n=o. The convergence is said to be "uniform" in an interval | the interval between-c and c, was given by Fourier, viz, we is, after specification of e, the same number n suffices at all have (values of m which exceed n. The numbers n corresponding to any €, however small, are all finite, but, when e is less than some fixed | The interval between - c and c may be called the “periodic finite number, they may have an infinite superior limit ($ 7); interval,” and we may replace it by any other interval, e.g. that when this is the case there must be at least one point, a, of the between o and 1, without any restriction of generality. When interval which has the property that, whatever number N we this is done the sum of the series takes the form take, e can be taken so small that, at some point in the neigh Li bourhood of c, n must be taken > N to make 1/(x) - Sm(r)]<e 2 when m>n; then the series docs 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 Li f(z) dz. (ii.) finite; choose e first and allow a to vary, then the number n Fourier's theorem is that, if the periodic interval can be divided becomes a function of a, which may tend to become infinite, or into a finite number of partial intervals within each of which the may remain below a fixed number; if such a fixed number function is ordinary (8 14), the series represents the function exists, however small e may be, the convergence is uniform. within each of those partial intervals. In Fourier's time a For example, the series sin x-1 sin 2x+} sin 3x-... is conver- function of this character was regarded as completely arbitrary, gent for all real values of x, and, when >x>-* its sum'is š.m; By a discussion of the integral (ii.) based on the Second Theorem but, when x is but a little less than 7, the number of terms which of the Mean (§ 15) it can be shown that, if f(x) has restricted oscilla. must be taken in order to bring the sum at all near to the value of He is very large, and this number tends to increase indefinitely, as f(x-0) at any point x within the interval, and that it is equal to tion in the interval ($11), the sum of the series is cqualto i \|(x+o)+ w approaches a. This series does not converge uniformly in the histo)+(1-0) at each end of the interval. (See the article neighbourhood of x =. Another example is afforded by the series Fourier's Series.) It therefore represents the function at any 2 (n+1)x rompti an+1)x2 +1, of which the remainder after n terms point of the periodic interval at which the function is continuous (except possibly the end-points), and has a definite value at each is nx/(n*x*+1). If we put *= 1/", for any value of n, however point of discontinuity. The condition of restricted oscillation great, the remainder is and the number of terms required to be includes all the functions contemplated in the statement of the taken to make the remainder tend to zero depends upon the value of theorem and some others. Further, it can be shown that, in any x when x is near to zero-it must, in fact, be large compared with partial interval throughout which y(x) is continuous, the series is. The series does not converge unisormly in the neighbourhood converges uniformly, and that no series of the form (í), with coof x=0. efficients other than those determined by Fourier's rule, can represent the function at all points, except points of discontinuity, in the same As regards series whose terms represent continuous functions periodic interval. The result can be extended to a function f(x) we have the following theorems: which tends to become infinite at a finite number of points a of the (1) If the series converges uniformly in an interval it represents interval, provided (1) f(x) tends to become determinately infinite at each of the points a, (2) the improper definite integral of f(x) a function which is continuous throughout the interval. through the interval is convergent, (3) f(x) has not an infinite number (2) If the series represents a function which is discontinuous of discontinuities or of maxima or minima in the interval. in an interval it cannot converge uniformly in the interval. 24. Representation of Continuous Functions by Series.- If the (3) A series which does not converge uniformly in an interval series for }(x) formed by Fourier's rule converges at the point may nevertheless, represent a function which is continuouse of the periodic interval, and if f(x) is continuous at a, the throughout the interval. sum of the series is /(a); but it has been proved by P. du Bois (4) A power series converges uniformly in any interval con- Reymond that the function may be continuous at e, and yet the tained within its domain of convergence, the end-points being series formed by Fourier's rule may be divergent at c. Thus excluded. some continuous functions do not admit of representation by (5) If & fr(x)=f(x) converges uniformly in the interval Fourier's series. All continuous functions, however, admit of being represented with arbitrarily close approximation in either between a and b of two forms, which may be described as terminated Fourier's Ss(x)dx=, Saso(+)dx, series " and "terminated power series,” according to the two following theorems: or a series which converges unformly may be integrated term by (1) If |(x) is continuous throughout the interval between o and 27, and if any positive number e however small is specified, (6) If fr(x) converges uniformly in an interval, then 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 3 [y(x) converges in the interval, and represents a continuous for f(x), formed by Fourier's rule with periodic interval from differentiable function, $(x); in fact we have o to 27, shall be less than e at all points of the interval. This result can be extended to a function which is continuous in any $'(x)=, 351() given interval. or a series can be differentiated term by term if the series of (2) If y(x) is continuous throughout an interval, and any derived functions converges unisormly. positive number e however small is specified, it is possible to A series whose terms represent functions which are not con find an integer n and a polynomial in x of the nth degree, so tinuous throughout an interval may converge uniformly in the that the difference between the value of f(x) and the value of the interval. If & $(x),=f(x), is such a series, and if all 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 functions fr(x) have limits at a, then /(x) has a limit at Q, which a given interval, polynomials in x of finite degrees can be found, is Ê LI [(x). A similar theorem holds for limits on the left so as to form an infinite series of polynomials whose sum is equal --0-8 to /(x) at all points of the interval. Methods of representation or on the right. 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 Particular interest attaches to continuous functions which are not differentiable. Wcierstrass gave as an example the function 00+ 3. (a.cosami+ba sino-1). represented by the series & a* cos (b*xx), where a is positive and less and the rule for determining the coefficients ang bn of such a than unity, and b is an odd integer exceeding (1 +fr)la. It can be series, in order that it may represent a given function f(x) in shown that this series is uniformly convergent in every interval, term. (i.) and that the continuous function [(x) represented by it has the asymptotic expansions for the sum, difference, product, quotient, property that there is, in the neighbourhood of any point to, an or integral, as the case may be. infinite aggregate of points x', having to as a limiting point, for 26. Interchange of the Order of Limiting Operalions.-When which fix')-f(xo)]/(' - x) tends to become infinite with one sign when x' - *, approaches zero through positive values, and we require to perform any limiting operation upon a function infinite with the opposite sign when x'- xc approaches zero through which is itself represented by the result of a limiting process, negative values. Accordingly the function is not differentiable at the question of the possibility of interchanging the order of the any point. The definite integral of such a function f(x) through the two processes always arises. In the more elementary problems interval between a fixed point and a variable point x, is a continuous of analysis it generally happens that such an interchange is differentiable function F(x), for which F'(x) = S(x); and, if f(-x) is one-signed throughout any interval F(x) is monotonous throughout possible; but in general it is not possible. In other words, the that interval, but yet F(x) cannot be represented by a curve. In performance of the two processes in different orders may lead any interval, however small, the tangent would have to take the to two different results; or the performance of them in one of the same direction for infinitely many points, and yet there is no interval in which the tangent has everywhere the same direction. Further, two orders may lead to no result. The fact that the interchange it can be shown that all functions which are everywhere continuous is possible under suitable restrictions for a particular class of and nowhere differentiable are capable of representation by series of operations is a theorem to be proved. the form Sandn(x), where Ean is an absolutely convergent series of Among examples of such interchanges we have the differentiation numbers, and on(x) is an analytic sunction whose absolute value and integration of an infinite series term by term ($ 22), and the never exceeds unity. differentiation and integration of a definite integral with respect to a parameter by performing the like processes upon the subject of 25. Calculations with Divergent Series.-When the series integration (19). As a last example we may take the limit of the described in (1) and (2) of $ 24 diverge, they may, nevertheless, sum of an infinite series of functions at a point in the domain of be used for the approximate numerical calculation of the values convergence. Suppose that the series 2 fo(x) represents a function of the function, provided the calculation is not carried beyond a certain number of terms. Expansions in series which have the fx) in an interval containing a point a, and that cach of the functions If we first put x=d, and then sum the series, property of representing a function approximately when the L.() has a limit atq; we have the value f(a); if we first sum the series for any x, and expansion is not carried 100 lar are called “ asymptotic expan- afterwards take the limit of the sum at r=a, we ha the limit of sions.” Sometimes they are called “semi-convergent series "; f(x) at a; if we first replace each function 8,(x) by its limit at a, and but this term is avoided in the best modern usage, because then sum the series, we may arrive at a value different from either it is often used to describe series whose convergence depends second results are equal; if the functions f.(r) are all continuous at of the foregoing. If the function f(x) is continuous at a, the first and upon the order of the terms, such as the series 1-1+}-... a, the first and third results are equal; is the series is uniformly In ! neral, let fo(x) +f1(x)+. . be a series of functions which convergent, the second and third results are equal. This last case does not converge in a certain domain. It may happen that, if any is an example of the interchange of the order of two limiting operanumber e, however small, is first specified, a number n can after. tions, and a sufficient, though not always a necessary, condition, wards be found so that, at a point a of the domain, the value f(a) of for the validity of such an interchange will usually be found in some a certain function f(x) is connected with the sum of the first nti suitable extension of the notion of uniform convergence. AUTHORITIES.--Among the more important treatises and memoirs terms of the series by the relation 13(a) – €8,(a)/<«. It must connected with the subject are: R. Baire, Fonctions discontinues (Paris, 1905): 0. Biermann, Analytische Functionen (Leipzig, 1887); also happen that, if any number N, however great, is specified, a Ệ. Borel, Théorie des fonctions (Paris, 1898) (containing an intronumber n'(>) can be found so that, for all values of m which exceed ductory account of the Theory of Aggregates), and Séries divergentes (Paris, 1901), also Fonctions de variables réelles (Paris, 1905); T. J. n', 138,(@)]>N. The divergent series fo(x) +fi(x)+... is then an I'A. Bromwich, Introduction to the Theory of Infinite Scries (London, 1908); H. $. Carslaw, Introduction to the Theory of Fourier's Senes asymptotic expansion for the function f(x) in the domain. and Integrals (London, 1906): U. Dini, Functionen e. reellen Grosse The best known example of an asymptotic expansion is Stirling's (Leipzig: 1892), and Serie di Fourier (Pisa, 1880): A. Genacchi formula for n! when n is large, viz. u. G. Peano. Diff. u. Int.-Rechnung (Leipzig, 1899); J. Harkness n! = V (27)***+le-ntonin, and •F. Morley, Introduction to the Theory of Analytic Functions where is some number lying between o and 1. This formula is (London, 1898); A. Harnack, Diff. and Int. Calculus (London, 1891); included in the asymptotic expansion for the Gamma function. Theory of Fourier's Series (Cambridge, 1907); C. Jordan, Cours E. W. Hobson. The Theory of Functions of a real Variable and the We have in fact d'analyse (Paris, 1893-1896); L. Krunecker, Theorie d. einfachen log (r(x)} = (x-1) log x-x+1 log 25+ (x), u. vielfachen Integrale (Leipzig, 1894); H. Lebesgue, Leçons sur where ®(x) is the function defined by the definite integral l'intégration (Paris, 1904); M. Pasch, Dif. u. Int.-Rechnung (Leipzig. 1882); E. Picard, Traité d'analyse (Paris, 1891); 0. @(x) = $.*(1-2-4-4-4-4-}}{lertsdi. Stolz, Allgemeine Ari!hmetik (Leipzig, 1885), and Dif. u. int.. The multiplier of e- under the sign of integration can be expanded (Paris, 1886); W. H. and G. C.'Young. The Theory of Sets of Points Rechnung (Leipzig, 1893-1899); J. Tannery, Théorie des fonctions in the power series (Cambridge, 1906); Brodén,“ Stetige Functionen e. reellen Včränder. B, B2 lichen," Crelle, Bd. cxviii.; G. Cantor, A series of memoirs on the 2! * Theory of Aggregates and on “ Trigonomctric series" in Acta where Bı, Ba,... are " Bernoulli's numbers " given by the formula Math. it. ii., vii., and Math. Ann. Bde.iv.-xxiji.; Darboux, "Fonctions discontinues," Ann. Sci. École normale sup. (2). t. iv.; Dedekind, Br = 2.2m! (2x)** (p=2m). 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 a (x) takes the form Allgemeine Functionentheorie (Tübingen, 1882), and many memoirs B. 1 B, i B, 1 in Crelle and in Math. Ann.; Heine, Functionenlehre," Crelle, 1.2 * 3. + x 5.6 x Bd. Ixxiv.; J. Pierpont, The Theory of Functions of a rcal Variable (Boston, 1905); F. Klein, “ Allgemeine Functionsbegriff," Math. This series is divergent; but, if it is stopped at any term. the difference Ann. Bd. xxii.; W. F. Osgood, " On Uniform Convergence," Amer. between the sum of the series so terminated and the value of a(x) is J. of Math. vol. xix.; Pincherle. “Funzioni analitiche secondo less than the last of the retained ternis. Stirling's formula is obtained Weierstrass," Giorn. di mal. t. xviii.; Pringsheim, " Bedingungen by retaining the first term only. Other well-known examples of asymp- d. Taylorschen Lehrsatzes," Math. Ann. Bd. xliv.; Riemann, totic expansions are afforded by the descending series for Bessel's "Trigonometrische Reihe," Ges. Werke (Leipzig, 1876). Schoenflics, functions. Methods of obtaining such expansions for the solutions of Entwickelung d. Lehre v. d. Punkt mannigfaltigkeiten," Jahresber. linear differential equations of the second order were investigated by d. deutschen Math.- Vereinigung. Bd. viii; Study, Memoir on G. G. Stokes (Math, and Phys. Papers, vol, ii. P-329), and a general " Functions with Restricted Oscillation," Math. Ann. Bd. xlvii.; theory of asymptotic expansions has been developed by H. Poincaré. Weierstrass, Memoir on "Continuous Functions that are not Differ. A still more general theory of divergent series, and of the conditions entiable," Ges. malh. Werke, Bd. ii. p. 91 (Berlin, 1895), and on the in which they can be used, as above, for the purposes of approximate : Representation of Arbitrary Functions," ibid, Bd. ill. p. 1; W. H. calculation has been worked out by E. Borcl. The great merit of Young. "On Uniform and Non-uniform Convergence," Proc. London asymptotic expansions is that they ad it or addition, subtraction, Math. Soc. (Ser. 2) t. 6. Further information and very full references multiplication and division, terın by term, in the same way as will be found in the articles by Pringsheim, Schoenflies and loss in absolutely convergent series, and they admit also of integration the Encyclopädie der malh. Wissenschaften, Bdc. i., ii. (Leipzig, 1898, term by term; that is to say, the results of such operations are 1899). (A. E. H, L.) treatment. periods; (§ 23), Geomelrical applications of Elliptic Functions, II.-FUNCTIONS OF COMPLEX VARIABLES shows that any plane curve of deficiency unity can be expressed In the preceding section the doctrine of functionality is dis- by elliptic functions, and gives a geometrical proof of the addition cussed with respect to real quantities; in this section the theory theorem for the function P(x); (§ 24). Integrals of Algcbraic when complex or imaginary quantities are involved receives Functions in connexion with the theory of plane curves, discusses The following abstract explains the arrangement the generalization to curves of any deficiency, (§ 25), Monogenic of the subject matiet: ( 1), Complex numbers, states what a Functions of several independent variables, describes briefly the complex number is, ($ ), Plotting of simple expressions involving beginnings of this theory, with a mention of some fundamental complex numbers, illustrates the meaning in some simple cases, theorems: (§ 26), Multiply-Periodic Functions and the Theory introducing the notion of conformal representation and proving of Surfaces, attempts to show the nature of some problems now that an algebraic equation has complex, if not real, roots; (§ 3), being actively pursued. Limiting operations, defines certain simple functions of a complex Beside the brevity necessarily attaching 10 the account here variable which are obtained by passing to a limit, in particular given of advanced parts of the subject, some of the more clethe exponential function, and the generalized logarithm, here mentary results are stated only, without proof, as, for instance: denoted by 1(s); (8 4), Funclions of a complex variable in general, the monogeneity of an algebraic function, no reference being after explaining briefly what is to be understood by a region of made, morcover, to the cases of differential equations whose the complex plane and by a path, and expounding a logical integrals are monogenic, that a function possessing an algebraic principle of some importance, gives the accepted definition of a addition theorem is necessarily an elliptic function (or a particular function of a complex variable, establishes the existence of a case of such); that any arca can be conformally represented on complex integral, and proves Cauchy's theorem relating thereto; a half plane, a theorem requiring further much more detailed (8 5), Applications, considers the differentiation and integration consideration of the meaning of arca ihan we have given; while of series of functions of a complex variable, proves Laurent's the character and properties, including the connectivity, of a theorem, and establishes the expansion of a function of a complex Riemann surface have not been referred to. The iheta functions variable as a power series, leading, in (§ 6), Singular points, to are referred to only once, and the principles of the theory of a definition of the region of existence and singular points of a Abelian Functions have been illustrated only by the developfunction of a complex variable, and thence, in (8 7), Monogenic ments given for elliptic functions. Functions, to what the writer believes to be the simplest definition § 1. Complex Numbers.--Complex numbers are numbers of of a function of a complex variable, that of Weierstrass; (8 8), the form rtiy, where x, y are ordinary real numbers, and i is a Some clementary properties of single valued functions, first discusses symbol imagined capable of combination with itself and the The meaning of a pole, proves that a single valued function with ordinary real numbers, by way of addition, subtraction, multionly poles is rational, gives Mittag-Leffler's theorem, and Wcier- plication and division, according to the ordinary commutative, strass's theorem for the primary factors of an integral function associative and distributive laws; the symbol i is further such stating generalized forms for these, leading to the theorem of that 1 =-1. ($ 9), The construction of a monogenic function with a given region of Taking in a plane two rectangular axes O.r. Oy, we assume that existence, with which is connected ($10), Expression of a monogenic every point of the plane is definitely associated with two real numbers function by rational functions in a given region, of which the associated with a single complex number; in particular, for every method is applied in ($11), Expression of (1-2)" by polynomials, point of the axis Ox, for which y=0, the associated number is an to a definite example, used here to obtain ($ 12), An expansion ordinary real number; the complex numbers, thus include the real of on arbitrary function by means of a series of polynomials, over numbers. The axis Ox is often called the real axis, and the axis Oy á star region, also obtained in the original manner of Mittag. variable :=x+iy, the distance OP be called ,, and the positive the imaginary axis. If P be the point associated with the complex Leftler; (§ 13), Application of Cauchy's theorem to the determination angle less than 2x between O.x and OP be called 0, we may write of definite integrals, gives iwo examples of this method; (§ 14), :=rcos e +ı sin ); thenis called the modulus or absolute value Doubly Periodic Functions, is introduced at this stage as surnish- of and often denoted by island o is called the phase or amplitude of s, and often denoted by ph (s); strictly the phase is ambiguous ing an excellent example of the preceding principles. by additive multiples of 21. 11 ' = x'tiy be represented by P. reader who wishes to approach the matter from the point of view the complex argument s'+z is represented by a point P' obtained of Integral Calculus should first consult the section (§ 20) below, by drawing from P'. a line equal to and parallel to OP; the geodealing with Elliptic Integrals; ($ 15), Potential Functions, metrical representation involves for its validity certain properties Conformal representation in general, gives a sketch of the con- the possibility of constructing a parallelogram (with OP'asdiagonal). nexion of the theory of potential functions with the theory of it is important constantly to bear in mind, what is capable of easy conformal representation, enunciating the Schwarz-Christoffel algebraic proof (and geometrically is Euclid's proposition III. 7). theorem for the representation of a polygon, with the application that the modulus of a sum or difference of two complex numbers is to the case of an equilateral triangle; (§ 16), Multiple-valued moduli, and is greater than (or cqual to) the difference of their Functions, Algebraic Functions, deals for the most part with moduli; the former statement thus holds for the sum of any number algebraic functions, proving the residue theorem, and establishing of complex numbers. We shall write E(io) for cos Oti sin 0; it is that an algebraic function has a definite Order; (§ 17), Integrals at once verified that Elia). E(B) = Eli(a+)so that the phase of a of Algebraic Functions, enunciating Abel's theorem; (8 18), product of complex quantities is obtained by addition of their respective phases. 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 $ = (2-1)/(x+i), and, putting elliptic integral the number of these is two; ($ 10), Reversion of $ =$+in, take a new plane upon which &, ņ are rectanguan algebraic integral, mentions a problem considered below in iar co-ordinates, the cquations $=(x2 + y2-1)/(x2+(y+1)'), detail for an elliptic integral; ($ 20), Elliplic Integrals, considers n=-2xy (r?+(+1)] 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 := -i, that is, r=0, 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 a 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 $ = 0, and that there Functions, and á hint of the connexion with the theory of lincar is a definite correspondence of point to point between points differential equations; ($ 22), A property of integral functions, in the 2-plane which are above the real axis and points of the deduced from the theory of modular functions, proves that there s-plane which are interior to this circle; in particular sei 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 Morcover, 5 being a rational function of z, both and , are con. the modulus of the elliptic functions in terms of the ratio of the Itinuous differentiable functions of x and y, save when s is infinite; The =0; dx drit in z, functions writing s=f(x, y) =f(2-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 ajlax tioflay=0, and hence to shown that for 151 = 2, every term is also less than Min absolute value, ajon og on af 22€ namely, lanl < Mr. ll in every arbitrarily small neighbourhood of дх ду' ду ду? 2=0 there be a point for which two converging power series Ean:", Ebrz* agree in value, then the series are identical, or QA = baithus also so that ş is not any arbitrary function of x, y, and when & is known il Eanz* vanish at :=o there is a circle of finite radius about :=oas is determinate save for an additive constant. Also, in virtue of centre within which no other points are found for which the sum of these equations, if Si Ś be the values of 5 corresponding to two the series is zero. Considering a power series |(z) = ?uns" of radius of near values of 2, say ? and :', the ratio (5-5)/(2'-2) has a definite limit when :'=2, independent of the ultimate phase of :'-2, this convergence R, if Izol <R and we put : =20 +1 with 11 <R-120, the resulting series E9. (20+!)* may be regarded as a double scries limit being therefore equal to asiax, that is, o/artidndx, Geo- in zo and i, which, since 201+i<R, is absolutely convergenti metrically this fact is interpreted by saying that if two curves in the it may then be arranged according to powers of l. Thus we may 2-plane intersect at a point P, at which both the differential co write |(2) = SA,!"; hence Ao = f(20), and we have ((zo+1)-1(2:0)/= eficients og/dx, Onləx are not zero, and P', P' be two points near 2 Anta-s, wherein the continuous series on the right reduces to A, to P on these curves respectively, and the corresponding points of the $-plane be Q. Q. Q', then (1) the ratios PP/PP', QQQQ are for 1=0; thus the ratio on the left has a definite limit when 1=0, ultimately equal, (2) the angle P'PP' is equal to Q'oo"; (3) the equal namely to A, or £10.2*-! In other words, the original series rotation from PP to PP' is in the same sense as from QD' to QQ"; may legitimately be differentiated at any interior point zo of its circle it being understood that the axes of $. n in the one plane are related of convergence. Repeating this process wc find jizo+1) = 2/" f) (20) {11!, as are the axes of x, y. Thus any diagram of the 3-plane becomes a where fini (20) is the nth differential coefficient. Repeating for this diagram of the s-plane with the same angles; the magnification, power series, in l, the argument applied about z=for Zanz", we [:) however, which is equal to (5):11 inser that for the series (2) every point which reduces it to zero is varies from point to an isolated point, and of such poinis only a finite number lic within point. Conversely, it appears subsequently that the expression a circle which is within the circle of convergence of f(s). of any copy of a diagram (say, a map) which preserves angles requires 21/37... of which the radius of convergence is infinite. By As another illustration consider the case when 5 is a polynomial when r, yare real, and :=x+iy, exp (3) Ecxp (x) exp (iy). Now the 3= pain + P12+...+pei H being an arbitrary real positive number, it can be shown that a Ve=sin y, Vo=1 -cos y, U, = y-sin y, radius R can be found such for every 1212 R we have !!! > H; V. = {y-1+cos y, U, = {y-y+sin y; V;= y* - 4y+i-cos y,.. consider the lower limit of 15l for 12/<R; as stop is a real continuous function of x, y for 1:1 <R, there is a point (x, y), the first is the preceding one; as a function of a real variable) is all vanish lor y = 0, and the differential coeficient of any one after say (ro. yo), at which is l'is least, say equal top and therefore increasing when its differential coefficient is positive, we inler, for within a circle in the s-plane whose centre is the origin, of radius a there are no points s representing values corresponding 101:! <R. limit we hence infer that y positive, that each of these functions is positive; proceeding to a But if so be the value of 5 corresponding to (xo. yo), and the expression of s-so near 20= xo +iyo, in terms of 2-20. be A (3-2)" + cos y=1-}y? +iliyo -..., sin y=y-dy'tiboys-..., B(s—20)*+1 +..., where A is not zero, to two points near 10 (xo, yo), for positive, and hence, for all values of y. We thus have exp (iy) = say («. 1) or 21 and 2=2+(21-50) (cosmati sinn). will corre cos y +: sin y, and exp (2) = exp (x). (cos y+isin y). In other words, the modulus of exp (2) is exp (x) and the phase is y. Hence also spond two points ncar to so, say 31, and 250 - 5'1, situated so that so exp (z+211)=exp (x){cos (y+28) +i sin (y+2x)] is between ihem. One of these must be within the circle (p). We which we express by saying that exp. (?) has the period ari, inser then that p=0, and have proved that every polynomial in and hence also the period akni, where k is an arbitrary integer. z vanishes for some value of 2, and can therefore be written as a From the fact that the constantly increasing function exp (x) can product of ors of the form :-a, where a denotes a complex vanish only for x=0, we at once prove that exp (3) has no other This proposition alone suffices to suggest the importance periods. of complex numbers. Taking in the plane of z an infinite strip lying between the lines $ 3. Limiting Operations.-In order that a complex number9; y = 24 and plotting the function $. cxp (-) upon a new plane, s=&tin may have a limit it is necessary and sufficient that each of s arises when : takes in turn all positions in this strip, and that of & and 7 has a limit. Thus an infinite series w+witwe+ no value arises twice over. The equation $ =exp (2) thus defines 2, whose terms are complex numbers, is convergent is the real regarded as depending upon with only an additive ambiguity serics formed by taking the real parts of its terms and that 2kai, where k is an integer. We write :=1(3); when $ is real this formed by the imaginary terms are both convergent. The 2kri, where k is an integer; and when $ describes a closed circuit series is also convergent is the real series formed by the moduli surrounding the origin the phase of 5 increases by 27, or k increases of its terms is convergent; in that case the series is said 10 bc by unity. Differentiating the series for $ we have dj/dz = 5, so absolutely convergent, and it can be shown that its sum is that 2, regarded as depending upon $, is also differentiablc, with unaltered by taking the terms in any other order. Generally did. On the other hand, consider the series 5-1-1(3-1):+ the necessary and sufficient condition of convergence is that, is-115 1; its differential coefficient is, however, 1-(3-1) + for a given real positive e, a number m exists such that for every. (5-1)-.. that is, (1+5-1)Wherefore if (s) denote this »>m, and every positive P, the batch of terms wn+watif series, for $-11 <1, the difference (5) -9(5), regarded as a .+wn+p is less than e in absolute value. If the terms depend take the value of 1(s) which vanishes when $ +1 we inser thence function of and 7. has vanishing differential coefficients; if we upon a complex variable 2, the convergence is called uniform for a range of values of 3,' when the inequality holds, for the that for 18-11<1, 2(3) = 24) (3-1)". It is to be remarked same e and m, for all the points z of this range. that it is impossible for $ while subject to 15-11 <1 to make a The infinite series of most importance are those of which the circuit about the origin. For values of $ for which 15-11*1, we general term is anz", wherein an is a constant, and z is regarded as can also calculate 1 () with the help of infinite serics, utilizing the variable, n=0, 1, 2, 3,... Such a series is called a power series. fact that 1155') = 1(5)+1(5). Il a rcal and positive number M exists such that for : =20 and every The function (5) is required to define qe when 5 and a are complex , ! @nio" | <M, a condition which is satisfied, for instance, if the numbers; this is defined as exp (a1(5). that is as E a"|^(5))"in!. series converges for 2 = 2, then it is at once proved that the series when a is a real integer the ambiguity of 1(5) is immaterial here, converges, absolutely for every : for which 121 <1201, and converges uniformly over every range 121 < lor which r'< 1.01. where a'is a positive integer, there are q values possible for sale, of since exp (a (5)+2kani)=exp la1(5)); when a is of the form 119, To cvery power series there belongs then a circle of convergence within which it converges absolutely and unisorınly; the function the form exp with k=0, 1,...9-1, all other of z represented by it is ihus continuous within the circle (this being the result of a general property of uniformly convergent series of values of k leading to one of these; the oth power of any one of continuous functions); the sum for an interior point z is, however, these values is si when a =pla, where p q are integers without continuous with the sum for a point zo on the circumference, as : common factor, 9 being positive, we have fple (510)". The approaches to zo provided the series converges for 2= ko, as can be definition of the symbol aço is thus a generalization of the ordinary shown without much difficulty. Within a common circle of con- definition of a power, when the numbers are real. As an exainplc, vergence two power series {anz", kbnt" can be multiplicd together let it be required to find the meaning of ; the number i is of according to the ordinary rule, this being a consequence of a theorem modulus uniiy and phase \r; thus 1(1) = {(\:+2kr); thus for absolutely convergent scries. Il be less than the radius of si = exp(-11-2km) = exp(-1) cxp (-27), convergence of a series Santa and for a l=n, the sum of the scries is always real, but has an infinite number of values. number. The function exp (z) is used also to define a generalized form of definite finite real value attached to every interior or boundary the cosine and sine functions when z is complex; we write, namely, point of the region, say f(x,y). It may have a finite upper limit H COS 2 = lexp (13) + exp(-iz) and sin 2 = - belexp (12) - exp (-iz)). I lor the region, so that no point (x,y) exists for which f(x,y) > H. It will be found that these obey the ordinary relations holding when but points (x,y) exist for which f(x,y) > H-e. however small é mnay z is rcal, except that their moduli are not inferior to unity. For be; is not we say that its upper limit is infinite. There is then at example, cos i = 1+1/2!+1/4+...is obviously greater than unity. Icast one point of the region such that, for points of the region within $4. Of Functions of a Complex Varieble in General. -We have a circle about this point, the upper limit of f(x,y) is ů, however small the radius of the circle be taken; for if not we can put about in what precedes shown how to generalize the ordinary rational, every point of the region a circle within which the upper limit of algebraic and logarithmic functions, and considered more Mix.y) is less than H; then by the result (B) above the region general cases, of functions expressible by power series in 2. consists of a finite number of sub-regions within each of which the With the suggestions furnished by these cases we can frame a upper limit is less than 1; this is inconsistent with the hypothesis that the upper limit for the whole region is H. A similar statement general definition. So far our use of the plane upon which z is holds for the lower limit. A case of such a function f(x3) is ite represented has been only illustrative, the results being capable radius ro of the neighbourhood proper to any point 20. spokea of of analytical statement. In what follows this representation is above. We can hence prove the statement (A) above. vital to the mode of expression we adopt; as then the properties lower limit of zo is zero? Let thens be a point such that the lower Suppose the property (3,20) extensive, and, il possible, that the of numbers cannot be ultimately based upon spatial intuitions, limit of ro is zero for points zo within a circle about s however small: it is necessary to indicate what are the geometrical ideas requiring let r be ihe radius of the neighbourhood proper to si take zo so clucidation. that. 1z0-$1<br; the property (2,20), being extensive, holds within a circle, centre zo. of radius 7-180-51, which is greater Consider a square of side a, to whose perimeter is attached a definite direction of description, which we take to be counter. than 120-$1, and increases to ? as 120 - $ 1 diminishes; this bcing clockwise; another square, also of side a, may be added to this, so true for all points zo near $, the lower limit of ro is not zero for the that there is a side common; this common side being erased we ncighbourhood of $, contrary to what was supposed. This proves have a composite region with a definite direction of perimeter: 1 (A). Also, as is here shown that ror-120-51, may similarly be to this a third square of the same size may be atiached, so shown that rro-120-$1. Thus ro diffcrs arbitrarily little from that there is a side common to it and one of the foriner squares, r when I 20-51 is sufficiently small; that is, to varies conting Next suppose the and this common side may be erased. If this process be continued ously, with 26 unction /(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 rcal, so that about every point 24. or more polygonal closed lines, no two of which intersect; and at tion, which is such that the region is on the left of the describing for all points of the region interior to this circle, we have each portion of the perimeter there is a definite direction of descrip within or upon the boundary of the region, n being an arbitrary real point. Similarly we may construct a region by piccing together 11(x,y) -f(x0,90) <!n. and therefore (x', y") being any other point triangles, so that every consecutive two have a side in common, it being understood that there is assigned an upper limit for the interior to this circle, if(x',y') -- {(x,y)<n. We can then apply greatest side of a triangle, and a lower limit for the smallest angle. :he result (A) obtained above, taking for the neighbourhood proper by lines through its centre parallel to its sides; in the latter method (x.y). ? (z',>'), we have 1 S(x",y') - S(x,y)}<n. This is clearly an cach triangle may be divided into four others by lines joining the extensive property. Thus, a number 1 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), (x,y'), within a circle =r about any point 20, we have 1/(x,y')-f(x,y) /<. the contrary is stated, we shall suppose it capable of being generated and in particular, 1/(x,y) = {(x0,30) 15 7. there in 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 Take now any path in the region, whose extreme points are as, lower limit to the size of an angle of the triangle. We shall also and let 21, ... En– be intermediate points of the path, in order; require to speak of a path in the plane; this is to be understood as denote the continuous function f(x,y) by f(:), and let f, denote any capable of arising as a limit of a polygonal path of finite length, quantity such that 13.-()1 3 11 (3+1)-1(0)1; consider the sum there being a definite direction or sense of description at every point (31-)fo+(2-2)fit...+(2-2n-1)fn-1. of the paih, which therefore never mects itself. From this the By the definition of a path we can suppose, n being large enough, meaning of a closed path is clear. The boundary points of a region that the intermediate points 21...mi are so taken that if form one or more closed paths, but, in general, it is only in a limiting 2:+1 be any two points intermediate, in order, to z and 3741, we have sense that the interior points of a closed path are a'region, 1 2+1-2, 1 <1271-E1; we can thus suppose ! 3:- 21.12-, ... There is a logical principle also which must be referred to. We li s-2m, Tall to converge constantly to zero. This being so, we can frequently have cases where, about every, interior or boundary, show that the sum above has a definite limit. For this it is sufficient, point 20 of a ccrtain region a circle can be put, say of radius ro, such as in the case of an integral of a function of one real variable, to that for all points : of the region which are interior to this circle, prove this to be so when the convergence is obtained by taking new for which, that is, 12-20]<7o, a certain property holds. Assuming points of division intermediate to the former ones. '11, however, that to ro is given the value which is the upper limit for 20, of the 2r,1, 95,2... 27.-! be intermediate in order to s and Ertl, and possible values, we may call the points 13-21<0, the neighbour: 11.-14.1) I=1/(1,1+1)-1(4,1) 1, the difference between 2(+1-4), hood belonging to or proper too, and may speak of the property and as the property (3,20). The value of To will in general vary with coil 2{(20,1—?)$7.0+(24,2—2,1) fr.1+...+(+1-2m-1){r,m-1). 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) which is cqual to This lower limit is certainly greater than zero provided the property Eg (r,itd-81,0)(fr,-fr); (3,2) is of a kind which we may call extensive; such, namely, that if it holds, for some position of zo and all positions of z, within a certain is, when 127+1–2r! is small enough, to ensure 1/(x+1)-1() [<, less in absolute value than region, then the property (2,21) holds within a circle of radius R about any interior point of this region for all points 2 for which 2299137,1+1–27.11. the circle 15--11= R is within the region. Also in this case to which, if S be the upper limit of the perimeter of the polygon from varies continuously with 2. (B) Whether the property is of this which the path is generated, is < 2,5, and is therefore arbitrarily extensive character or not we can prove that the region can be divided small. into a finite number of sub-regions such that, for every oncol these, the property holds, (1) for some point zo within or upon the boundary The limit in question is called fis(e)dz . In particular when of the sub-region, (2) for every point : within or upon the boundary S(z) = J, it is obvious from the definition that its value is :-*; of the sub-region. We prove these statements (A), (B) in reverse order. To prove value is 1 (2-2); these results will be applied immediately. when H), by taking f, = }(+1-2), it is equally clear that its (B) Ict a region for which the property (:,-) holds for all pointsz and some point of the region, be called suitable; if cach of the triangles certain region there belong two definite finite numbers f(za). F(), Suppose now that to every interior and boundary point zs of a of which the region is built up be suitable, what is desired is proved; such that, whatever real positive quantity n may be, a rcal positive if not let an unsuitable triangle be subdivided into four, as before number e cxists for which the condition explained; is 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, cach contained in the which we describe as the condition (3,20). is satisfied for every point , preceding, which converge to a point, say si after a certain stage within or upon the boundary of the region, satisfying the limitation all these will be interior to the proper region of si this, however, is 12--20 <e. Then f(z) is called a differentiable function of the contrary to the supposition that they are all unsuitable. complex variable zo over this region, its differential cocfficicnt being We now make some applications of this result (B). Suppose a F(zu). The function f(20) is thus a continuous íunction of the real | 12) =) -F(0)|< |