding one; as a function (of a real variable) is y positive, that each of these functions is positive; proceeding to a limit we hence infer that

cos y=1-+-..., sin y-y-ly3+loys — ..., 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 (2) is exp (x) and the phase is y. Hence also exp (2+2xi) = exp (x)[cos (y+2x)+i sin (y+2′′)}, which we express by saying that exp (2) has the period 2ri, and hence also the period akai, where k is an arbitrary integer. From the fact that the constantly increasing function exp (x) can vanish only for x=o, we at once prove that exp (2) has no other periods.

Taking in the plane of z an infinite strip lying between the lines Yo, y=2 and plotting the function exp (2) upon a new plane, It follows at once from what has been said that every complex value of arises when z takes in turn all positions in this strip, and that no value arises twice over. The equation exp (2) thus defines 2, regarded as depending upon, with only an additive ambiguity 2kri, where k is an integer. We write = X(); when 5 is real this becomes the logarithm of ; in general X(5) = log || +i ph (5)+ 2kri, where k is an integer; and when describes a closed circuit surrounding the origin the phase of increases by 27, or k increases by unity. Differentiating the series for we have dy/dz-, so that 2, regarded as depending upon, is also differentiable, with did. On the other hand, consider the series -1- (5 −1)2 + } ( − 1 ) 3 — . . . ; it converges when =2 and hence converges for -11; its differential coefficient is, however, 1-(-1)+ (3 − 1)2 — .., that is, (1-1). Wherefore if (t) denote this series, for 1 <1, the difference (5)-(5), regarded as a take the value of X() which vanishes when 1 we infer thence function of and . has vanishing differential coefficients; if we (-1)". It is to be remarked that for 15-1] <1, λ(3) = 2· that it is impossible for while subject to |-1|<1 to make a circuit about the origin. For values of 5 for which 1-111, we can also calculate A() with the help of infinite series, utilizing the fact that λ(5') =λ(3) +λ(5′).

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The function () is required to define when and a are complex numbers; this is defined as exp [ax(5)], that is as a"[X()}"/n!. When a is a real integer the ambiguity of A() is immaterial here, where a is a positive integer, there are q values possible for 14, of since exp [ax()+2kari]=exp [ax(5)]; when a is of the form 1/9,

the form exp •[()]exp (2), with k=o, 1,...q-1, all other

values of k leading to one of these; the gth power of any one of these values is ; when a p/q, where p q are integers without common factor, 9 being positive, we have = (1/4)P. The definition of the symbol 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 ; the number i is of modulus unity and phase; thus λ(i) =i(}+2k); thus

i=exp(--2kx)=cxp (−1) exp (−2kx),

is always real, but has an infinite number of values.

The function exp (2) is used also to define a generalized form of the cosine and sine functions when s is complex; we write, namely, cos z=[exp (is) + exp (−iz)] and sin z=i[exp (iz) —exp (−iz)]. It will be found that these obey the ordinary relations holding when s 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 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.

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definite finite real value attached to every interior or boundary point of the region, say f(x,y). It may have a finite upper limit H for the region, so that no point (x,y) exists for which f(x,y) > H, but points (x,y) exist for which f(x,y)> H-e, however smalle may be; if not we say that its upper limit is infinite. There is then at Icast one point of the region such that, for points of the region within small the radius of the circle be taken; for if not we can put about a circle about this point, the upper limit of f(x,y) is H, however every point of the region a circle within which the upper limit of f(x,y) is less than H; then by the result (B) above the region consists of a finite number of sub-regions within each of which the upper limit is less than H; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement holds for the lower limit. A case of such a function f(x,y) is the radius ro of the neighbourhood proper to any point so, spoken of above. We can hence prove the statement (A) above. lower limit of ro is zero. Let then be a point such that the lower Suppose the property (8,20) extensive, and, if possible, that the limit of 70 is zero for points zo within a circle about however small; let r be the radius of the neighbourhood proper to ; take so that -r; the property (2,2), being extensive, holds Consider a square of side a, to whose perimeter is attached a within a circle, centre zo, of radius 7-120-31, which is greater definite direction of description, which we take to be counterthan -, and increases to r as 20-I diminishes; this being clockwise; another square, also of side a, may be added to this, so true for all points 20 near , the lower limit of ro is not zero for the that there is a side common; this common side being erased we neighbourhood of , contrary to what was supposed. This proves have a composite region with a definite direction of perimeter; (A). Also, as is here shown that ro-2-1, may similarly be shown that rro-120-1. Thus ro differs arbitrarily little from 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; ously with zo 7 when 0-1 is sufficiently small; that is, ro varies continuand this common side may be erased. If this process be continued Next suppose the function f(x,y), which has a definite finite value at every point of the region considered, to be any number of times we obtain a region of the plane bounded by one continuous but not necessarily real, so that about every point z or more polygonal closed lines, no two of which intersect; and at each portion of the perimeter there is a definite direction of descrip- within or upon the boundary of the region, being an arbitrary real tion, which is such that the region is on the left of the describing for all points z of the region interior to this circle, we have positive quantity assigned beforehand, a circle is possible, so that point. Similarly we may construct a region by piecing together triangles, so that every consecutive two have a side in common, f(x,y)-f(x,y) <n, and therefore (x,y) being any other point it being understood that there is assigned an upper limit for the interior to this circle,|f(x,y')−f(x,y)[<n. We can then apply the result (A) obtained above, taking for the neighbourhood proper 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 to any point 20 the circular area within which, for any two points by lines through its centre parallel to its sides; in the latter method (x,y),(x,y), we have f(x',y')—f(x,y)|<n. This is clearly an each triangle may be divided into four others by lines joining the extensive property. Thus, a number is assignable, greater than middle points of its sides; this halves the sides and preserves the zero, such that, for any two points (x,y), (xy) within a circle angles. When we speak of a region of the plane in general, unlessr about any point 20, we have f(x,y′)—f(x,y) \<n. the contrary is stated, we shall suppose it capable of being generated and, in particular, f(x,y)-f(xo, yo), where n is an arbitrary in this latter way by means of a finite number of triangles, there real positive quantity agreed upon beforehand. 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 of a certain region a circle can be put, say of radius ro, such that for all points z of the region which are interior to this circle, for which, that is, 12-<70, a certain property holds. Assuming that to ro is given the value which is the upper limit for 2, of the possible values, we may call the points 13-201<ro, the neighbourhood belonging to or proper too, and may speak of the property as the property (2,20). The value of ro will in general vary with so; what is in most cases of importance is the question whether the lower limit of ro for all positions is zero or greater than zero. (A) This lower limit is certainly greater than zero provided the property (2,20) is of a kind which we may call extensive; such, namely, that if it holds, for some position of zo and all positions of 2, within a certain region, then the property (2,2) holds within a circle of radius R about any interior point of this region for all points for which the circle 12-R is within the region. Also in this case ro varies continuously with zo. (B) Whether the property is of this extensive character or not we can prove that the region can be divided into a finite number of sub-regions such that, for every one of these, the property holds, (1) for some point 2, within or upon the boundary of the sub-region, (2) for every point s 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 (2,20) holds for all points 2 and some point 20of the region, be called suitable: if each of the triangles of which the region is built up be suitable, what is desired is proved; if not let an unsuitable triangle be subdivided into four, as before explained; if one of these subdivisions is unsuitable let it be again subdivided; and so on. Either the process terminates and then what is required is proved; or else we obtain an indefinitely continued sequence of unsuitable triangles, each contained in the preceding, which converge to a point, say ; after a certain stage all these will be interior to the proper region of this, however, is contrary to the supposition that they are all unsuitable.

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Take now any path in the region, whose extreme points are zo, z, and let 1,... be intermediate points of the path, in order: denote the continuous function f(x,y) by f(z), and let f, denote any quantity such that | fr-f()| = 1ƒ(2+1)-f(2) |; consider the sum (21-20)fo+(22-21)f1+...+(3-2-1)fm-1. By the definition of a path we can suppose, n being large enough, that the intermediate points 21,... are so taken that if 2. be any two points intermediate, in order, to 2, and 21, we have 2641—8i | < | &+1=1; we can thus suppose - 201, 12-21.... 2-2-1 (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, 2, 1, 2, 2,... 2r,m-1 be intermediate in order to and 41, and ||fr-f(2.) |<[ƒ(2r,i+1)—ƒ(2,1), the difference between Z(441-4)fi and which is equal to {(2,1-2)fr.0+(2,3-2,1) fr.1+...+(24+1−2,m−1)fr.m-1}.

We now make some applications of this result (B). Suppose a

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is, when +1-21 is small enough, to ensure |f(2+1)−ƒ(3) | < n. less in absolute value than

E2n2|2,1+1=2,i|, which, if S be the upper limit of the perimeter of the polygon from which the path is generated, is < 27S, and is therefore arbitrarily small.

The limit in question is called fif(e)ds. In particular when f(z) =1, it is obvious from the definition that its value is zvalue is (22-03); these results will be applied immediately. when f(x)=2, by taking f,(+1-2), it is equally clear that its Suppose now that to every interior and boundary point of a such that, whatever real positive quantity may be, a real positive certain region there belong two definite finite numbers ƒ(20). F(2) number e exists for which the condition f(3) − f(0) − F(20) | < n.

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2-20

which we describe as the condition (2,20), is satisfied for every point within or upon the boundary of the region, satisfying the limitatio 2-20] <e. Then f(zo) is called a differentiable function of the complex variable 20 over this region, its differential coefficient bein F(s). The function f() is thus a continuous function of the re

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